Chapter 8: Current Electricity
In the middle of the 1820s, a schoolteacher in Cologne named Georg Simon Ohm set out to answer a question that sounds simple and turned out not to be. If you push electricity through a wire, how much of it flows?
The difficulty was the pushing. The chemical cells of his day grew weaker while he watched, so a reading taken at the start of an afternoon could not be trusted against a reading taken at the end. Ohm eventually used a different source, a junction of two metals kept hot at one end and cold at the other, whose push stayed steady for as long as he held the temperatures. To detect the flow he hung a magnetic needle on a fine thread beside the wire and watched how far it twisted.
Then he changed one thing at a time. He fitted wires of the same metal but different lengths into the same circuit and recorded the needle each time. Slowly a pattern appeared: double the push and the flow doubled; lengthen the wire and the flow fell away in a regular manner.
Published in 1827, his result was received coldly in Germany, and he waited years for recognition. Today every physics student meets it in the first week of this topic, and the unit of resistance carries his name.
Diagram
Fig. 8.0-1 — Ohm Testing Wires Of Different Lengths Simple period-style illustration of Ohm’s arrangement on a wooden bench: at the left, a bent metal strip joining two different metals, one end held in a small flame and the other end standing in a vessel of ice-water, labelled ‘source of steady push (hot–cold metal junction)’; two thick connecting strips lead from it to a pair of terminals labelled ‘test wire fitted here’, with three loose wires of visibly different lengths lying beside the bench labelled ‘wires of the same metal, different lengths’; at the right, a small magnetic needle suspended by a thread above a scale, placed parallel to and just below the connecting wire, labelled ‘suspended needle shows how much flows’. Add a curved arrow at the needle labelled ‘twist’. Keep the drawing clean and uncluttered; no modern instruments.
8.1 Electric Charge Revisited
Everything in this chapter — current, potential difference, resistance, power — is built on one idea you already met in Class IX. So it is worth spending a page putting that idea back in place, in the exact form the rest of the chapter will use it.
That idea is electric charge, the property of a body that makes it able to attract or repel another charged body. Charge is the thing that moves when a current flows, and a current is nothing more than charge on the move.
8.1.1 The Two Kinds Of Charge
When two non-conducting bodies are rubbed together — glass with silk, or ebonite with fur — electrons are transferred from one body to the other. Neither body creates anything. Electrons simply change owner, as shown in Fig. 8.1-1.
The body that gains electrons now has more electrons than it needs to balance its own protons, and it is said to be negatively charged. The body that loses electrons is left short of electrons, and it is positively charged.
Diagram
Fig. 8.1-1 — Transfer Of Electrons On Rubbing Two-panel illustration. Left panel labelled ‘before rubbing’: a glass rod and a silk cloth side by side, each drawn with equal numbers of small ‘+’ and ‘−’ symbols inside, captioned ‘both neutral’. Right panel labelled ‘after rubbing’: the two are shown apart, with three small ‘−’ symbols and a curved arrow running from the glass rod to the silk cloth labelled ‘electrons transferred’; the glass rod now shows surplus ‘+’ symbols labelled ‘positively charged (electrons lost)’ and the silk shows surplus ‘−’ symbols labelled ‘negatively charged (electrons gained)’. Keep symbols large and countable.
So there are two kinds of charge: positive charge and negative charge. Two like charges repel each other, while two unlike charges attract each other. Charge is denoted by the symbol \(q\) (or \(Q\)).
8.1.2 The Coulomb And Its Smaller Units
The SI unit of charge is the coulomb, whose symbol is C. In a laboratory the coulomb is a rather large amount of charge, so smaller units are used as well:
\[1\\ \\text{mC} = 10^{-3}\\ \\text{C},\\qquad 1\\ \\mu\\text{C} = 10^{-6}\\ \\text{C},\\qquad 1\\ \\text{nC} = 10^{-9}\\ \\text{C}\]
These are read as the milli-coulomb, the micro-coulomb and the nano-coulomb. Whenever a charge is given in one of these smaller units, convert it to coulomb before putting it into any relation in this chapter.
8.1.3 Charge As A Multiple Of The Electronic Charge
How much charge a body carries is decided entirely by how many electrons it is short of, or how many extra electrons it holds. Since electrons come in whole numbers, charge does too. A body can carry the charge of five electrons or of six electrons, but never of five and a half.
The charge on a single electron is \(-1.6 \\times 10^{-19}\\ \\text{C}\), and this quantity is written as \(e\). The charge on a body is therefore always a whole-number multiple of it:
\[q = \\pm, n e\]
where \(q\) is the charge on the body in coulomb (C), \(n\) is a whole number — the number of electrons in deficit or in excess — and \(e\) is the magnitude of the charge on an electron, \(1.6 \\times 10^{-19}\\ \\text{C}\). The plus sign is taken when the body is short of \(n\) electrons, and the minus sign when it holds \(n\) extra electrons.
Turning that relation around tells us how many electrons make up one coulomb:
\[n = \\frac{q}{e} = \\frac{1}{1.6 \\times 10^{-19}} = 6.25 \\times 10^{18}\]
So one coulomb of positive charge means a deficit of \(6.25 \\times 10^{18}\) electrons. It is an enormous number, which is exactly why the coulomb feels like a large unit.
8.1.4 Static Charge And Mobile Charge
There is one more thing to settle before current can be discussed, and it is the difference between a material that lets charge travel and one that does not.
In a non-conductor, or insulator, the charge stays exactly where it was put. If a spot on a rubbed ebonite rod has extra electrons, those electrons sit at that spot; they do not spread out and they do not travel along the rod. Charge held like this, at rest, is called static charge, and it does not flow.
A conductor, such as a metal, behaves quite differently. Taken as a whole, a piece of copper is electrically neutral — it has as many electrons as protons. But a large number of its electrons are not tied to any one atom, and these are free to wander throughout the whole volume of the metal. Charge in a conductor can therefore move, and it is this movement that the rest of the chapter is about. Their arrangement inside a metal is examined more closely in Section 8.4.
8.2 Electric Current
In Section 8.1 charge was mostly sitting still. Now let it move. A charge in motion is what we call an electric current, and everything that follows in this chapter is a way of describing that motion — how much of it there is, what pushes it, and what holds it back.
8.2.1 Current As The Rate Of Flow Of Charge
Picture a wire and imagine slicing straight across it, so that you are looking at one full cross section of the metal. Charge is drifting past that slice. The question a physicist asks is not how much charge has gone past in total, but how much goes past each second — a steady trickle and a heavy rush are two different things even if, given enough time, the same total charge crosses.
That is exactly what electric current measures: the amount of charge flowing per second across any cross section of the conductor. In words, the more charge that crosses in a given time, the larger the current; and the same charge crossing in a shorter time also means a larger current.
Current is denoted by the letter \(I\) (sometimes \(i\)). If a charge \(Q\) crosses a cross section normal to its direction of motion in a time \(t\), then
\[I = \\frac{Q}{t}\]
where \(Q\) is the charge that crosses in coulomb (C), \(t\) is the time taken in second (s), and \(I\) is the current in ampere (A). The cross section is taken normal — that is, at right angles — to the flow, so that all the charge moving along the wire has to pass through it.
Diagram
Fig. 8.2-1 — Charge Crossing A Cross Section Of A Wire A short length of cylindrical wire drawn in perspective, cut open at one place to show a shaded flat disc across it labelled ‘cross section, at right angles to the flow’. Several small circles marked ‘−’ are drawn inside the wire with arrows all pointing the same way through the disc, labelled ‘charge Q crosses in time t’. An arrow along the outside of the wire is labelled ‘current I = Q/t’. Keep it simple, no circuit around it.
8.2.2 The Ampere And Its Smaller Units
Since charge is measured in coulomb and time in second, current is measured in coulomb per second. That unit has been given a name of its own, the ampere (symbol A), in honour of the French physicist André Ampère.
\[1\\ \\text{A} = \\frac{1\\ \\text{C}}{1\\ \\text{s}} = 1\\ \\text{C s}^{-1}\]
Currents in everyday circuits are often far smaller than an ampere, so two smaller units are in common use:
\[1\\ \\text{mA} = 10^{-3}\\ \\text{A}\\qquad \\text{and} \\qquad 1\\ \\mu\\text{A} = 10^{-6}\\ \\text{A}\]
read as the milli-ampere and the micro-ampere. As with charge, a current given in milli-ampere is converted to ampere before it is used in any relation.
8.2.3 What Carries The Current In Different Conductors
Charge cannot flow unless something is there to carry it, and the carrier is not the same in every material.
In metals, the moving charges are the electrons that are not tied to any one atom. These wander through the metal, and it is their movement along the wire that makes up the current. Nothing else in a metal moves — the atoms themselves stay in place.
In electrolytes, such as an acid or a salt dissolved in water, and in ionised gases, the carriers are ions. Positively charged ions, called cations, move one way, and negatively charged ions, called anions, move the opposite way. Both movements add to the current, because a positive charge going one way and a negative charge going the other way both transfer charge in the same sense.
If, in a time \(t\), a number \(n\_1\) of cations each of charge \(q\_1\) move in one direction while \(n\_2\) anions each of charge \(q\_2\) move in the opposite direction, the current in the direction in which the cations move is
\[I = \\frac{n\_1 q\_1 + n\_2 q\_2}{t}\]
where the charges are in coulomb (C), the time is in second (s), and \(I\) is in ampere (A).
8.2.4 Current In Terms Of The Number Of Electrons
In a metal it is often more natural to count electrons than to measure charge. If \(n\) electrons pass through a cross section of a conductor in a time \(t\), then, using the result of Section 8.1.3, the total charge that has passed is
\[Q = n \\times e\]
and the current in the conductor is
\[I = \\frac{Q}{t} = \\frac{ne}{t}\]
where \(n\) is the number of electrons crossing the section, \(e = 1.6 \\times 10^{-19}\\ \\text{C}\) is the magnitude of the charge on one electron, \(t\) is the time in second (s), and \(I\) is the current in ampere (A).
This lets us say what one ampere means in terms of electrons. One coulomb is carried by \(6.25 \\times 10^{18}\) electrons, so a current of one ampere in a conductor means that \(6.25 \\times 10^{18}\) electrons pass in one second across any cross section of it.
8.2.5 Direction Of Current And Its Measurement
Current is a scalar quantity — it has size but is not treated as a vector, and currents meeting at a junction are added as ordinary numbers, not as arrows.
Even so, we do speak of the direction of a current in a wire, and it is worth being clear about what that phrase means. By agreement, the direction marked for a current is the direction in which positive charge would move, out of the positive terminal of the source and round the circuit. In a metal the carriers are electrons, which are negative, so the direction marked for the current is opposite to the direction in which the electrons are actually moving. So when a diagram shows the current flowing from left to right along a wire, the electrons in that wire are drifting from right to left.
Measuring The Current
The current in a circuit is measured with an ammeter, an instrument that must be connected in series in that circuit — that is, joined end to end with the other components, so that the very current being measured passes through it. Its terminal marked \(+\) is connected towards the positive terminal of the source of current, so that the needle moves the correct way across the scale.
Diagram
Fig. 8.2-2 — An Ammeter Connected In Series Circuit diagram using standard symbols, drawn as a single rectangular loop with no branches: a cell (long line positive, short line negative) at the bottom, a key (switch) beside it, a resistor on the right-hand side, and an ammeter (circle with letter A) in the top wire. Mark the ammeter’s left terminal ‘+’ and show it facing the positive terminal of the cell. Add an arrow on the top wire labelled ‘current I’ running from the positive terminal of the cell round the circuit. Label the components: cell, key, resistor, ammeter.
8.3 Electric Potential And Potential Difference
Section 8.2 described the flow, but not the push. Charge in a wire does not move on its own; something has to drive it. That something is described by two closely related quantities — potential, and potential difference.
8.3.1 Work Done In Moving A Charge
Two like charges repel each other. So if a charge A is brought near another charge B of the same kind, someone has to push A forward against that repulsion, and work has to be done in moving it.
Two unlike charges attract each other. If a charge A is brought near an unlike charge B, A is pulled in on its own, and this time the attractive force does the work.
Either way, moving a charge in the neighbourhood of another charge always involves work. Building up a whole collection of charges at one place therefore takes a great deal of work, and it is this stored-up work that the idea of potential is designed to describe.
To describe a region tidily we use a test charge — a charge of known, small value, small enough that bringing it in does not disturb the charges already present in the region.
8.3.2 Potential At A Point
The electric potential at a point is measured by the work needed to bring a test charge from a point of zero potential up to that point. That raises a question: where is the potential zero?
The force between two charges falls off as they are moved apart, and at infinite separation it is zero. A charge at infinity therefore feels nothing from the charges in our region, and infinity is taken as the point of zero potential.
So we bring the test charge from infinity, as in Fig. 8.3-1, and measure the work done. Dividing that work by the size of the test charge expresses it for one unit of charge, and the result is the potential at that point.
Diagram
Fig. 8.3-1 — Bringing A Test Charge From Infinity To A Point A small shaded sphere on the left marked ‘+’ labelled ‘positively charged body’. To its right, at a short distance, a point marked P. Further right, a small circle labelled ‘+Q’ with a horizontal arrow pointing left, from the right edge of the frame towards P, labelled ‘test charge brought in from infinity’. A dotted line joins the charged body through P to the arrow. Label the work done along the arrow as ‘work W done in bringing the charge to P’. Keep the drawing flat and simple.
If \(W\) joule of work is done in bringing a test charge \(Q\) coulomb from infinity to a point P, then the electric potential at P is
\[V = \\frac{W}{Q}\]
where \(W\) is the work done in joule (J), \(Q\) is the charge moved in coulomb (C), and \(V\) is the potential at that point in volt (V). Rearranged, the same relation tells us the work needed to move a charge \(Q\) from infinity to a point where the potential is \(V\):
\[W = QV\]
What The Sign Of A Potential Tells You
The sign of a potential is not a piece of bookkeeping. It says who does the work.
Near a positive charge, the potential is positive. A positive test charge is repelled, so work has to be done on it to push it in from infinity. Near a negative charge, the potential is negative. A positive test charge is attracted, so it is drawn in and the work is done by the field itself, not by us.
8.3.3 The Volt
The unit of potential follows straight from the relation \(V = W/Q\):
\[\\text{unit of potential} = \\frac{\\text{unit of work}}{\\text{unit of charge}} = \\frac{\\text{joule}}{\\text{coulomb}}\]
This unit, the joule per coulomb, is given the name volt, symbol V, in honour of the scientist Alessandro Volta.
\[1\\ \\text{volt} = \\frac{1\\ \\text{joule}}{1\\ \\text{coulomb}} = 1\\ \\text{J C}^{-1}\]
In words: the potential at a point is 1 volt when 1 joule of work is done in bringing a charge of 1 coulomb from infinity to that point.
8.3.4 Potential Difference Between Two Points
When we study a current between two points A and B of a circuit, we only care about charge moving from one of them to the other. Whether the potential at A happens to be 200 V or 12 V does not matter by itself; what drives the charge along is the fact that A is at a higher potential than B.
For that reason it is enough to know the potential difference, shortened to p.d., between the two points — the work done per unit charge in moving a positive test charge from one point to the other. Like potential, it is a scalar quantity, and it is also measured in volt.
If \(W\) joule of work is done in moving a positive test charge of \(Q\) coulomb from a point A to a point B, then the potential difference between A and B is
\[V\_{\\text{A}} - V\_{\\text{B}} = \\frac{W}{Q}\]
where \(W\) is in joule (J), \(Q\) is in coulomb (C), and the potential difference is in volt (V). A potential difference of 1 volt between two points means that 1 joule of work is done in moving a charge of 1 coulomb from one point to the other.
8.3.5 Measurement Of Potential Difference
The potential difference between two points in a circuit is measured with a voltmeter. Since a potential difference belongs to a pair of points, the voltmeter is connected across those two points — that is, in parallel with the part of the circuit between them, as in Fig. 8.3-2. This is the opposite of an ammeter, which is cut into the circuit in series because a current belongs to one path.
Its terminal marked \(+\) is connected to the point at the higher potential, so that the needle moves up the scale rather than against its stop.
Diagram
Fig. 8.3-2 — A Voltmeter Connected Across A Resistor Circuit diagram in standard symbols: a rectangular loop containing a cell (long line positive, short line negative) at the bottom, a key beside it, and a resistor labelled R in the top wire, with the two ends of the resistor marked A and B. A voltmeter, drawn as a circle containing the letter V, is drawn above the resistor and joined by two short wires, one to A and one to B, forming a parallel branch across the resistor only. Mark the voltmeter’s terminals ‘+’ and ‘−’, with ‘+’ joined to the end nearer the positive terminal of the cell, labelled ‘higher potential’. Add an arrow in the main loop labelled ‘current I’.
8.4 Resistance Of A Conductor
A potential difference across the ends of a wire drives a current through it. But the wire does not hand that current over freely. Every conductor puts up some opposition to the charge trying to move through it, and different conductors put up different amounts of it.
8.4.1 What Resistance Means
When a current flows through a conductor — say a metal wire — the wire offers some obstruction to that flow. It resists the movement of charge through itself. This property is called its resistance.
Saying that a conductor has resistance is easy. Saying why it has resistance takes a little longer, and it is worth the trouble, because almost everything in the next few sections — why a long wire resists more than a short one, why a hot wire resists more than a cold one — comes straight out of that explanation.
8.4.2 Free Electrons In A Metal
When a metal is formed, its atoms give up the electrons in their outermost orbits. What is left behind is a large number of positive ions, fixed in place, together with an equal number of loosed electrons that no longer belong to any one atom. These wandering electrons are called free electrons, because they move almost freely inside the metal.
Notice that the metal as a whole is still electrically neutral: the negative charge of all those electrons exactly balances the positive charge of all those ions. Nothing has been added to the metal or taken away from it.
With no potential difference across the ends of the wire, the free electrons are not idle. They move at random in every direction at once, colliding with one another and with the fixed positive ions, as Fig. 8.4-1 (a) shows. Because these random motions point every which way, as many electrons cross any section one way as cross it the other way. The net flow of electrons in any particular direction is zero, and so no current flows.
Diagram
Fig. 8.4-1 — Flow Of Current In A Metal Wire Two stacked panels, each showing the same short length of metal wire drawn as a horizontal cylinder in cut-away view. Inside each, draw a regular grid of circles marked ‘+’ labelled ‘positive ions (fixed)’, and a scatter of small solid dots labelled ‘free electrons’. Panel (a), captioned ‘when no potential difference is applied’: short arrows on the electrons pointing in many different directions, with a label ‘random motion; no net flow, so no current’. Panel (b), captioned ‘when a potential difference is applied’: a cell drawn above the wire with its ‘+’ terminal joined to the right-hand end and its ‘−’ terminal to the left-hand end; the electron arrows now all lean towards the right-hand (positive) end though still zig-zagging between ions, labelled ‘overall drift towards the positive end’. Mark one electron path as a zig-zag with small collision stars where it meets ions, labelled ‘collisions’.
8.4.3 Drift Of Electrons And The Cause Of Resistance
Now join the ends of the wire to a cell, so that a potential difference is applied across it. Every free electron inside the wire at once feels a force pushing it from the end at lower potential towards the end at higher potential. As a result the electrons begin to drift towards the positive end, as in Fig. 8.4-1 (b). Drift means a slow, overall movement in one direction, laid on top of the random motion that was already there.
You might expect that force to make the electrons go faster and faster, without limit. It does not, and the reason is the collisions. While drifting, an electron runs into the fixed positive ions and into other electrons. In each collision it loses part of its kinetic energy, and its speed drops. That lost energy does not disappear — it passes to the ions, which vibrate more strongly, and the wire warms up.
After a collision the electron is accelerated again towards the positive end by the potential difference that is still there, so its speed rises once more, only to fall in the next collision. The process repeats endlessly along the whole length of the wire.
The result is that the electrons do not move in bulk with a continuously increasing speed. Instead there is a steady overall drift of electrons towards the positive terminal. Every one of those collisions is a hindrance to the flow, and this is precisely the resistance the wire offers.
That gives us the physical picture behind the whole idea: the resistance of a conductor depends on the number of collisions the electrons suffer with the positive ions while travelling from one end of it to the other. More collisions on the journey means more resistance.
8.5 Ohm’s Law
We now have all three quantities in place: the current in a conductor, the potential difference driving it, and the resistance opposing it. The next question is the one Georg Simon Ohm set himself — exactly how are they linked?
8.5.1 Why A Steady Potential Difference Is Needed
Take two conductors at different potentials and join them by a metallic wire. Electrons begin to move at once, from the conductor that has an excess of electrons to the one that has a deficit. That movement is a current.
But it does not last. As electrons pass across, the two conductors come closer and closer in potential, and once they reach the same potential the flow stops altogether. A single transfer of charge gives only a momentary current.
For a current to keep flowing in a wire, then, the two conductors at its ends must be held at a definite difference of potential. In other words, a constant potential difference has to be maintained across the ends of a conductor if the current in it is to be continuous. This is exactly the job of a cell or a battery, which is taken up in Section 8.12.
8.5.2 Statement Of Ohm’s Law
In 1826 the German scientist Georg Simon Ohm, working through experiments of his own, found the relationship between the potential difference applied across a conductor and the current flowing in it.
The condition at the end of that sentence is not decoration. Ohm’s law is a statement about a conductor kept in the same state throughout — same temperature, same shape, same stretch. Change the state of the conductor while the experiment runs and the law no longer describes what you see.
Direct proportionality has a plain meaning. Double the potential difference across the ends of the conductor and the current flowing in it also doubles; halve the potential difference and the current halves.
8.5.3 The Relation V = IR
Let a current \(I\) flow in a conductor when the potential difference across its ends is \(V\). Ohm’s law says
\[I \\propto V \\qquad \\text{or} \\qquad \\frac{V}{I} = \\text{constant}\]
The constant here is a property of the conductor itself, and it is exactly the obstruction described in Section 8.4. We call it the resistance \(R\) of the conductor. Putting the constant equal to \(R\) gives the relation used more than any other in this chapter:
\[V = IR\]
where \(V\) is the potential difference across the ends of the conductor in volt (V), \(I\) is the current flowing in it in ampere (A), and \(R\) is its resistance in ohm (\(\\Omega\)). The relation holds at a given temperature, at which \(R\) is a constant for that conductor.
Notice one useful special case. If \(I = 1\\ \\text{A}\), then \(V = R\). So the resistance of a conductor is numerically equal to the potential difference that must be applied across its ends to drive unit current through it.
8.5.4 The Ohm And Its Larger Units
The unit of resistance follows from \(V = IR\), rearranged as \(R = V/I\):
\[\\text{unit of } R = \\frac{\\text{unit of } V}{\\text{unit of } I} = \\frac{\\text{volt}}{\\text{ampere}}\]
The volt per ampere (\(\\text{V A}^{-1}\)) is named the ohm after the scientist, and is written with the Greek letter \(\\Omega\) (omega).
Many resistors used in electronics have resistances of thousands or millions of ohms, so two larger units are in use:
\[1\\ \\text{kilo-ohm} = 1\\ \\text{k}\\Omega = 10^{3}\\ \\Omega,\\qquad 1\\ \\text{mega-ohm} = 1\\ \\text{M}\\Omega = 10^{6}\\ \\Omega\]
8.5.5 Conductance
Resistance says how hard a conductor makes it for a current to pass. It is sometimes handier to say the opposite — how easily current passes. That quantity is the reciprocal of resistance, and it is called conductance, denoted by \(G\).
\[\\text{conductance} = \\frac{1}{\\text{resistance}}, \\qquad \\text{that is} \\qquad G = \\frac{1}{R}\]
where \(R\) is the resistance in ohm (\(\\Omega\)) and \(G\) is the conductance in ohm\(^{-1}\) (\(\\Omega^{-1}\)), also called the siemens, symbol S. An older name for this unit, the mho, is no longer used.
8.5.6 The I–V Graph And Its Slope
For a metallic conductor the ratio \(V/I\) has the same value whatever readings of \(V\) and \(I\) are taken. So if the current \(I\) is plotted against the potential difference \(V\), the points fall on a straight line passing through the origin, as shown in Fig. 8.5-1.
The line passes through the origin because the two quantities are directly proportional: with no potential difference across the conductor, there is no current in it.
Diagram
Fig. 8.5-1 — I–V Graph For A Metallic Conductor Graph with the horizontal axis labelled ‘potential difference V (volt)’ and the vertical axis labelled ‘current I (ampere)’, origin marked O. Plot six crosses lying on a straight line rising from the origin, and draw the best-fit straight line through them, extended a little beyond. On the line mark two points, one lower left and one upper right; from them draw a horizontal dashed line and a vertical dashed line forming a right-angled triangle, with the horizontal side labelled ‘ΔV’ and the vertical side labelled ‘ΔI’. Place the label ‘slope = ΔI ÷ ΔV = 1 ÷ R’ beside the triangle.
The slope of this line carries physical meaning. Taking two points on the line and reading the increase \(\\Delta I\) in current for an increase \(\\Delta V\) in potential difference,
\[\\text{slope} = \\frac{\\Delta I}{\\Delta V} = \\frac{1}{\\text{resistance of the conductor}}\]
so the slope of an I–V graph is the reciprocal of the resistance. A steep line means a small resistance, because a small increase in potential difference produces a large increase in current.
8.5.7 Limitation Of Ohm’s Law
Ohm’s law is not a law of nature that every conductor must obey under all conditions. It is obeyed only when the temperature of the conductor remains constant.
The reason is already in front of us. Section 8.4 explained resistance in terms of collisions between drifting electrons and the fixed positive ions. If the conductor heats up while the current flows, those ions vibrate more vigorously, collisions become more frequent, and the resistance changes during the very experiment in which it was supposed to stay constant. The ratio \(V/I\) then no longer holds steady, and the graph is no longer a straight line. Section 8.7 looks at conductors of exactly this kind.
8.6 Experimental Verification Of Ohm’s Law
Ohm’s law claims that the current in a conductor is proportional to the potential difference across it. A claim of that kind is settled in the laboratory, by taking many pairs of readings from one wire and seeing whether the ratio really does stay put.
8.6.1 The Electric Circuit Used
The circuit used for the verification is shown in Fig. 8.6-1.
Diagram
Fig. 8.6-1 — Electric Circuit For The Verification Of Ohm’s Law Circuit diagram in standard symbols. Main loop, clockwise from bottom left: a battery labelled B with its terminals marked ‘+’ and ‘−’; a key labelled K; a rheostat drawn as a resistor with a sliding arrow on it, labelled Rh; an ammeter drawn as a circle containing A, with its terminals marked ‘+’ and ‘−’ and its ‘+’ terminal facing the positive terminal of the battery; then the resistance wire drawn as a resistor labelled R, captioned ‘resistance wire (say nichrome)’; and back to the battery. A voltmeter, drawn as a circle containing V with terminals marked ‘+’ and ‘−’, is connected as a separate branch across the two ends of R only, its ‘+’ terminal on the side nearer the positive terminal of the battery. Add an arrow in the main loop labelled ‘current I’.
The battery B, the key K, the rheostat Rh, the ammeter A and the resistance wire R are all joined one after the other, in series. The voltmeter V is connected in parallel, across the ends of the resistance wire alone.
Each part has one job.
- The battery B drives the current round the circuit. - The key K is used to start and stop the current, so that the circuit is closed only while readings are being taken. - The rheostat Rh changes the current in the circuit. It is a resistor whose resistance can be varied by sliding a contact along it, which is what allows several different currents to be obtained from one battery. - The ammeter A measures the current in the circuit, and is in series so that the whole current passes through it. Its terminal marked \(+\) is kept towards the positive terminal of the battery. - The resistance wire R — a nichrome wire, say — is the conductor under test. Its resistance is the unknown of the experiment. - The voltmeter V measures the potential difference across the ends of R, and is in parallel across R because a potential difference belongs to a pair of points. Its terminal marked \(+\) is also kept towards the positive terminal of the battery.
8.6.2 Procedure
The key K is closed, and current flows in the circuit. The rheostat is first adjusted to give the smallest reading that is not zero on the ammeter A and the voltmeter V, and this pair of readings — the current \(I\) and the potential difference \(V\) — is written down.
The sliding terminal of the rheostat is then moved so as to increase the current a little, and the new pair of readings on the ammeter and the voltmeter is written down. Moving the slider again gives a third pair, and so on.
In this way several different sets of values of \(I\) and \(V\) are collected for the same wire. Readings are taken briskly and the key is opened between them, so that the wire does not warm up — Ohm’s law holds only while the temperature stays constant.
8.6.3 Observations And Result
The readings are entered in a table like Table 8.2. For each set of values of \(I\) and \(V\), the ratio \(V/I\) is worked out in the last column.
Table 8.2 — Observations For The Verification Of Ohm’s Law
| S. No. | Ammeter reading \(I\) (in ampere) | Voltmeter reading \(V\) (in volt) | Resistance \(R = V/I\) (in ohm) |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 |
Average \(R = \\ldots\\ldots\\ldots\) ohm
Reading down the last column shows the result of the experiment: the ratio \(V/I\) is almost the same for every observation, even though the current has been changed several times over. Its average value is taken as the resistance \(R\) of the wire.
The small differences between one row and the next come from the ordinary limits of reading a scale, not from any failure of the law.
8.6.4 The V–I Graph And Its Slope
The same readings can be shown as a graph. Here the potential difference \(V\) is plotted on the Y-axis and the current \(I\) on the X-axis. The points are found to lie on a straight line, as in Fig. 8.6-2, and this straight line verifies Ohm’s law.
Diagram
Fig. 8.6-2 — Graph Of V Against I For A Resistance Wire Graph with the horizontal X-axis labelled ‘current I (ampere)’ and the vertical Y-axis labelled ‘potential difference V (volt)’, origin marked O. Draw a straight line rising from the origin. Mark two points on the line, Q lower down and P higher up. From P draw a horizontal dashed line PA to the Y-axis, meeting it at A, and a vertical dashed line PC to the X-axis, meeting it at C. From Q draw a horizontal dashed line QB to the Y-axis, meeting it at B, and a vertical dashed line QD to the X-axis, meeting it at D. Label the Y-axis values at A and B as V-A and V-B, and the X-axis values at C and D as I-C and I-D. Mark the vertical gap between the two dashed horizontals as ‘ΔV’ and the horizontal gap between the two dashed verticals as ‘ΔI’.
To find the slope of that line, two points P and Q are taken on it. From P and Q, normals PA and QB are drawn to the Y-axis and normals PC and QD to the X-axis. The potentials \(V\_{\\text{A}}\) at A and \(V\_{\\text{B}}\) at B are read off, and their difference gives
\[\\Delta V = V\_{\\text{A}} - V\_{\\text{B}}\]
Similarly the currents \(I\_{\\text{C}}\) at C and \(I\_{\\text{D}}\) at D are read off, and their difference gives
\[\\Delta I = I\_{\\text{C}} - I\_{\\text{D}}\]
The slope of the line is then \(\\Delta V/\\Delta I\), and this slope gives the resistance \(R\) of the conductor:
\[R = \\frac{\\Delta V}{\\Delta I} = \\text{slope of the } V \\text{ against } I \\text{ graph}\]
where \(\\Delta V\) is in volt (V), \(\\Delta I\) is in ampere (A), and \(R\) is in ohm (\(\\Omega\)). It follows that the greater the slope of a V–I graph, the greater the resistance of the conductor.
8.7 Ohmic And Non-Ohmic Resistors
Ohm’s law was verified in Section 8.6 for a nichrome wire. Try the same experiment on the filament of a bulb, or on a junction diode, and the readings behave quite differently. Conductors therefore fall into two groups, according to whether they obey the law or not.
8.7.1 Ohmic Resistors
The conductors which obey Ohm’s law are called ohmic resistors, or linear resistances. The word linear is a reminder of what their graph looks like.
Examples of ohmic resistors are all metallic conductors — such as silver, aluminium, copper and iron — nichrome, a copper sulphate solution with copper electrodes, and dilute sulphuric acid, all taken at a constant temperature.
For such a resistor, a graph plotted for the potential difference \(V\) against the current \(I\) is a straight line passing through the origin, exactly as in Fig. 8.6-2. Its resistance \(R\) is the same whatever the value of \(V\) or \(I\); that is, the ratio \(V/I\) is constant for all values of \(V\) or \(I\).
8.7.2 Non-Ohmic Resistors And Dynamic Resistance
The conductors which do not obey Ohm’s law are called non-ohmic resistors, or non-linear resistances. Examples are an LED, a solar cell, a junction diode, a transistor and the filament of a bulb.
For these conductors, a graph plotted for the potential difference \(V\) against the current \(I\) is not a straight line but a curve. Fig. 8.7-1 shows such a graph for a junction diode.
Diagram
Fig. 8.7-1 — V–I Graph For A Non-Ohmic Conductor Graph with the horizontal X-axis labelled ‘current I (ampere)’ and the vertical Y-axis labelled ‘potential difference V (volt)’, origin marked O. Draw a smooth curve starting at the origin, rising steeply at first and then bending over so that it becomes much flatter as I increases — clearly not a straight line — captioned ‘junction diode’. At one point on the curve, draw a straight tangent line touching it, and on that tangent mark a small right-angled triangle with its vertical side labelled ‘ΔV’ and its horizontal side labelled ‘ΔI’. Beside the triangle place the label ‘slope of the tangent = dynamic resistance at this point’.
Because the graph is a curve, the ratio \(V/I\) is not the same at all points on it. So the resistance of a non-ohmic conductor is different for different values of \(V\) or \(I\); the conductor does not have one single resistance the way a nichrome wire does.
The resistance at a particular value of \(V\) or \(I\) is found by drawing a tangent to the curve at the corresponding point and finding the slope of that tangent. This value \(\\Delta V/\\Delta I\) is called the dynamic resistance, and its value is different for different values of \(V\) or \(I\).
Note one further difference. For an ohmic resistor it is necessary that the straight line on the V–I graph passes through the origin. For a non-ohmic resistor, it is not necessary that the curve on the V–I graph passes through the origin at all.
8.7.3 Ohmic Compared With Non-Ohmic Resistors
The two kinds of conductor are set side by side in Table 8.3.
Fig. 8.7-2 shows the V–I graphs obtained in four different cases, and reading them is a good test of the distinction.
Diagram
Fig. 8.7-2 — V–I Graphs Obtained In Four Different Cases Four small graphs side by side, each with the horizontal axis labelled ‘I (ampere)’ and the vertical axis labelled ‘V (volt)’, origin marked O on each. (a) A straight line through the origin, captioned ‘pure metal’. (b) A straight line through the origin with a different, steeper slope, captioned ‘copper sulphate solution with copper electrodes’. (c) A curve rising from the origin and bending upward, captioned ‘filament of a bulb’. (d) A curve that rises steeply at first and then flattens, captioned ‘junction diode’. Keep all four the same size and style.
The first two graphs are straight lines through the origin, so those two are ohmic resistors. The last two are curves, so those two are non-ohmic resistors.
8.8 Factors Affecting The Resistance Of A Conductor
Two wires can have very different resistances. What decides the value? The resistance of a conductor depends on four things: the material it is made of, its length, its thickness, and its temperature. Each of the four is explained by the same picture of drifting electrons colliding with fixed ions that was built up in Section 8.4.
8.8.1 Dependence On The Material
Different materials have different concentrations of free electrons — that is, different numbers of free electrons packed into the same volume. The resistance of a conductor therefore depends on the material it is made of.
Among the metals, silver, copper, gold, aluminium and lead have concentrations of free electrons in decreasing order. Identical wires made of these five metals therefore offer resistance in the increasing order: a silver wire offers the least, and a lead wire of the same length and thickness offers the most.
Substances that have a large concentration of free electrons offer little resistance, and these are called good conductors of electricity. Substances such as rubber, glass and wood have a negligible concentration of free electrons, offer a very high resistance, and are the insulators met in Section 8.1.
8.8.2 Dependence On The Length
Picture an electron making the whole journey from one end of a wire to the other. In a long conductor that journey is longer, so the electron runs into the positive ions more times on the way. More collisions mean more opposition, and so a longer conductor offers proportionately more resistance.
In fact the resistance of a conductor is directly proportional to its length:
\[R \\propto l\]
where \(l\) is the length of the conductor in metre (m) and \(R\) is its resistance in ohm (\(\\Omega\)). This holds for a conductor of unchanged material and unchanged area of cross section.
So if the length of a wire of the same radius is doubled, the resistance of the wire gets doubled.
8.8.3 Dependence On The Thickness
In a thick conductor the electrons get a larger area of cross section to flow through than in a thin one, much as a wide road carries traffic more easily than a narrow lane. A thick conductor therefore offers less resistance.
The resistance of a conductor is inversely proportional to its area of cross section \(a\), taken normal to the direction of flow of current:
\[R \\propto \\frac{1}{a}\]
where \(a\) is the area of cross section in \(\\text{m}^2\) and \(R\) is the resistance in ohm (\(\\Omega\)). This holds for a conductor of unchanged material and unchanged length.
Take care here, because thickness is usually given as a radius while the resistance depends on an area. If the radius of a wire of the same length is doubled, its area of cross section becomes four times as large, so its resistance becomes one-fourth. If the radius is tripled, the area of cross section becomes nine times as large, and the resistance becomes one-ninth. And so on.
Diagram
Fig. 8.8-1 — How Length And Thickness Change The Resistance Three wires drawn as horizontal cylinders, one below the other, all shaded the same to show the same material. Top: a short thin wire labelled ‘length l, area a’, captioned ‘resistance R’. Middle: a wire of the same thickness but twice as long, labelled ‘length 2l, area a’, captioned ‘resistance 2R’. Bottom: a wire of the same length as the top one but with twice the radius, labelled ‘length l, radius doubled so area 4a’, captioned ‘resistance R ÷ 4’. Show the cross section of each as a small circle at its right-hand end, drawn to scale.
8.8.4 Dependence On The Temperature
With an increase in the temperature of a conductor, the random motion of its free electrons increases. The fixed positive ions also vibrate more vigorously about their places. As a result, the number of collisions of electrons with the positive ions increases. Hence the resistance of a conductor increases with an increase in its temperature.
The filament of a bulb is the everyday proof of this. Its resistance is more when it is glowing — that is, when it is at a high temperature — than when it is not glowing, that is, when it is cold.
8.8.5 V–I Graph At Two Temperatures
The effect of temperature shows up clearly on a graph. Fig. 8.8-2 shows two straight lines A and B on the V–I graph for one and the same conductor, taken at two different temperatures \(T\_1\) and \(T\_2\), where \(T\_1 > T\_2\).
Diagram
Fig. 8.8-2 — V–I Graph Of A Conductor At Two Temperatures Graph with the horizontal X-axis labelled ‘current I (ampere)’ and the vertical Y-axis labelled ‘potential difference V (volt)’, origin marked O. Draw two straight lines starting from the origin and rising to the right: the upper, steeper line labelled A and marked ‘T-1’, and the lower, less steep line labelled B and marked ‘T-2’. Place the note ‘T-1 is greater than T-2’ near the top left. Keep the two lines clearly separated.
The straight line A is steeper than the line B. Since the slope of a V–I graph gives the resistance, the steeper line means the larger resistance — and it belongs to the higher temperature \(T\_1\). This is the temperature effect of Section 8.8.4, seen on a graph: the resistance of the conductor is more at the high temperature \(T\_1\) than at the low temperature \(T\_2\).
8.9 Specific Resistance Or Resistivity
Section 8.8 gave four separate statements about resistance. Three of them — material, length and area — can now be gathered into a single relation, and doing so brings out a new quantity that belongs to the material itself rather than to any particular piece of wire.
8.9.1 How Resistance Depends On Length And Area
Experiment shows two things about a wire of a given material, kept at a fixed temperature.
First, the resistance of a wire is directly proportional to its length:
\[R \\propto l\]
Second, the resistance of a wire varies inversely as its area of cross section:
\[R \\propto \\frac{1}{a}\\qquad \\text{or} \\qquad R \\propto \\frac{1}{\\pi r^2}\]
where \(r\) is the radius of the wire in metre (m), since the area of cross section of a round wire is \(a = \\pi r^2\).
Combining the two results,
\[R \\propto \\frac{l}{a} \\qquad \\text{or} \\qquad R = \\rho,\\frac{l}{a} = \\rho,\\frac{l}{\\pi r^2}\]
Here \(R\) is the resistance in ohm (\(\\Omega\)), \(l\) is the length in metre (m), \(a\) is the area of cross section in \(\\text{m}^2\), and \(\\rho\) (the Greek letter rho) is a constant of proportionality. It is the same for every piece of one material and different for different materials, because it reflects the characteristics of the material of the wire. It is called the specific resistance, or the resistivity, of that material.
8.9.2 Defining Specific Resistance
Rearranging the relation above makes \(\\rho\) the subject:
\[\\rho = \\frac{R,a}{l}\]
Now put \(l = 1\) and \(a = 1\). Then \(\\rho = R\). This gives the meaning of the quantity in plain terms: the specific resistance of a material is the resistance of a wire of that material having unit length and unit area of cross section.
Said another way, the specific resistance of a material is the resistance between the opposite faces of a unit cube of that material, as in Fig. 8.9-1.
Diagram
Fig. 8.9-1 — Specific Resistance As The Resistance Of A Unit Cube A cube drawn in simple perspective, with each edge labelled ‘1 m’. Shade the front face and the opposite back face lightly, and label them ‘opposite faces’. Draw an arrow passing straight through the cube from the front face to the back face, labelled ‘current’. Below the cube, place the caption line ‘resistance measured between these two faces = specific resistance ρ of the material, in ohm metre’.
8.9.3 Unit Of Specific Resistance
The unit follows from \(\\rho = Ra/l\):
\[\\text{unit of } \\rho = \\frac{\\text{unit of } R \\times \\text{unit of } a}{\\text{unit of } l} = \\frac{\\text{ohm} \\times \\text{metre}^2}{\\text{metre}} = \\text{ohm} \\times \\text{metre}\]
So the SI unit of specific resistance is the ohm metre, written \(\\Omega\\ \\text{m}\).
8.9.4 Factors Affecting The Specific Resistance
Two things govern the specific resistance of a substance, and neither of them is its shape.
First, the specific resistance is a characteristic property of the substance. It is different for different substances. It is very low for metals and alloys, of the order of \(10^{-8}\\ \\Omega\\ \\text{m}\); much higher for semiconductors; and very high for insulators, of the order of \(10^{13}\\ \\Omega\\ \\text{m}\) and beyond.
Second, the specific resistance of a substance depends on its temperature. It increases with an increase in temperature for metals, but it decreases with an increase in temperature for semiconductors. For some alloys, such as constantan and manganin, it remains practically unchanged with a change in temperature — a property that turns out to be extremely useful, as Section 8.10 shows.
The specific resistance of the material of a conductor at a given temperature is a constant, and it does not depend on the shape and size of the conductor.
8.9.5 Specific Resistance Of Some Substances
Table 8.4 gives the specific resistance of some common substances at \(20^\\circ\\text{C}\).
Table 8.4 — Specific Resistance Of Some Substances At 20 °C
| Substance | Specific resistance (\(\\Omega\\ \\text{m}\)) | Substance | Specific resistance (\(\\Omega\\ \\text{m}\)) |
|---|---|---|---|
| Metals | Semiconductors | ||
| Silver | \(1.63 \\times 10^{-8}\) | Graphite | \(0.2 \\times 10^{-5}\) |
| Copper | \(1.73 \\times 10^{-8}\) | Carbon | \(3.5 \\times 10^{-5}\) |
| Gold | \(2.44 \\times 10^{-8}\) | Germanium (pure) | about \(0.5\) |
| Aluminium | \(2.63 \\times 10^{-8}\) | Silicon (pure) | about \(2.3 \\times 10^{3}\) |
| Lead | \(2.8 \\times 10^{-8}\) | ||
| Tungsten | \(5.5 \\times 10^{-8}\) | Insulators | |
| Iron | \(9.8 \\times 10^{-8}\) | Wood | \(10^{8} - 10^{11}\) |
| Tin | \(11.3 \\times 10^{-8}\) | Glass | \(10^{10} - 10^{14}\) |
| Steel | \(20 \\times 10^{-8}\) | Diamond | nearly \(10^{12}\) |
| Alloys | Amber | \(5 \\times 10^{14}\) | |
| Brass | \(6.6 \\times 10^{-8}\) | Mica | \(10^{11} - 10^{15}\) |
| Manganin | \(44 \\times 10^{-8}\) | Polythene | nearly \(10^{16}\) |
| Constantan | \(49 \\times 10^{-8}\) | ||
| Nichrome | \(100 \\times 10^{-8}\) |
Two patterns in this table are worth reading off. Among the metals, silver has the lowest value and is therefore the best conductor of the lot. Among the alloys, every value is far higher than that of the pure metals — nichrome is about sixty times as resistive as copper — which is why alloys, and not pure metals, are chosen wherever a wire is wanted for its resistance rather than in spite of it.
8.9.6 Stretching A Wire Or Doubling It On Itself
A wire can be reshaped without adding or removing any material. When that happens the specific resistance stays exactly as it was, but the resistance need not, because both the length and the area of cross section have changed. Fig. 8.9-2 shows the two cases that matter.
Diagram
Fig. 8.9-2 — Stretching A Wire And Doubling It On Itself Two panels, one above the other. Panel (a), captioned ‘stretched to twice its length’: on the left, a short thick wire labelled ‘length l, area a, resistance R’; an arrow labelled ‘stretched, same volume of metal’ points to the right, where a wire twice as long and visibly thinner is labelled ‘length 2l, area a ÷ 2, resistance 4R’. Panel (b), captioned ‘doubled on itself’: on the left, the same original wire labelled ‘length l, area a, resistance R’; an arrow labelled ‘folded in half, both halves side by side’ points to the right, where the wire is drawn bent double into a hairpin with the two halves touching along their length, labelled ‘length l ÷ 2, area 2a, resistance R ÷ 4’, with the two free ends marked ‘new ends’.
A Wire That Is Stretched
Suppose a given wire is stretched so that its length gets doubled. The volume of the metal cannot change, so if the length becomes twice as great, the area of cross section becomes half of what it was.
Now both changes push the resistance the same way. The resistance increases twice over because of the increase in length, and then twice again because of the decrease in the area of cross section. So the resistance increases to four times its previous value.
Similarly, if the length is increased to three times by stretching the wire, its area of cross section becomes one-third, and its resistance becomes nine times its previous value.
A Wire That Is Doubled On Itself
If a wire is doubled on itself — folded in half so that the two halves lie side by side — the length between the new ends is halved and the area of cross section carrying the current is doubled. The resistance between the new ends therefore becomes one-fourth of its previous value.
In each of these cases the specific resistance remains unchanged, because nothing about the material itself has been altered.
8.9.7 Conductivity
Just as conductance is the reciprocal of resistance, the reciprocal of specific resistance is called conductivity. It is represented by the symbol \(\\sigma\) (sigma):
\[\\sigma = \\frac{1}{\\rho} = \\frac{l}{R,a}\]
where \(\\rho\) is the specific resistance in ohm metre (\(\\Omega\\ \\text{m}\)), and \(\\sigma\) is the conductivity of the material. Its SI unit is therefore
\[\\frac{1}{\\text{ohm} \\times \\text{metre}} = \\text{ohm}^{-1}\\ \\text{metre}^{-1}\]
written \(\\Omega^{-1}\\ \\text{m}^{-1}\), and also called siemens per metre (\(\\text{S m}^{-1}\)). A material with a high conductivity is one that lets current pass easily — silver has the highest conductivity of the metals in Table 8.4, precisely because it has the lowest specific resistance.
8.10 Choice Of Material Of A Wire
Every wire in a circuit is chosen, not just picked up. The choice of the material of a wire depends on the purpose for which the wire is to be used, and the properties that decide it are exactly the ones set out in Sections 8.8 and 8.9 — specific resistance, melting point, and how the resistance behaves when the wire gets hot.
8.10.1 Connecting And Power Transmission Wires
The wires used for electrical connections and for power transmission should possess the least possible resistance.
There are two reasons for that. With a low, or negligible, resistance in the connecting wires, the current in the circuit remains unaffected by the wires themselves — they do their carrying quietly and leave the circuit to behave as designed. And the loss of energy due to heating along the way is prevented, which matters enormously over the many kilometres of a transmission line.
Hence such wires are made of materials such as copper or aluminium, whose specific resistance is very small. Further, they are made thick, so that with a larger area of cross section their resistance becomes lower still.
The specific resistance of silver is less than that of copper or aluminium, so silver would in principle be the best choice of all. But because of its high cost, silver is not used.
8.10.2 Resistance Wires And Standard Resistors
Resistance wires, and the standard resistors used in a laboratory, need the opposite of what a connecting wire needs. Such wires are of high resistance, and their resistance must not change due to a change in temperature — a standard resistor that drifted while it was being used would be of no use as a standard at all.
They are therefore made from alloys such as manganin and constantan, for which the specific resistance is high and the effect of a change in temperature on their resistance is negligible.
8.10.3 Fuse Wire
A fuse wire is made from an alloy of lead and tin, and two properties of that alloy are put to work at once.
Its melting point is low, so it melts before the rest of the circuit comes to any harm. And its specific resistance is more than that of copper or aluminium, so that the resistance of a short and thin fuse wire is more than that of the connecting wire it protects — meaning the heating happens in the fuse and not elsewhere.
In normal use the fuse simply permits current up to its safe limit to pass through it. But an excessive current melts it, so that it blows off and the circuit is broken.
Diagram
Fig. 8.10-1 — A Fuse Wire In A Circuit Simple drawing of a fuse holder: two metal terminals set into an insulating base, joined by a short, visibly thin wire labelled ‘fuse wire, short and thin, alloy of lead and tin’. Thick wires labelled ‘connecting wire (copper, thick)’ run away from each terminal to the left and right, drawn noticeably thicker than the fuse wire. Above the fuse wire, place the label ‘highest resistance in the path, and the lowest melting point’. To the right of the drawing, show a second small panel of the same holder with the fuse wire melted through and a gap in the middle, captioned ‘an excessive current melts it and the circuit is broken’.
8.10.4 Filament Of An Electric Bulb
For the filament of an electric bulb, a tungsten wire is used, because it has a high melting point. The filament has to be raised to a temperature at which it glows brightly, and tungsten glows to provide light without melting.
8.10.5 Heating Elements Of Appliances
For the heating element in appliances such as a heater, a toaster or an oven, a nichrome wire is used. The specific resistance of nichrome is high, and its resistance increases to a great extent with an increase in temperature.
Table 8.5 gathers the five choices together.
8.11 Superconductors
Section 8.8 established that the resistance of a metal falls as its temperature falls. Push that cooling far enough and something more dramatic happens.
It is observed experimentally that the resistance of some substances, such as tin and lead, decreases tremendously with a decrease in temperature, and becomes almost zero in the low temperature range near absolute zero. Such substances are called superconductors.
Zero resistance is a stronger claim than it may look. Since conductivity is the reciprocal of resistance, zero resistance means infinite conductivity: once a current starts flowing in a superconductor, it persists even when there is no potential difference across it. In an ordinary wire the current dies away the moment the cell is disconnected; in a superconducting ring it simply keeps going.
Each superconductor has its own temperature below which this happens. Mercury below \(4.2\\ \\text{K}\), lead below \(7.25\\ \\text{K}\) and niobium below \(9.2\\ \\text{K}\) are superconductors. Fig. 8.11-1 shows what the change looks like on a graph.
Diagram
Fig. 8.11-1 — Resistance Of A Superconductor Against Temperature Graph with the horizontal axis labelled ‘temperature T (kelvin)’ marked from 0 to about 12 with the values 4.2 and 9.2 shown, and the vertical axis labelled ‘resistance R (ohm)’, origin marked O. Draw a curve that starts at the upper right, falls gently as the temperature drops, and then plunges vertically to the horizontal axis at one temperature, after which it runs exactly along the axis at zero all the way to 0 K. Label the vertical plunge ‘resistance falls suddenly to zero’ and mark the temperature at which it happens with a dashed vertical line. Label the flat portion along the axis ‘superconducting: R = 0’.
Superconductors are not in common use, since it is very difficult to achieve such a low temperature. Where the cost is worth paying they are already used — the powerful magnets of a hospital MRI scanner are coils of superconducting wire kept extremely cold, because only a current that never dies away can hold so strong a magnetic field steady.
If it became possible to obtain superconductors at room temperature, the consequences would be large. The size of computers could then be reduced to a few centimetre, and power transmission lines could be made as thin as a single wire, since a wire with no resistance wastes no energy however much current it carries and needs no thickness to keep its resistance down.
8.12 Electro-Motive Force Of A Cell
Section 8.5 left one thing unfinished. A current keeps flowing in a conductor only if a potential difference is maintained across its ends — and maintaining it, second after second, takes a device built for the job. That device is the electric cell.
8.12.1 The Electric Cell
An electric cell has two conductors of suitable material, kept in an appropriate electrolyte taken in a vessel. The two conductors are called the electrodes, or the terminals, of the cell. The electrolyte is the liquid between them, a solution that conducts by means of ions.
The electric cell maintains a constant difference in potential between the two electrodes in the electrolyte, by a chemical reaction, so as to obtain a continuous flow of charge between them. In other words a cell stores chemical energy, and can be used as a source of current. While the cell is in use, that chemical energy changes into electrical energy.
Fig. 8.12-1 shows a simple cell having two electrodes, one of zinc and the other of copper, placed in an electrolyte of dilute sulphuric acid, together with the symbol used for a cell in a circuit diagram.
Diagram
Fig. 8.12-1 — A Cell In Open Circuit Two panels side by side. Panel (a), captioned ‘the cell’: a rectangular glass vessel containing a shaded liquid labelled ‘dilute sulphuric acid’; two flat plates dipping into it, the left one labelled ‘copper’ and the right one labelled ‘zinc’, each rising above the liquid to a terminal at the top. Mark the copper terminal ‘+’ and the zinc terminal ‘−’, and draw a double-headed arrow between the two terminals above the vessel labelled ‘e.m.f. ε’. Show nothing joining the two terminals, and add the note ‘no current is drawn — open circuit’. Panel (b), captioned ‘the symbol’: the standard cell symbol, a long thin line marked ‘+’ and a short thick line marked ‘−’, with a double-headed arrow above it labelled ‘ε’.
8.12.2 How A Cell Builds A Potential Difference
The electrolyte, here sulphuric acid (\(\\text{H}\_2\\text{SO}\_4\)) in water, dissociates into positive ions (\(\\text{H}^+\)) and negative ions (\(\\text{SO}\_4^{,2-}\)). These ions move towards the electrodes.
There, two things happen at once. The negative ions (\(\\text{SO}\_4^{,2-}\)) impart electrons to the zinc electrode, while the positive ions (\(\\text{H}^+\)) gain electrons from the copper electrode.
This chemical action creates a difference in the concentration of electrons between the two electrodes of the cell. The zinc electrode, piling up electrons, acquires a negative potential, and is called the anode. The copper electrode, losing electrons, acquires a positive potential, and is called the cathode. A potential difference has therefore been created between the two electrodes, and this potential difference is the electro-motive force, written e.m.f., of the cell.
The e.m.f. of a cell is generally denoted by the symbol \(\\varepsilon\) (epsilon), and its unit is the volt (V). It can be measured by connecting a voltmeter of high resistance across the cell, or more accurately by a potentiometer. The voltmeter must be of high resistance for a reason worth remembering: it must draw as little current as possible, since e.m.f. is the reading of a cell from which no current is being taken.
8.12.3 Defining Electro-Motive Force
Now join some electrical component, such as a bulb, across the terminals of the cell. Electrons flow in the external circuit from the zinc electrode to the copper electrode — which is the same as saying that current flows in the external circuit from the copper electrode to the zinc electrode. This current is maintained by the chemical action in the electrolyte, through the flow of ions within it.
Why does the current flow at all? Because the e.m.f. between the terminals provides the energy for the charge to travel through the component joined in the circuit. And the charge must also be carried through the electrolyte inside the cell, which takes energy too. The definition of e.m.f. counts both.
If \(W\) work is done in taking a test charge \(q\) around the complete circuit of the cell, then the e.m.f. of the cell is
\[\\varepsilon = \\frac{W}{q}\]
where \(W\) is the work done in joule (J), \(q\) is the charge carried round in coulomb (C), and \(\\varepsilon\) is the e.m.f. in volt (V).
Flowchart
FC 8.1 — Energy Conversion In A Cell Driving A Circuit Chemical energy stored in the electrode–electrolyte system → chemical reaction separates charge, creating an e.m.f. between the terminals → work done on the charge as it is carried round the complete circuit → electrical energy delivered to the component in the external circuit → converted there into heat, light or motion
8.12.4 Factors Affecting The E.M.F. Of A Cell
The e.m.f. of a cell depends on just two things: the material of the electrodes, and the electrolyte used in the cell. Both are matters of chemistry, which fits, since it is a chemical reaction that produces the e.m.f. in the first place.
It is independent of everything else about how the cell is built — of the shape of the electrodes, of the distance between the electrodes, and of the amount of electrolyte in the cell. A large cell and a small cell of the same kind have the same e.m.f.; the large one simply lasts longer.
The e.m.f. is therefore a characteristic property of a cell, and it is different for different kinds of cell. The e.m.f. of a voltaic cell is \(1.08\\ \\text{V}\), that of a Leclanche cell is \(1.5\\ \\text{V}\), and that of a Daniel cell is \(1.08\\ \\text{V}\).
8.13 Terminal Voltage Of A Cell
The e.m.f. of a cell is what it offers when nothing is being taken from it. Connect something across it and start drawing current, and the voltage available at its terminals is no longer the same number. That second number has a name of its own.
8.13.1 Terminal Voltage Defined
When current is drawn from a cell — that is, when the cell is in a closed circuit, as in Fig. 8.13-1 — the potential difference between the electrodes of the cell is known as its terminal voltage. It is generally denoted by the letter \(V\), and like the e.m.f. it is expressed in volt (V).
Diagram
Fig. 8.13-1 — A Cell In Closed Circuit Rectangular glass vessel containing shaded liquid labelled ‘dilute sulphuric acid’, with two plates dipping into it, the left labelled ‘Cu’ and marked ‘+’ at its terminal, the right labelled ‘Zn’ and marked ‘−’. From the two terminals, wires run up and across the top to a resistor drawn in the top wire, labelled R. Between the two terminals, below the wires, draw a double-headed arrow labelled ‘terminal voltage V’. Add an arrow on the top wire labelled ‘current I’, running from the copper terminal round to the zinc terminal through R. Beside the drawing, place a small inset showing the circuit symbol version: cell symbol, resistor R, and the same double-headed arrow labelled V across the cell.
Compare that definition with the one for e.m.f. in Section 8.12.3 and one word does all the work: external. The e.m.f. counts the work done all the way round — outside the cell and through the electrolyte inside it. The terminal voltage counts only the work done outside.
If \(W'\) is the work done in carrying a test charge \(q\) around the circuit connected across the terminals of the cell, that is, only outside the cell and not inside it, then the terminal voltage of the cell is
\[V = \\frac{W'}{q}\]
where \(W'\) is in joule (J), \(q\) is in coulomb (C), and \(V\) is in volt (V).
8.13.2 Voltage Drop Inside The Cell
From the two definitions it is clear that the terminal voltage \(V\) of a cell is less than its e.m.f. \(\\varepsilon\), by the amount of energy spent in the flow of charge through the electrolyte inside the cell.
Carrying charge through the electrolyte is not free. If \(w\) is the work done, that is the energy spent, in carrying a test charge \(q\) through the electrolyte inside the cell, then the quantity \(w/q\) is called the voltage drop in the cell, denoted by \(v\):
\[v = \\frac{w}{q}\]
where \(w\) is in joule (J), \(q\) is in coulomb (C), and \(v\) is in volt (V).
So the work done per unit charge in carrying the positive test charge through the electrolyte is called the voltage drop in the cell. It is called a drop because this voltage is not available to us for use. In the conversion of chemical energy to electrical energy, an amount of energy \(w\) is lost inside the cell, and it ultimately appears as heat energy — which is why a cell supplying a large current grows warm.
8.13.3 Relation Between E.M.F. And Terminal Voltage
The link between the three quantities comes straight out of the law of conservation of energy.
Let \(W\) be the work done in moving a test charge \(q\) outside as well as inside the cell, \(W'\) the work done in moving it outside the cell, and \(w\) the work done in moving it inside the electrolyte of the cell. Since the journey round the complete circuit is just the outside journey plus the inside journey,
\[W = W' + w\]
Dividing both sides by the charge \(q\),
\[\\frac{W}{q} = \\frac{W'}{q} + \\frac{w}{q}\]
Each of these three terms has already been named. From the definitions above, \(\\dfrac{W}{q} = \\varepsilon\), \(\\dfrac{W'}{q} = V\) and \(\\dfrac{w}{q} = v\). Therefore
\[\\varepsilon = V + v \\qquad \\text{or} \\qquad V = \\varepsilon - v\]
where \(\\varepsilon\) is the e.m.f. of the cell, \(V\) its terminal voltage and \(v\) the voltage drop inside it, all in volt (V).
Thus, when current is drawn from a cell, its terminal voltage \(V\) is less than its e.m.f. \(\\varepsilon\) by an amount equal to the voltage drop \(v\) inside the cell.
There is more to read from this. When a heavy current is drawn from the cell, a large number of charge carriers flow through the electrolyte, and hence more work is done inside the cell. That makes the voltage drop \(v\) larger, and so leaves the terminal voltage \(V\) smaller. The terminal voltage of a cell therefore also depends on the amount of current \(I\) drawn from it — unlike the e.m.f., which does not.
8.13.4 E.M.F. Compared With Terminal Voltage
The two quantities are measured in the same unit and are easily confused, so Table 8.6 sets them side by side.
8.14 Internal Resistance Of A Cell
Section 8.13 showed that some of a cell’s e.m.f. is spent inside the cell itself. Spent on what? On pushing charge through the electrolyte — and anything that opposes the flow of current is a resistance. That is the quantity this section names and measures.
8.14.1 What Internal Resistance Is
When a cell is in a closed circuit, current flows in two quite different ways along its path.
Inside the cell, the current flows through the electrolyte, due to the flow of ions — both cations and anions. Outside the cell, it flows through the external circuit, due to the flow of free electrons in the connecting wires.
In the external circuit the current flows from the terminal at positive potential to the terminal at negative potential; inside the cell it is carried on as ions through the electrolyte, so that a constant potential difference is maintained and the flow can continue without interruption.
The electrolyte, however, does not hand the current across freely, any more than a wire does.
It is because of this internal resistance that the voltage of a cell drops on drawing current from it. When a current \(I\) is drawn from a cell of internal resistance \(r\), the voltage drop inside the cell is
\[v = Ir\]
where \(I\) is the current in ampere (A), \(r\) is the internal resistance in ohm (\(\\Omega\)), and \(v\) is the voltage drop in volt (V). This is simply Ohm’s law applied to the inside of the cell.
For the purpose of circuit analysis, the internal resistance \(r\) of a cell is considered to be connected in series with the cell, as shown in Fig. 8.14-1. The cell is then drawn as a perfect source of e.m.f. \(\\varepsilon\) with a small resistance \(r\) sitting beside it, both inside the same dashed outline.
Diagram
Fig. 8.14-1 — Representation Of A Cell With Its Internal Resistance A dashed rectangle drawn to enclose two symbols joined end to end in series: on the left the standard cell symbol (long thin line marked ‘+’, short thick line marked ‘−’) labelled ‘ε’, and immediately to its right a small resistor symbol labelled ‘r’. A wire leaves each end of the dashed rectangle and ends in a solid dot marked ‘terminal’. Label the dashed rectangle ‘the real cell’. Add the note ‘the internal resistance is treated as being in series with the cell’.
8.14.2 Relation Between E.M.F., Terminal Voltage And Internal Resistance
Let a cell of e.m.f. \(\\varepsilon\) and internal resistance \(r\) be used to send a current in an external resistance \(R\) connected across the cell, as shown in Fig. 8.14-2.
Diagram
Fig. 8.14-2 — Current In An External Resistance From A Cell Circuit diagram as a single rectangular loop. In the top wire, inside a dashed rectangle, draw the cell symbol labelled ‘ε’ in series with a small resistor labelled ‘r’. In the bottom wire, draw a resistor labelled R. On the left-hand wire, mark an arrow labelled ‘I’. Between the two ends of the dashed rectangle, draw a long double-headed arrow across the loop labelled ‘terminal voltage V, the same as the p.d. across R’.
Since \(r\) and \(R\) are in series, the current has to pass through both, so
\[\\text{total resistance of circuit} = R + r\]
The current drawn from the cell is then found by applying Ohm’s law to the whole circuit — the e.m.f. is the total driving voltage, and \(R + r\) is the total resistance it works against:
\[I = \\frac{\\text{e.m.f. of cell}}{\\text{total resistance}} = \\frac{\\varepsilon}{R + r}\]
which can equally be written
\[\\varepsilon = I(R + r)\]
The terminal voltage of the cell is the potential difference across the external resistance alone, so
\[V = IR\]
and the voltage drop due to the internal resistance is
\[v = Ir\]
In all of these, \(\\varepsilon\), \(V\) and \(v\) are in volt (V), \(I\) is in ampere (A), and \(R\) and \(r\) are in ohm (\(\\Omega\)).
Putting the last three relations together, \(I(R + r) = IR + Ir\) gives back the result of Section 8.13:
\[\\varepsilon = V + v \\qquad \\text{or} \\qquad v = \\varepsilon - V\]
Finally, the internal resistance itself can be found from measurements taken outside the cell. Since \(v = Ir\),
\[r = \\frac{v}{I} = \\frac{\\varepsilon - V}{I} = \\frac{\\varepsilon - V}{V/R} = \\left(\\frac{\\varepsilon}{V} - 1\\right)R\]
where the middle step uses \(I = V/R\) from the external resistance. This last form is useful in the laboratory, because \(\\varepsilon\) is read on open circuit, \(V\) is read while the cell supplies current, and \(R\) is known.
8.14.3 Factors Affecting The Internal Resistance
The internal resistance of a cell depends on four factors, and each of them is about the journey the ions have to make through the electrolyte.
- The surface area of the electrodes — the larger the surface area of the electrodes, the less is the internal resistance, because the ions have a wider path to arrive at.
- The distance between the electrodes — the more the distance between the electrodes, the greater is the internal resistance, since the ions must travel further through the electrolyte.
- The nature and concentration of the electrolyte — the less ionic the electrolyte, or the higher the concentration of the electrolyte, the greater is the internal resistance.
- The temperature of the electrolyte — the higher the temperature of the electrolyte, the less is the internal resistance.
Notice that the first two of these are exactly the length and area effects met in Section 8.8, now applied to a column of liquid instead of a wire.
8.15 Combination Of Resistors
In order to have a desired resistance in a circuit, we sometimes have to connect two or more resistors together. A box of resistors rarely contains the exact value a circuit needs, but almost any value can be built from the ones it does contain.
Resistors can be combined in three ways: in series, in parallel, and both in series and in parallel.
The choice between the first two is made on simple grounds. When the resistance of the combination is to be increased, the resistors are combined in series. When the resistance of the combination is to be decreased, in order to pass a heavy current in the circuit, they are combined in parallel.
8.15.1 Resistors In Series
In the series combination, the resistors are joined one after the other, so that there is a single unbroken path from one end of the combination to the other.
In Fig. 8.15-1, three resistors \(R\_1\), \(R\_2\) and \(R\_3\) are connected in series between the points A and D. One end of \(R\_1\) is connected to the point A and its other end to \(R\_2\); the other end of \(R\_2\) is connected to one end of \(R\_3\); and the other end of \(R\_3\) goes to the point D. The ends A and D of the combination are then connected to the terminals of the battery through a key K.
Diagram
Fig. 8.15-1 — Series Combination Of Resistors Circuit diagram as a rectangular loop. Along the top wire, from left to right, place three resistors in a row, labelled R-1, R-2 and R-3. Mark the junctions on the top wire: A at the left-hand end before R-1, B between R-1 and R-2, C between R-2 and R-3, and D at the right-hand end after R-3. In the bottom wire place a battery, its ‘+’ terminal on the side joined through the left-hand wire to A, together with a key labelled K. Draw arrows labelled ‘I’ on the left-hand wire, on the right-hand wire and between the resistors, all pointing the same way round the loop, to show that the same current passes through all three.
When the circuit is closed, the same current \(I\) drawn from the battery passes through each of \(R\_1\), \(R\_2\) and \(R\_3\), because this is the only path available for the current to flow.
The end A of the resistor \(R\_1\) is connected to the positive terminal of the battery, and so is at a higher potential. The potential decreases in steps across each resistor, and at the end D of the last resistor \(R\_3\) the potential is a minimum, equal to that of the negative terminal of the battery. So if \(V\_A\), \(V\_B\), \(V\_C\) and \(V\_D\) are the potentials at the points A, B, C and D respectively, then
\[V\_A > V\_B > V\_C > V\_D\]
The potential difference across the resistor \(R\_1\) is \(V\_1 = V\_A - V\_B\), that across \(R\_2\) is \(V\_2 = V\_B - V\_C\), and that across \(R\_3\) is \(V\_3 = V\_C - V\_D\).
8.15.1.1 Two Facts About The Series Combination
Two facts are to be noted about the series combination, and every series problem is solved using them.
a. The current has a single path for its flow. Hence the same current passes through each resistor, so the potential difference across any resistor is directly proportional to its resistance. Thus the potential difference is different across different resistances — the largest resistor takes the largest share.
b. The potential difference across the entire circuit is equal to the sum of the potential differences across the individual resistors, that is,\[V = V\_1 + V\_2 + V\_3 + \\ldots\]
8.15.1.2 Expression For The Equivalent Resistance
If a current \(I\) is drawn from the battery, the current through each resistor is also \(I\). Applying Ohm’s law to each resistor in turn,
\[V\_1 = V\_A - V\_B = IR\_1, \\qquad V\_2 = V\_B - V\_C = IR\_2, \\qquad V\_3 = V\_C - V\_D = IR\_3\]
Adding these three,
\[V = V\_1 + V\_2 + V\_3 = V\_A - V\_D = I(R\_1 + R\_2 + R\_3) \\qquad \\ldots \\text{(i)}\]
Now suppose the whole combination were replaced by a single resistor of resistance \(R\_s\) that drew the same current from the same battery. This \(R\_s\) is called the equivalent resistance between A and D, and for it
\[V = V\_A - V\_D = I R\_s \\qquad \\ldots \\text{(ii)}\]
The left-hand sides of (i) and (ii) are the same potential difference, so the right-hand sides must be equal:
\[I R\_s = I(R\_1 + R\_2 + R\_3)\]
Cancelling the current \(I\), which is common to both sides,
\[R\_s = R\_1 + R\_2 + R\_3\]
Thus, in the series combination, the equivalent resistance is equal to the sum of the individual resistances. In general, if \(n\) resistances joined in series are \(R\_1, R\_2, \\ldots, R\_n\), the equivalent resistance is
\[R\_s = R\_1 + R\_2 + \\ldots + R\_n\]
and if there are \(n\) equal resistances, each of value \(R\), connected in series,
\[R\_s = R + R + \\ldots\\ n \\text{ times} = nR\]
where every resistance is in ohm (\(\\Omega\)). For example, three resistors of \(2\\ \\Omega\), \(3\\ \\Omega\) and \(5\\ \\Omega\) joined in series have an equivalent resistance \(R\_s = 2 + 3 + 5 = 10\\ \\Omega\) — larger than any one of them, which is exactly why series is the choice when more resistance is wanted.
8.15.2 Resistors In Parallel
In the parallel combination, one end of each resistor is connected at one point, say A, and the other end of each resistor is connected together at the other point, say B. The battery, along with a key K, is then connected in between the points A and B, as shown in Fig. 8.15-2.
Diagram
Fig. 8.15-2 — Parallel Combination Of Resistors Circuit diagram. Mark two junction dots, A on the left and B on the right. Between them draw three separate branches, one above the other, each containing a single resistor, labelled R-1 (top), R-2 (middle) and R-3 (bottom); every branch starts at A and ends at B. Below this arrangement, complete the loop through a battery with its ‘+’ terminal towards A, and a key labelled K. Mark the main current arrow ‘I’ on the wire approaching A, and mark ‘I-1’, ‘I-2’ and ‘I-3’ with arrows on the three branches. Mark the main current arrow ‘I’ again on the wire leaving B.
If the circuit is closed, the current \(I\) drawn from the battery divides itself at the point A into three parts: \(I\_1\) in \(R\_1\), \(I\_2\) in \(R\_2\) and \(I\_3\) in \(R\_3\). The current in a resistor is inversely proportional to its resistance — the less the resistance, the more is the current in it.
If \(V\_A\) and \(V\_B\) are the potentials at the points A and B, then the point A, connected to the positive terminal of the battery, is at a higher potential than the point B, connected to the negative terminal, so \(V\_A > V\_B\). Every one of the three resistors has one end at A and the other at B, so the potential difference across each resistor is the same, \(V = V\_A - V\_B\).
8.15.2.1 Two Facts About The Parallel Combination
a. The potential difference across each resistor is the same, equal to $V\_A - V\_B = V$, which is also equal to the potential difference across the terminals of the battery, or source.
b. The current in a resistor is inversely proportional to its resistance. The sum of the currents $I\_1, I\_2, I\_3, \\ldots$ in the separate branches of the parallel circuit is equal to the total current $I$ drawn from the battery, that is,\[I = I\_1 + I\_2 + I\_3 + \\ldots\]
8.15.2.2 Expression For The Equivalent Resistance
Let \(I\_1\), \(I\_2\) and \(I\_3\) be the currents through the resistances \(R\_1\), \(R\_2\) and \(R\_3\) respectively. Then the total current drawn from the battery is
\[I = I\_1 + I\_2 + I\_3 \\qquad \\ldots \\text{(i)}\]
If the potential difference between the two ends A and B is \(V\), then applying Ohm’s law to each branch separately — legitimate because each branch has that same \(V\) across it —
\[I\_1 = \\frac{V}{R\_1}, \\qquad I\_2 = \\frac{V}{R\_2}, \\qquad I\_3 = \\frac{V}{R\_3}\]
On adding these,
\[I\_1 + I\_2 + I\_3 = \\frac{V}{R\_1} + \\frac{V}{R\_2} + \\frac{V}{R\_3} \\qquad \\ldots \\text{(ii)}\]
If the equivalent resistance of the combination between the points A and B is \(R\_p\), then the total current drawn from the source is
\[I = \\frac{V}{R\_p} \\qquad \\ldots \\text{(iii)}\]
Substituting the values of \(I\) from (iii) and of \(I\_1 + I\_2 + I\_3\) from (ii) into (i),
\[\\frac{V}{R\_p} = V\\left(\\frac{1}{R\_1} + \\frac{1}{R\_2} + \\frac{1}{R\_3}\\right)\]
and cancelling the potential difference \(V\), which is common to every term,
\[\\frac{1}{R\_p} = \\frac{1}{R\_1} + \\frac{1}{R\_2} + \\frac{1}{R\_3}\]
Thus, in the parallel combination, the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the individual resistances. In general, for \(n\) resistances joined in parallel,
\[\\frac{1}{R\_p} = \\frac{1}{R\_1} + \\frac{1}{R\_2} + \\frac{1}{R\_3} + \\ldots + \\frac{1}{R\_n}\]
If there are \(n\) equal resistances, each of value \(R\), connected in parallel, then \(\\dfrac{1}{R\_p} = \\dfrac{1}{R} + \\dfrac{1}{R} + \\ldots\\ n\) terms \(= \\dfrac{n}{R}\), so
\[R\_p = \\frac{R}{n}\]
For the common case of just two resistances \(R\_1\) and \(R\_2\) in parallel,
\[\\frac{1}{R\_p} = \\frac{1}{R\_1} + \\frac{1}{R\_2} = \\frac{R\_2 + R\_1}{R\_1 R\_2} \\qquad \\text{or} \\qquad R\_p = \\frac{R\_1 R\_2}{R\_1 + R\_2}\]
with every resistance in ohm (\(\\Omega\)). This last form — product over sum — is worth memorising, but note that it works for two resistors only.
In the parallel combination the equivalent resistance \(R\_p\) is less than even the smallest resistance connected. Adding one more resistor in parallel opens one more path for the current, so the combination can only get easier to pass through.
As an example, connect three resistances of \(1\\ \\Omega\), \(10\\ \\Omega\) and \(100\\ \\Omega\) in parallel. Then
\[\\frac{1}{R\_p} = \\frac{1}{1} + \\frac{1}{10} + \\frac{1}{100} = \\frac{111}{100} \\qquad \\text{so} \\qquad R\_p = \\frac{100}{111} = 0.9009\\ \\Omega \\approx 0.9\\ \\Omega\]
which is less than \(1\\ \\Omega\), the smallest of the three.
Table 8.7 sets the two combinations side by side.
8.15.3 Resistors Both In Series And In Parallel
Sometimes, to obtain a desired value of resistance from the resistors we have, some resistors are connected in parallel and this parallel combination is then connected in series with some more resistors. A branch of a parallel combination may itself have two or more resistors in series. Similarly, there can be more than one parallel combination in series.
In such a case the work is done in a fixed order.
- First determine the resistance of each branch of the parallel combination, adding up any resistors that are in series within that branch.
- Then determine the equivalent resistance of each parallel combination.
- After this, find the total resistance by adding each resistance and the equivalent resistance of each parallel combination that are connected in series.
For example, suppose an equivalent resistance of \(1.5\\ \\Omega\) is to be obtained using three resistors each of \(1\\ \\Omega\). Connect a parallel combination of two of them in series with the third, as shown in Fig. 8.15-3.
Diagram
Fig. 8.15-3 — Combination Of Three Resistors Each Of 1 Ω For An Equivalent Resistance Of 1.5 Ω Circuit fragment drawn between two open terminals. On the left, two parallel branches between a pair of junction dots, each branch containing a resistor labelled ‘1 Ω’; from the right-hand junction dot a single wire leads to a third resistor labelled ‘1 Ω’ in series, and on to the right-hand terminal. Underneath, add two small labels: ‘parallel part = 0.5 Ω’ beneath the two-branch portion, and ‘total = 0.5 + 1 = 1.5 Ω’ beneath the whole.
The equivalent resistance of the parallel combination is
\[R\_p = \\frac{1 \\times 1}{1 + 1} = 0.5\\ \\Omega\]
and the total resistance of this in series with the third resistor is
\[R\_s = 0.5\\ \\Omega + 1\\ \\Omega = 1.5\\ \\Omega\]
8.15.4 Series And Parallel On The I–V Graph
For two given resistors joined in series, the equivalent resistance is more than when the same two are joined in parallel. This difference can be seen at once on an I–V graph, as in Fig. 8.15-4.
Diagram
Fig. 8.15-4 — I–V Graphs For Series And Parallel Combinations Graph with the horizontal axis labelled ‘potential difference V (volt)’ and the vertical axis labelled ‘current I (ampere)’, origin marked O. Draw two straight lines from the origin: the upper, steeper line labelled ‘A (parallel combination)’ and the lower, less steep line labelled ‘B (series combination)’. Keep both lines clearly separated and extend them to the same value of V.
The reason lies in what the slope of a line on an I–V graph means. From Section 8.5.6, that slope is the conductance, which is the reciprocal of the resistance:
\[\\text{slope of the } I \\text{ against } V \\text{ graph} = \\frac{1}{\\text{resistance}}\]
So the larger the resistance, the smaller the slope. The straight line A, with the greater slope, corresponds to the parallel combination of the resistors, while the straight line B, with the smaller slope, corresponds to the series combination — more resistance, less slope.
8.16 Electrical Energy
Energy exists in various forms — mechanical energy, heat energy, chemical energy, electrical energy, light energy, nuclear energy and others. According to the law of conservation of energy, energy can be transformed from one form to another, but it can neither be created nor destroyed.
That law is the reason electricity is worth generating at all. Electrical energy is not useful in itself; it is useful because it can be carried along a wire to wherever we are, and turned there into whatever form we happen to need.
8.16.1 Electrical Energy Converted Into Other Forms
In daily life we use many devices in which electrical energy is used by converting it into other forms such as heat energy, light energy, mechanical energy, chemical energy, sound energy and magnetic energy. Six examples show the range.
- On passing electric current through the heating element of an electric heater, oven or geyser, the element gets heated up because of its resistance, and the electrical energy gets converted into heat energy, which is used for heating purposes.
- On passing electric current through an electric lamp, the filament of the bulb is heated to such an extent that it glows. The electrical energy thus changes into light energy and heat energy.
- On passing electric current through an electric motor, such as the one used in a fan, mixer or juicer, its coil and the attachment on the axle of the coil begin to rotate, and at the same time the coil gets slightly heated up. The electrical energy thus changes mostly into mechanical energy, and a small part into heat energy.
- On passing electric current through an electrolyte, such as dilute sulphuric acid, while charging a lead accumulator, a chemical reaction takes place and the electrical energy gets converted into chemical energy.
- On passing electrical pulses from a microphone to a loudspeaker, the electrical energy gets converted into sound energy.
- On passing electric current in a coil wound around a soft iron bar, the bar gets magnetised and becomes an electromagnet. Thus the electrical energy gets converted into magnetic energy.
Notice how often heat appears in that list even when heat is not what was wanted. Every one of these devices contains conductors with resistance, and Section 8.4 explained why a current in a resistance always produces some heat.
Flowchart
FC 8.2 — Energy Conversions In An Electric Fan Electrical energy supplied to the motor → magnetic effect of the current in the coil → rotation of the coil and the axle, that is mechanical energy → kinetic energy of the moving air, with a small part of the electrical energy becoming heat energy in the windings
8.16.2 Expression For Electrical Energy
Let a current \(I\) flow through a conductor of resistance \(R\) for a time \(t\), when a source of potential difference \(V\) is connected across its ends, as shown in Fig. 8.16-1. How much electrical energy has the source supplied?
Diagram
Fig. 8.16-1 — Current Flowing In A Resistance Circuit diagram as a rectangular loop. In the top wire, a resistor labelled R. In the bottom wire, a cell with its terminals marked ‘+’ and ‘−’, and a key. Across the resistor, draw a double-headed arrow labelled ‘V’. Mark an arrow labelled ‘I’ on the left-hand wire showing the direction of the current round the loop, and add the note ‘current flows for a time t’.
Two results we already have are all that is needed.
Because of the flow of current \(I\) for time \(t\), the amount of charge passed is \(Q = I \\times t\), from Section 8.2.
By the definition of potential difference in Section 8.3, the work needed to move a charge \(Q\) through a potential difference \(V\) is
\[W = QV\]
Substituting \(Q = I \\times t\) into it,
\[W = VIt\]
where \(V\) is the potential difference in volt (V), \(I\) is the current in ampere (A), \(t\) is the time for which the current flows in second (s), and \(W\) is the work done, in joule (J).
This relation gives the work done, or the electrical energy \(W\) supplied by the source — a battery or the mains — in providing the current \(I\) for the time \(t\) in the conductor under a potential difference \(V\).
The SI unit of electrical energy is the joule, symbol J.
8.16.3 Other Expressions Using Ohm’s Law
The relation \(W = VIt\) needs all three of \(V\), \(I\) and \(t\). Often only two of them are known, and Ohm’s law lets us do without the third.
By Ohm’s law, \(V = IR\). Substituting this for \(V\) in \(W = VIt\),
\[W = I^2 R t\]
Alternatively, Ohm’s law can be written \(I = V/R\). Substituting this for \(I\) instead,
\[W = V \\times \\frac{V}{R} \\times t = \\frac{V^2 t}{R}\]
Collecting all the forms together, the electrical energy supplied by the source is
\[W = QV = VIt = I^2 R t = \\frac{V^2 t}{R}\]
where \(Q\) is in coulomb (C), \(V\) in volt (V), \(I\) in ampere (A), \(R\) in ohm (\(\\Omega\)), \(t\) in second (s), and \(W\) in joule (J). The last two forms hold for a conductor obeying Ohm’s law; all four give the same answer, so use whichever matches the quantities you have been given.
8.17 Electrical Power
Two lamps may both convert the same total energy — one over an hour, the other over a minute. What separates them is not how much energy, but how fast. That is the quantity called power.
8.17.1 Power As The Rate Of Supply Of Energy
In an electrical circuit, electric power is defined as the rate at which the electrical energy is supplied by the source.
8.17.2 Expressions For Electrical Power
Let a source of potential difference \(V\) send a current \(I\) for a time \(t\) in a resistance \(R\). The charge passed is \(Q = It\), and the energy supplied by the source is \(W = QV\), as in Section 8.16.
Then, by the definition just given,
\[P = \\frac{\\text{energy supplied}}{\\text{time}} = \\frac{W}{t} = \\frac{QV}{t}\]
Since \(Q = It\), this becomes
\[P = \\frac{VIt}{t} = VI\]
where \(V\) is the potential difference in volt (V), \(I\) is the current in ampere (A), and \(P\) is the power in watt (W). Notice that the time has cancelled out — power does not depend on how long the current flows, only on how vigorously energy is being converted while it does.
As with energy, Ohm’s law gives two further forms. Substituting \(I = \\dfrac{V}{R}\),
\[P = V \\times \\frac{V}{R} = \\frac{V^2}{R}\]
and substituting \(V = IR\) instead,
\[P = \\frac{(IR)^2}{R} = I^2 R\]
Collecting them,
\[P = \\frac{W}{t} = VI = \\frac{V^2}{R} = I^2 R\]
where \(W\) is the electrical energy in joule (J), \(t\) the time in second (s), \(V\) in volt (V), \(I\) in ampere (A), \(R\) in ohm (\(\\Omega\)) and \(P\) in watt (W). The last two forms apply to a conductor obeying Ohm’s law.
8.17.3 Units Of Electrical Power
The SI unit of potential difference is the volt and that of current is the ampere, so the SI unit of electrical power is the volt ampere (\(\\text{V A}\)). This is given the name watt, symbol W, and since a volt is a joule per coulomb and an ampere is a coulomb per second, it is also the joule per second (\(\\text{J s}^{-1}\)).
In the relation \(P = VI\), if \(V = 1\\ \\text{volt}\) and \(I = 1\\ \\text{ampere}\), then \(P = 1\\ \\text{watt}\).
A watt is a small unit for describing heaters, motors or power stations, so bigger units are used:
\[1\\ \\text{kilowatt (kW)} = 1000\\ \\text{W} = 10^3\\ \\text{W}\] \[1\\ \\text{megawatt (MW)} = 1000\\ \\text{kW} = 10^6\\ \\text{W}\] \[1\\ \\text{gigawatt (GW)} = 1000\\ \\text{MW} = 10^9\\ \\text{W}\]
A ceiling fan is rated in tens of watt, a room heater in kilowatt, and the output of a large power station in gigawatt.
8.18 Commercial Units Of Electrical Energy
The SI unit of energy is the joule. It is the right unit for a laboratory and a clumsy one for a household, as a moment’s arithmetic shows: a single room heater running for an evening uses tens of millions of joule. So electricity is bought and sold in units of a more comfortable size.
If an electrical appliance of power \(P\) watt is used for a time \(t\) second, the electrical energy consumed is
\[W = \\text{power} \\times \\text{time} = P \\times t\]
in joule, that is in watt second. So the watt second can itself be used as a unit of electrical energy — and it is simply another name for the joule.
In practice, the electrical energy is measured in bigger units, namely the watt-hour (Wh) and the kilowatt-hour (kWh). These are the commercial units of electrical energy.
8.18.1 Watt-Hour
Its value in joule follows at once, since one hour is \(60 \\times 60 = 3600\) second:
\[1\\ \\text{watt-hour} = 1\\ \\text{W} \\times (60 \\times 60\\ \\text{s}) = 3600\\ \\text{J}\]
so that
\[1\\ \\text{Wh} = 3600\\ \\text{J}\]
8.18.2 Kilowatt-Hour
Working out its value in joule step by step,
\[1\\ \\text{kilowatt-hour} = 1\\ \\text{kilowatt} \\times 1\\ \\text{hour} = 1000\\ \\text{watt} \\times 1\\ \\text{hour}\]
\[= 1000\\ \\text{J s}^{-1} \\times (60 \\times 60\\ \\text{s}) = 3.6 \\times 10^{6}\\ \\text{J}\]
so that
\[1\\ \\text{kWh} = 3.6 \\times 10^{6}\\ \\text{J}\]
The electrical energy, or electricity, consumed by the various appliances in our houses and in industries is measured in the unit kWh, and its cost is paid to the electric company. One kilowatt-hour is what an electricity bill calls one unit.
8.19 Power Rating Of Common Electrical Appliances
Generally an electrical appliance such as an electric bulb, a geyser or a heater is rated with its power and voltage. That pair of numbers, printed on the appliance itself, is more informative than it looks.
8.19.1 What A Power Rating Tells You
For example, a bulb is rated as \(100\\ \\text{W} - 220\\ \\text{V}\). It means that if the bulb is lighted on a \(220\\ \\text{V}\) supply, the electric power consumed by it is \(100\\ \\text{W}\) — that is, \(100\\ \\text{J}\) of electrical energy is consumed by the bulb in \(1\\ \\text{s}\).
Diagram
Fig. 8.19-1 — The Rating Marked On An Appliance Two small drawings side by side. On the left, an electric bulb drawn with its glass envelope and cap, with the marking ‘100 W 220 V’ printed on the cap, and a pointer to it labelled ‘power rating and voltage rating’. On the right, a rectangular rating plate as found on the body of an appliance, printed with three lines: ‘ROOM HEATER’, ‘1000 W’, ‘220 V’. Beneath both, place the caption line ‘the power is consumed only when the appliance is used at its rated voltage’.
From the rating of an appliance, we can calculate the following two quantities:
- the resistance of its filament or element when it is in use, and
- the safe limit of current which can flow through the appliance while in use.
8.19.2 Resistance Of The Element While In Use
From the relation \(P = \\dfrac{V^2}{R}\) of Section 8.17, the resistance of the element of the appliance while in use is \(R = \\dfrac{V^2}{P}\), that is
\[R = \\frac{(\\text{voltage rating on the appliance})^2}{\\text{power rating on the appliance}}\]
where the voltage rating is in volt (V), the power rating is in watt (W), and \(R\) is in ohm (\(\\Omega\)).
For the bulb rated \(100\\ \\text{W} - 220\\ \\text{V}\), the resistance of its filament while glowing is
\[R = \\frac{V^2}{P} = \\frac{(220)^2}{100} = 484\\ \\Omega\]
The words ‘while in use’ are essential here. The actual resistance of the filament of a bulb when it is not glowing — that is, when it is cold — is much less than this value, because the resistance of the filament increases when its temperature increases on glowing, as Section 8.8.4 explained.
8.19.3 Safe Current Through An Appliance
The current through the element of the appliance when in use follows from \(P = VI\):
\[I = \\frac{P}{V}\]
If the current exceeds this value, the power supplied at voltage \(V\) will exceed the rated power of the appliance, and the appliance may get damaged. So this value of current is called the safe current which can flow through the appliance at voltage \(V\):
\[\\text{safe current } I = \\frac{\\text{power rating on the appliance}}{\\text{voltage rating on the appliance}}\]
For the same bulb, the safe limit of current through the filament while glowing is
\[I = \\frac{P}{V} = \\frac{100}{220} = 0.454\\ \\text{A} \\approx 0.5\\ \\text{A}\]
Thus the bulb can withstand a current of \(0.5\\ \\text{A}\). If the current exceeds \(0.5\\ \\text{A}\), it may cause the bulb to fuse. At a voltage higher than the rated value, an excessive current flows through the filament, which raises its temperature to the extent that the filament melts and the bulb gets fused.
8.19.4 Working At A Voltage Other Than The Rated Voltage
The power consumed by a bulb will be equal to the power rating only when the bulb is connected exactly at the voltage rated on it. If the voltage at which the bulb is connected is slightly more, the bulb will glow more brightly — that is, it will consume more power. If the voltage is low, the bulb will glow less, with a faint dull red light, because it will consume less power.
Take the bulb rated \(100\\ \\text{W}\), \(220\\ \\text{V}\) again. It consumes \(100\\ \\text{W}\) at \(220\\ \\text{V}\), and the resistance of its filament while glowing is \(\\dfrac{(220)^2}{100} = 484\\ \\Omega\).
Now if this bulb is lighted at \(110\\ \\text{V}\), it will consume power equal to
\[P = \\frac{V^2}{R} = \\frac{(110)^2}{484} = 25\\ \\text{W}\]
But if the same bulb is lighted at \(250\\ \\text{V}\), and assuming it does not get fused, it will consume power equal to
\[P = \\frac{(250)^2}{484} = 129.1\\ \\text{W}\]
Two assumptions have been made in these two calculations, and both are worth stating plainly. The first is that the resistance of the filament remains the same when the bulb is used at \(110\\ \\text{V}\) and at \(250\\ \\text{V}\). In fact this is not true, because the resistance of the filament changes with temperature. At \(110\\ \\text{V}\) the resistance will be less than \(484\\ \\Omega\), so the power consumed will be more than \(25\\ \\text{W}\), though still less than \(100\\ \\text{W}\). Similarly at \(250\\ \\text{V}\) the resistance will be more than \(484\\ \\Omega\), so the power consumed will be less than \(129\\ \\text{W}\), though still more than \(100\\ \\text{W}\). The second assumption is that the current in the bulb remains less than its safe value; that is true at the low voltage, but at the high voltage the current exceeds the safe limit.
8.19.5 Power Ratings Of Some Common Appliances
Table 8.8 gives the power ratings of some appliances found in an ordinary household.
Table 8.8 — Power Rating Of Some Common Appliances
| Appliance | Power (in watt) | Voltage (in volt) |
|---|---|---|
| Car (filament) bulb | 20 | 12 |
| Electric bulb | 15 – 200 | 220 |
| LED television set | 30 – 80 | 220 |
| Fluorescent tube | 40 | 220 |
| Electric fan | 60 – 100 | 220 |
| Refrigerator | 150 | 220 |
| Electric iron | 700 | 220 |
| Electric mixer | 750 | 220 |
| Room heater | 1000 | 220 |
| Geyser | 1500 | 220 |
| Electric kettle | 2000 | 220 |
| Electric oven | 3000 | 220 |
Reading down the middle column shows a clear pattern: the appliances meant to produce heat sit at the bottom, with ratings of a kilowatt and more, while those meant to produce light, motion or sound sit at the top.
8.20 Household Consumption Of Electrical Energy
Everything in Sections 8.16 to 8.19 comes together in one familiar object: the electricity bill. Working out what a household owes is a straightforward piece of physics, and doing it once makes the bill readable for good.
8.20.1 The Electric Meter And The Unit Of Billing
An electric meter measures the amount of electrical energy consumed by the various appliances in a house in a given period of time. The electric meter is fixed at the mains board, or just outside the house, so that it counts everything that enters the house and nothing that does not.
The electrical energy is sold in units of kilowatt hour (kWh). Thus the unit in which the consumer pays the cost of electrical energy consumed is the kWh, and this is what a bill means when it says ‘units’.
Diagram
Fig. 8.20-1 — Where The Electric Meter Sits Simple block-style drawing. On the left, a pole with an incoming supply line labelled ‘supply from the electric company’. The line enters a rectangular box labelled ‘electric meter’, drawn with a small display panel reading ‘units (kWh)’. From the meter, a line goes to a second rectangle labelled ‘mains board of the house’, and from that, three lines branch away to small symbols of a lamp, a fan and a heater, labelled ‘appliances in the house’. Add the note under the meter: ‘measures all the electrical energy entering the house’.
8.20.2 Calculating The Energy Consumed And Its Cost
The electrical energy consumed by an appliance in a certain time can be calculated in kWh by the following relation:
\[\\text{energy (in kWh)} = \\text{power (in kW)} \\times \\text{time (in h)}\]
\[= \\frac{\\text{power (in watt)} \\times \\text{time (in hour)}}{1000} = \\frac{V\\ (\\text{volt}) \\times I\\ (\\text{ampere}) \\times t\\ (\\text{hour})}{1000}\]
The division by 1000 in the second and third forms converts watt into kilowatt. Time here is in hour, not in second — this is the one place in the chapter where a non-SI unit of time is the right one to use, because the kilowatt hour is defined with it.
From this relation it is clear that the total energy consumed by an electrical appliance in kWh can be calculated by multiplying its power rating in kW with the time duration in hour for which it was in use. The cost of electricity is then the product of the energy consumed in kWh with the rate in rupees per kWh:
\[\\text{cost of electricity} = \\text{electrical energy in kWh} \\times \\text{cost per kWh}\]
For instance, an electric iron of power \(700\\ \\text{W}\), that is \(0.7\\ \\text{kW}\), used for one hour each day, consumes \(0.7\\ \\text{kWh}\) per day, and in a month of 30 days consumes \(0.7\\ \\text{kW} \\times (30 \\times 1\\ \\text{h}) = 21\\ \\text{kWh}\). At a rate of ₹ 4.50 per kWh, its share of the bill is \(21 \\times ₹,4.50 = ₹,94.50\).
When Several Appliances Are Used
A household never runs one appliance alone. If we use appliances of different powers \(P\_1, P\_2, P\_3, \\ldots\) in kW for different durations \(t\_1, t\_2, t\_3, \\ldots\) in hour, the total energy consumed in kWh is the sum of the product of the power rating in kW and the time duration in hour for each appliance:
\[\\text{total energy consumed} = (P\_1t\_1 + P\_2t\_2 + P\_3t\_3 + \\ldots)\\ \\text{kWh}\]
Each appliance is worked out on its own and the results are added. The powers cannot be added first unless the appliances all run for the same length of time.
8.21 Heating Effect Of Current
Every section since 8.4 has hinted at this one. Drifting electrons collide with the fixed ions of a conductor, lose energy at every collision, and the conductor warms. Sometimes that heating is the whole purpose, as in a heater; sometimes it is a nuisance, as in a transmission line; and sometimes it is a safety device, as in a fuse. In each case the same question matters: how much heat is produced?
8.21.1 Factors On Which The Heat Produced Depends
The amount of heat produced in a wire on passing a current through it depends on the following three factors.
- The amount of current passing through the wire. The amount of heat \(H\) produced in a wire is directly proportional to the square of the current \(I\) passing through the wire, that is, \(H \\propto I^2\).
- The resistance of the wire. The amount of heat \(H\) produced in the wire is directly proportional to the resistance \(R\) of the wire, that is, \(H \\propto R\).
- The time for which current is passed in the wire. The amount of heat \(H\) produced in a wire is directly proportional to the time \(t\) for which current is passed in the wire, that is, \(H \\propto t\).
The first of these is the one to notice. The heat depends not on the current but on the square of the current, so doubling the current in a wire produces four times as much heat in it, not twice as much.
8.21.2 Joule’s Law Of Heating
Combining the three results above,
\[H \\propto I^2 R t\]
and taking the constant of proportionality as 1 when the heat is measured in joule,
\[H = I^2 R t\\ \\text{joule} \\qquad \\text{or} \\qquad H = 0.24, I^2 R t\\ \\text{calorie}\]
where \(I\) is the current in ampere (A), \(R\) is the resistance in ohm (\(\\Omega\)), and \(t\) is the time in second (s). The second form gives the same heat in calorie, since \(1\\ \\text{cal} = 4.18\\ \\text{joule}\) and \(1 \\div 4.18 = 0.24\).
This relation is known as Joule’s law of heating.
Notice that this is the same quantity as the electrical energy \(W = I^2Rt\) of Section 8.16. Nothing new has been created: in a conductor whose only job is to resist, every joule of electrical energy supplied appears as a joule of heat.
8.22 Summary
- There are two kinds of charge, positive and negative. Like charges repel and unlike charges attract. The SI unit of charge is the coulomb (C), with \(1\\ \\text{mC} = 10^{-3}\\ \\text{C}\), \(1\\ \\mu\\text{C} = 10^{-6}\\ \\text{C}\) and \(1\\ \\text{nC} = 10^{-9}\\ \\text{C}\).
- Charge exists only in whole multiples of the electronic charge: \(q = \\pm ne\), where \(q\) is the charge in coulomb (C), \(n\) is the number of electrons in deficit or excess, and \(e = 1.6 \\times 10^{-19}\\ \\text{C}\). One coulomb corresponds to \(6.25 \\times 10^{18}\) electrons.
- In an insulator the charge stays where it is placed and is static. In a conductor a large number of electrons are free to move through the whole volume, so charge can flow.
- Electric current is the rate of flow of charge across any cross section of a conductor: \[I = \\frac{Q}{t} = \\frac{ne}{t}\] where \(Q\) is the charge in coulomb (C), \(t\) the time in second (s) and \(I\) the current in ampere (A). One ampere is one coulomb per second.
- In metals the current is carried by free electrons; in electrolytes and ionised gases it is carried by cations and anions moving in opposite directions.
- Current is a scalar quantity. Its direction is taken as the direction in which positive charge would move, which is opposite to the direction of drift of the electrons in a metal. It is measured by an ammeter connected in series.
- The potential at a point is the work done per unit charge in bringing a positive test charge from infinity to that point: \[V = \\frac{W}{Q}\] where \(W\) is the work in joule (J), \(Q\) the charge in coulomb (C) and \(V\) the potential in volt (V). One volt is one joule per coulomb.
- The potential difference between two points is the work done per unit charge in moving a positive test charge from one point to the other. It is measured by a voltmeter connected in parallel across those points.
- Resistance is the obstruction offered by a conductor to the flow of current, measured in ohm (\(\\Omega\)). It arises from collisions of the drifting free electrons with the fixed positive ions of the metal.
- Ohm’s law: the current flowing in a conductor is directly proportional to the potential difference applied across its ends, provided the physical conditions and the temperature of the conductor remain constant. Hence \[V = IR\] with \(V\) in volt (V), \(I\) in ampere (A) and \(R\) in ohm (\(\\Omega\)).
- One ohm is the resistance of a conductor through which a current of one ampere flows when a potential difference of one volt is applied across its ends. Larger units are \(1\\ \\text{k}\\Omega = 10^{3}\\ \\Omega\) and \(1\\ \\text{M}\\Omega = 10^{6}\\ \\Omega\).
- Conductance is the reciprocal of resistance, \(G = 1/R\), measured in \(\\Omega^{-1}\) or siemens (S).
- The slope of an I–V graph is \(1/R\), while the slope of a V–I graph is \(R\) itself. Ohm’s law is verified experimentally by showing that \(V/I\) stays constant over many pairs of readings.
- Ohmic resistors obey Ohm’s law and give a straight line through the origin on a V–I graph; their resistance is the same at all values of \(V\) and \(I\). Non-ohmic resistors, such as a junction diode, an LED or the filament of a bulb, give a curve, and their resistance differs at different points.
- The resistance of a conductor depends on four factors: the material, the length, the thickness, and the temperature. It is directly proportional to the length, \(R \\propto l\), and inversely proportional to the area of cross section, \(R \\propto 1/a\). It increases with an increase in temperature for a metal.
- Combining the length and area effects gives \[R = \\rho,\\frac{l}{a}\] where \(l\) is the length in metre (m), \(a\) the area of cross section in \(\\text{m}^2\), \(R\) the resistance in ohm (\(\\Omega\)) and \(\\rho\) the specific resistance of the material in ohm metre (\(\\Omega\\ \\text{m}\)).
- Specific resistance, or resistivity, is the resistance of a wire of that material of unit length and unit area of cross section. It depends only on the material and its temperature, not on the shape or size of the conductor.
- A wire stretched to twice its length has four times its previous resistance; stretched to three times its length, nine times. A wire doubled on itself has one-fourth of its previous resistance. In every such case the specific resistance is unchanged.
- Conductivity is the reciprocal of specific resistance, \(\\sigma = 1/\\rho\), measured in \(\\Omega^{-1}\\ \\text{m}^{-1}\) or siemens per metre.
- The material of a wire is chosen for its purpose: copper or aluminium, made thick, for connecting and transmission wires; manganin or constantan for resistance wires and standard resistors; an alloy of lead and tin for a fuse wire; tungsten for a bulb filament; and nichrome for the heating element of an appliance.
- A superconductor is a substance of zero resistance, or infinite conductance, at a very low temperature. Mercury below \(4.2\\ \\text{K}\), lead below \(7.25\\ \\text{K}\) and niobium below \(9.2\\ \\text{K}\) are superconductors.
- The e.m.f. of a cell is the energy spent per unit charge in taking a positive test charge around the complete circuit of the cell, inside as well as outside it: \[\\varepsilon = \\frac{W}{q}\] with \(W\) in joule (J), \(q\) in coulomb (C) and \(\\varepsilon\) in volt (V). It depends only on the material of the electrodes and the electrolyte used.
- The terminal voltage of a cell is the work done per unit charge in carrying a positive test charge around the external circuit only, \(V = W'/q\). The voltage drop inside the cell is \(v = w/q\).
- The three are related by \[\\varepsilon = V + v \\qquad \\text{or} \\qquad V = \\varepsilon - v\] all in volt (V). The terminal voltage is less than the e.m.f. whenever current is drawn, and falls further as the current drawn increases.
- The internal resistance \(r\) of a cell is the resistance offered by its electrolyte to the flow of current, measured in ohm (\(\\Omega\)). With an external resistance \(R\), \[I = \\frac{\\varepsilon}{R + r},\\qquad V = IR,\\qquad v = Ir,\\qquad r = \\left(\\frac{\\varepsilon}{V} - 1\\right)R\]
- Internal resistance is less for electrodes of larger surface area, greater for electrodes further apart, greater for a less ionic or more concentrated electrolyte, and less at a higher temperature of the electrolyte.
- In a series combination the same current passes through each resistor, the potential differences add, and \[R\_s = R\_1 + R\_2 + \\ldots + R\_n\] which is larger than the largest resistance. For \(n\) equal resistances, \(R\_s = nR\).
- In a parallel combination the potential difference across each resistor is the same, the currents add, and \[\\frac{1}{R\_p} = \\frac{1}{R\_1} + \\frac{1}{R\_2} + \\ldots + \\frac{1}{R\_n}\] which is less than the smallest resistance. For two resistors, \(R\_p = \\dfrac{R\_1R\_2}{R\_1 + R\_2}\), and for \(n\) equal resistances, \(R\_p = R/n\).
- In a mixed network, each parallel combination is reduced first and the results are then added in series. On an I–V graph the parallel combination gives the steeper line, since it has the smaller resistance.
- Two resistors in parallel share the current as \(I\_1 = \\dfrac{IR\_2}{R\_1 + R\_2}\) and \(I\_2 = \\dfrac{IR\_1}{R\_1 + R\_2}\); two resistors in series share the voltage as \(V\_1 = \\dfrac{VR\_1}{R\_1 + R\_2}\) and \(V\_2 = \\dfrac{VR\_2}{R\_1 + R\_2}\).
- Electrical energy supplied by a source is \[W = QV = VIt = I^2Rt = \\frac{V^2t}{R}\] with \(Q\) in coulomb (C), \(V\) in volt (V), \(I\) in ampere (A), \(R\) in ohm (\(\\Omega\)), \(t\) in second (s) and \(W\) in joule (J).
- Electric power is the rate at which electrical energy is supplied: \[P = \\frac{W}{t} = VI = \\frac{V^2}{R} = I^2R\] measured in watt (W). One watt is the power consumed when a current of one ampere flows through a potential difference of one volt. \(1\\ \\text{kW} = 10^3\\ \\text{W}\), \(1\\ \\text{MW} = 10^6\\ \\text{W}\), \(1\\ \\text{GW} = 10^9\\ \\text{W}\).
- The commercial units of electrical energy are the watt hour and the kilowatt hour: \(1\\ \\text{Wh} = 3600\\ \\text{J}\) and \(1\\ \\text{kWh} = 3.6 \\times 10^{6}\\ \\text{J}\). The kilowatt hour is the unit in which electricity is billed.
- The rating of an appliance gives its power and its voltage. From it, the resistance of its element in use is \(R = \\dfrac{(\\text{voltage rating})^2}{\\text{power rating}}\) and the safe current is \(I = \\dfrac{\\text{power rating}}{\\text{voltage rating}}\). The rated power is consumed only at the rated voltage.
- Household consumption is measured by an electric meter in kWh, where energy (in kWh) \(=\) power (in kW) \(\\times\) time (in h). For several appliances, the total energy is \((P\_1t\_1 + P\_2t\_2 + \\ldots)\) kWh, and the cost is the energy in kWh multiplied by the cost per kWh.
- Joule’s law of heating: the heat produced in a wire is directly proportional to the square of the current, to the resistance and to the time, \[H = I^2Rt\\ \\text{joule} = 0.24,I^2Rt\\ \\text{calorie}\] with \(I\) in ampere (A), \(R\) in ohm (\(\\Omega\)) and \(t\) in second (s), where \(1\\ \\text{cal} = 4.18\\ \\text{J}\).
Flowchart
FC 8.3 — The Spine Of The Chapter, End To End Charge \(q\) in coulomb → charge in motion gives current \(I = Q/t\) in ampere → a potential difference \(V\) in volt is needed to keep it flowing → the conductor opposes it with resistance \(R\), giving Ohm’s law \(V = IR\) → \(R\) is fixed by material, length, area and temperature through \(R = \\rho l/a\) → a cell supplies the \(V\), its e.m.f. \(\\varepsilon\) shared between the circuit and its own internal resistance, \(\\varepsilon = V + v\) → resistors combined in series or parallel set the current the circuit draws → energy delivered \(W = VIt\) → rate of delivery \(P = VI\) → billed as kilowatt hours → and in every resistance, energy appears as heat, \(H = I^2Rt\)
8.23 Key Terms
- Ammeter — the instrument used to measure the current in a circuit; it is connected in series, and reads in ampere (A).
- Ampere — the SI unit of electric current; one ampere is one coulomb of charge passing in one second.
- Anions — negatively charged ions, which move through an electrolyte in the direction opposite to the cations and help to carry the current.
- Cations — positively charged ions, which move through an electrolyte and carry current in the direction of the conventional current.
- Conductance — the reciprocal of resistance, a measure of how easily a conductor passes current; SI unit ohm\(^{-1}\) (\(\\Omega^{-1}\)) or siemens (S).
- Conductivity — the reciprocal of specific resistance, a property of the material; SI unit ohm\(^{-1}\) metre\(^{-1}\) (\(\\Omega^{-1}\\ \\text{m}^{-1}\)) or siemens per metre.
- Conductor — a material such as a metal in which a large number of electrons are free to move, so that charge can flow through it.
- Coulomb — the SI unit of electric charge; the charge carried by \(6.25 \\times 10^{18}\) electrons.
- Drift — the slow overall movement of free electrons in one direction along a conductor when a potential difference is applied across it.
- Electric charge — the property of a body, arising from a deficit or an excess of electrons, that makes it attract or repel other charged bodies; SI unit coulomb (C).
- Electric current — the rate of flow of charge across any cross section of a conductor; SI unit ampere (A).
- Electric potential — the work done per unit charge in bringing a positive test charge from infinity to a point; SI unit volt (V).
- Electric power — the rate at which electrical energy is supplied by a source; SI unit watt (W).
- Electrical energy — the energy supplied by an electrical source, given by \(W = VIt\); SI unit joule (J), and in commerce the kilowatt-hour.
- Electrodes — the two conductors of a cell, dipping into the electrolyte, between which the cell maintains a potential difference.
- Electro-motive force — the energy spent per unit charge in taking a positive test charge around the complete circuit of a cell, inside it as well as outside; SI unit volt (V).
- Electrolyte — the conducting liquid in a cell, which carries current within the cell by the movement of its ions.
- Free electrons — the electrons in a metal that are not bound to any one atom and can move throughout its volume.
- Fuse wire — a short, thin wire of a lead–tin alloy placed in a circuit, which melts and breaks the circuit when the current exceeds a safe value.
- Good conductors — substances with a large concentration of free electrons, which therefore offer little resistance to a current.
- Insulator — a substance with a negligible concentration of free electrons, in which charge stays static and does not flow.
- Internal resistance — the resistance offered by the electrolyte inside a cell to the flow of current; SI unit ohm (\(\\Omega\)).
- Joule’s law of heating — the law that the heat produced in a conductor is proportional to \(I^2\), to \(R\) and to \(t\), so that \(H = I^2Rt\) in joule.
- Kilowatt-hour — the commercial unit of electrical energy; the energy consumed by an appliance of power one kilowatt used for one hour, equal to \(3.6 \\times 10^{6}\\ \\text{J}\).
- Non-ohmic resistors — conductors that do not obey Ohm’s law, whose V–I graph is a curve and whose resistance differs at different values of \(V\) or \(I\).
- Ohm — the SI unit of resistance; the resistance through which a current of one ampere flows when a potential difference of one volt is applied.
- Ohmic resistors — conductors that obey Ohm’s law, whose V–I graph is a straight line through the origin and whose resistance is constant.
- Parallel combination — an arrangement in which the resistors are joined between the same two points, so that each has the same potential difference across it.
- Potential difference — the work done per unit charge in moving a positive test charge from one point to another; SI unit volt (V).
- Resistance — the obstruction offered by a conductor to the flow of current through it; SI unit ohm (\(\\Omega\)).
- Resistivity — another name for specific resistance; SI unit ohm metre (\(\\Omega\\ \\text{m}\)).
- Safe current — the largest current an appliance can carry at its rated voltage without damage, equal to its power rating divided by its voltage rating; measured in ampere (A).
- Series combination — an arrangement in which the resistors are joined one after the other, so that the same current passes through each of them.
- Specific resistance — the resistance of a wire of a material of unit length and unit area of cross section, a property of the material alone; SI unit ohm metre (\(\\Omega\\ \\text{m}\)).
- Superconductor — a substance of zero resistance, or infinite conductance, at a very low temperature.
- Terminal voltage — the work done per unit charge in carrying a positive test charge around the external circuit connected across a cell; SI unit volt (V).
- Test charge — a charge of known small value, small enough that its presence does not disturb the charges already present in a region.
- Volt — the SI unit of potential and of potential difference; one volt is one joule per coulomb.
- Voltmeter — the instrument used to measure potential difference; it is connected in parallel across two points, and reads in volt (V).
- Watt — the SI unit of electric power; one watt is the power consumed when a current of one ampere flows through a potential difference of one volt, and equals one joule per second.
- Watt-hour — a commercial unit of electrical energy; the energy consumed by an appliance of power one watt used for one hour, equal to \(3600\\ \\text{J}\).
8.24 Points To Ponder
Free electrons drift along a wire far more slowly than a person walks, and yet a lamp lights the instant the switch is closed. If it is not the electrons from the switch that reach the lamp so quickly, what is it that travels along the wire — and how would you test your answer?
The e.m.f. of a cell does not depend on the size of its electrodes or on how much electrolyte it contains. Yet a large cell keeps a torch bright for longer, and its voltage sags less when a heavy current is drawn. Which quantities in this chapter carry the information that the e.m.f. leaves out?
Ohm’s law is not obeyed by everything that conducts. Suppose you are handed an unlabelled wire and asked to decide whether it is ohmic. What measurements would you take, how many, and what would you have to keep constant while taking them — and could any number of readings ever prove the law holds?
The appliances in a house are connected in parallel with one another, never in series. Work out what would go wrong in a series-wired house, considering both how the supply voltage would be shared and what would happen the moment one appliance failed.
Engineers reduce the energy wasted in long transmission lines by sending the power at a very high voltage rather than by making the cables endlessly thicker. Using \(P = VI\) and \(H = I^2Rt\) together, explain what is being traded against what, and why thickening the cable is not the whole answer.
Suppose a material were found that was superconducting at room temperature. Which of the ideas developed in this chapter would simply stop applying to a circuit built from it, and which would survive unchanged? In particular, what would happen to the relation \(V = IR\), and to the terminal voltage of a cell connected across such a wire?
8.25 Objective Questions
Multiple Choice Questions
- The charge on a body that has a deficit of \(2.5 \\times 10^{19}\) electrons is : (1 mark)
- \(-4.0\\ \\text{C}\)
- \(+4.0\\ \\text{C}\)
- \(-2.5\\ \\text{C}\)
- \(+1.6\\ \\text{C}\)
- The potential difference across a conductor is measured by a : (1 mark)
- voltmeter connected in series
- voltmeter connected in parallel
- ammeter connected in series
- ammeter connected in parallel
- One volt is equal to : (1 mark)
- one joule per second
- one coulomb per second
- one joule per coulomb
- one joule coulomb
- The slope of the graph of current \(I\) against potential difference \(V\) for a metallic conductor gives : (1 mark)
- the resistance of the conductor
- the conductance of the conductor
- the specific resistance of the material
- the power consumed in the conductor
- A wire is stretched so that its length becomes three times the original length. Its resistance becomes : (1 mark)
- three times
- six times
- nine times
- one-ninth
- The SI unit of specific resistance is : (1 mark)
- ohm
- ohm metre
- ohm metre\(^{-1}\)
- ohm\(^{-1}\) metre\(^{-1}\)
- Two resistors of \(6\\ \\Omega\) each are joined in parallel. Their equivalent resistance is : (1 mark)
- \(12\\ \\Omega\)
- \(6\\ \\Omega\)
- \(3\\ \\Omega\)
- \(1.5\\ \\Omega\)
- The terminal voltage of a cell in a closed circuit is : (1 mark)
- equal to its e.m.f.
- greater than its e.m.f.
- less than its e.m.f.
- zero
- The electrical energy consumed by an appliance is given by : (1 mark)
- \(VI\)
- \(V^2 / R\)
- \(I^2Rt\)
- \(I^2R\)
- One kilowatt hour is equal to : (1 mark)
- \(3600\\ \\text{J}\)
- \(3.6 \\times 10^5\\ \\text{J}\)
- \(3.6 \\times 10^6\\ \\text{J}\)
- \(10^3\\ \\text{J}\)
- The heat produced in a wire is doubled when : (1 mark)
- the current in it is doubled
- the resistance of the wire is doubled
- the current in it is halved
- the time of flow is halved
- A substance whose resistance becomes zero at a very low temperature is called : (1 mark)
- a good conductor
- a semiconductor
- an insulator
- a superconductor
Assertion And Reason
For each of the following, choose from the options : a. both A and R are true and R is the correct explanation of A b. both A and R are true but R is not the correct explanation of A c. A is false but R is true d. A is true but R is false
Assertion (A) : The equivalent resistance of a parallel combination is less than the smallest resistance in it. Reason (R) : Each additional resistor joined in parallel provides one more path for the current. (1 mark)
Assertion (A) : The filament of a bulb is a non-ohmic resistor. Reason (R) : The resistance of a metal increases when its temperature rises. (1 mark)
Assertion (A) : A fuse wire is made short and thin. Reason (R) : A short and thin wire has a lower melting point than a long thick one. (1 mark)
Match The Following
- Match each quantity in Column A with its SI unit in Column B. (1 mark)
| Column A | Column B |
|---|---|
| a. Specific resistance | i. volt |
| b. Electric power | ii. ohm metre |
| c. Terminal voltage | iii. siemens |
| d. Conductance | iv. watt |
Fill In The Blanks
The current in a conductor is the rate of flow of __________ through it. (1 mark)
The resistance of a wire is directly proportional to its __________ and inversely proportional to its __________. (1 mark)
When a wire is doubled on itself, its resistance becomes __________ of its previous value. (1 mark)
True Or False
Current is a vector quantity. (1 mark)
The e.m.f. of a cell depends on the amount of electrolyte in it. (1 mark)
In a series combination, the potential difference is greatest across the largest resistance. (1 mark)
Name The Following
The instrument that must be connected in series to measure the current in a circuit. (1 mark)
The alloy used for making the heating element of an electric toaster. (1 mark)
The metal used for the filament of an electric bulb. (1 mark)
Give One Word Or Phrase For
The resistance offered by the electrolyte inside a cell to the flow of current. (1 mark)
The resistance of a wire of unit length and unit area of cross section. (1 mark)
State The SI Unit Of
Electric charge. (1 mark)
Electrical energy. (1 mark)
Conductivity. (1 mark)
Odd One Out, And Give A Reason
Copper, nichrome, junction diode, iron. (1 mark)
Manganin, constantan, tungsten, brass. (1 mark)
8.26 Check Your Concepts
Define electric current and state its SI unit. (2 marks)
State the SI unit of potential difference and explain the meaning of the statement ‘the potential difference between two points is 1 volt’. (2 marks)
Give a reason why an ammeter is connected in series in a circuit while a voltmeter is connected in parallel. (2 marks)
Define specific resistance and state its SI unit. (2 marks)
Name two factors on which the e.m.f. of a cell depends, and two on which it does not. (2 marks)
State Ohm’s law, giving the condition under which it holds. (3 marks)
Explain, on the basis of the movement of free electrons, why a metal wire offers resistance to the flow of current through it. (3 marks)
Differentiate between an ohmic and a non-ohmic resistor, stating the basis of the difference, and give one example of each. (3 marks)
Give a reason why the connecting wires in a circuit are made of thick copper, while the fuse wire in the same circuit is made short and thin from an alloy of lead and tin. (3 marks)
Why does the terminal voltage of a cell fall when a heavy current is drawn from it? (3 marks)
An electric bulb is rated ‘60 W, 220 V’. What two quantities can be calculated from this rating, and how? (3 marks)
Differentiate between the e.m.f. and the terminal voltage of a cell, stating the basis of the difference. (4 marks)
Name the four factors on which the resistance of a given wire depends, and state how the resistance is affected by each of them. (4 marks)
What would happen if a blown fuse wire were replaced by a piece of thick copper wire of the same length? Explain your answer using the properties for which a fuse wire is chosen. (4 marks)
State Joule’s law of heating and explain, with a reason, why the coil of an electric heater glows red while the copper leads carrying the same current to it stay cool. (4 marks)
Derive an expression for the equivalent resistance of three resistors \(R\_1\), \(R\_2\) and \(R\_3\) connected in series across a source of potential difference \(V\). State the two facts about a series combination that your derivation uses. (5 marks)
Derive an expression for the equivalent resistance of three resistors joined in parallel, and hence explain why the equivalent resistance of a parallel combination is less than the smallest resistance in it. (5 marks)
A cell of e.m.f. \(\\varepsilon\) and internal resistance \(r\) sends a current through an external resistance \(R\). Obtain expressions for the current in the circuit, the terminal voltage of the cell, and the voltage drop inside it, explaining each step. (5 marks)
8.27 Diagrams, Graphs & Constructions
Draw a neat labelled circuit diagram of the arrangement used to verify Ohm’s law, containing a battery, a key, a rheostat, an ammeter, a voltmeter and an unknown resistance wire. Mark the direction of the current, and show clearly which meter is in series and which is in parallel. (4 marks)
Look at the four graphs of Fig. 8.7-2. Identify which of them represent ohmic resistors and which represent non-ohmic resistors, and give a reason for your answer in each case. (3 marks)
Sketch, on the same pair of axes, the V–I graphs of one conductor at two different temperatures \(T\_1\) and \(T\_2\), where \(T\_1 > T\_2\). Label the axes with their units and state which line corresponds to the higher temperature, with a reason. (3 marks)
Fig. 8.5-1 shows the I–V graph of a metallic conductor. Describe, step by step, how you would obtain the resistance of the conductor from this graph, naming the quantity the gradient represents. (3 marks)
Draw a circuit diagram showing a \(2\\ \\Omega\) resistor and a \(3\\ \\Omega\) resistor joined in parallel, this combination being in series with a \(4\\ \\Omega\) resistor and a cell. Place an ammeter so as to read the total current drawn from the cell, and a voltmeter so as to read the potential difference across the parallel combination. (4 marks)
Draw a labelled diagram to represent a cell of e.m.f. \(\\varepsilon\) and internal resistance \(r\) connected to an external resistance \(R\). Mark on it the current \(I\), the terminal voltage \(V\) and the voltage drop \(v\) inside the cell. (3 marks)
Sketch the shape of the graph you would expect for the resistance of a superconducting metal plotted against its temperature, labelling both axes and marking the region in which the substance is superconducting. (3 marks)
8.28 Practice With Numericals
A charge of \(90\\ \\text{C}\) passes through a conductor in \(1.5\) minutes. Calculate the current in the conductor. (2 marks)
A current of \(0.25\\ \\text{A}\) flows through a resistor of \(40\\ \\Omega\). Calculate the potential difference across its ends. (2 marks)
A wire of resistance \(3\\ \\Omega\) is stretched uniformly until its length becomes three times the original length. Calculate its new resistance. (2 marks)
A current of \(0.8\\ \\text{A}\) flows in a wire for \(5\\ \\text{s}\). Calculate the charge that passes through it and the number of electrons this represents. Take \(e = 1.6 \\times 10^{-19}\\ \\text{C}\). (3 marks)
A nichrome wire has a length of \(1.5\\ \\text{m}\) and an area of cross section \(3.0 \\times 10^{-7}\\ \\text{m}^2\). Taking the specific resistance of nichrome as \(100 \\times 10^{-8}\\ \\Omega\\ \\text{m}\), calculate its resistance. (3 marks)
An electric appliance is rated \(2200\\ \\text{W} - 220\\ \\text{V}\). Calculate the resistance of its element while in use and the safe current through it. (3 marks)
A lamp of power \(60\\ \\text{W}\) is used for \(5\) hours each day for \(30\) days. Calculate the electrical energy consumed in kWh and its cost at ₹ 4.50 per kWh. (3 marks)
A resistor of \(4\\ \\Omega\) and a resistor of \(6\\ \\Omega\) are joined in parallel across a battery of potential difference \(12\\ \\text{V}\) and negligible internal resistance. Calculate the equivalent resistance, the total current drawn from the battery, and the current in each resistor. (4 marks)
A cell of e.m.f. \(2.0\\ \\text{V}\) and internal resistance \(0.5\\ \\Omega\) is connected to an external resistance of \(4.5\\ \\Omega\). Calculate the current in the circuit, the terminal voltage of the cell and the voltage drop inside it. (4 marks)
A current of \(2.5\\ \\text{A}\) flows through a resistor of \(16\\ \\Omega\) for \(2\) minutes. Calculate the heat produced, in joule and in calorie. (4 marks)
A resistor of \(3\\ \\Omega\) is joined in series with a parallel combination of a \(6\\ \\Omega\) and a \(3\\ \\Omega\) resistor, and the arrangement is connected to a supply of \(10\\ \\text{V}\) of negligible internal resistance. Calculate the equivalent resistance of the arrangement, the current drawn from the supply, and the potential difference across the parallel combination. (4 marks)
A resistance wire has a resistance of \(2\\ \\Omega\) per metre. Calculate the length of this wire needed to make a resistance of \(8\\ \\Omega\). If a potential difference of \(12\\ \\text{V}\) is then applied across this length, calculate the current in it and the heat produced in it in one minute. (5 marks)
A house uses a heater of \(1.5\\ \\text{kW}\) for \(2\) hours a day, three lamps of \(60\\ \\text{W}\) each for \(6\) hours a day, and a fan of \(75\\ \\text{W}\) for \(8\) hours a day. Calculate the total electrical energy consumed in \(30\) days and its cost at ₹ 5.00 per kWh. (5 marks)