Chapter 12: Electromagnetic Induction
In the year 1831, at the Royal Institution in London, Michael Faraday sat with a soft iron ring on his bench. Around one side of the ring he had wound a coil of insulated wire and connected it to a battery through a switch. Around the opposite side he had wound a second, completely separate coil, joined to a galvanometer — an instrument that detects small currents. The two coils never touched each other. No wire ran from one circuit to the other.
Faraday closed the switch. The galvanometer needle jerked, and then settled back to zero, even though the battery kept driving a steady current through the first coil. He opened the switch. The needle jerked again — this time in the opposite direction.
That brief twitch of a needle was the whole discovery. Electricity could be produced from magnetism, but only while something was changing. A steady magnetic field gave nothing; a changing one gave a push to the electrons in a distant, unconnected wire.
Almost every generating station, transformer, induction cooktop and electric motor braking system in the world today rests on what that needle did in a few seconds. This chapter is the study of that moment.
Diagram
Fig. 12.0 – Faraday’s soft iron ring with two separate coils wound on opposite sides, one connected to a battery and switch, the other to a galvanometer with its needle deflecting.
By the end of this chapter, you will be able to answer each of these questions for yourself.
12.1 Introduction
The Question That Started This Chapter
So far in your study of physics, the two halves of electromagnetism have been kept separate. Charges that sit still produce electric fields around them. Charges that move — that is, an electric current — produce magnetic fields around them.
During the early decades of the nineteenth century, Oersted, Ampere and a few other scientists established firmly that electricity and magnetism are not two unrelated subjects at all. They are inter-related. A current in a wire can deflect a compass needle; a magnetic field can push on a current-carrying wire.
Once that link was established, a natural question arose. If moving electric charges produce magnetic fields, can the converse effect also happen? That is, can we produce an electric current by moving magnets?
This is not an obvious question to answer. Nature does not have to be symmetric. The only way to find out was to go into the laboratory and try.
Faraday and Henry: The Year 1831
In 1831, Faraday in England performed a series of experiments in connection with the generation of electric current by means of magnetic flux. In the same year, Joseph Henry (1799–1878) demonstrated that electric currents were indeed produced in closed circuits or coils when those circuits were subjected to time-varying magnetic fields — that is, magnetic fields whose strength at the coil changes from instant to instant.
The outcome of these experiments led to a very basic and important law of electromagnetism, known as Faraday’s law of induction. Stated in the simplest way, it says that an electromotive force (emf) — and therefore a current, if the circuit is closed — can be induced in any process that involves a change in magnetic flux.
Two words in that sentence need to be clear in your mind from the very beginning, because the whole chapter is built on them.
Electromotive force (emf) is not a force at all, in spite of its name. It is the work done in driving a unit positive charge once around a circuit, and it is measured in volt (V). A battery supplies emf by chemical action; here we are asking whether magnetism can supply emf instead.
Magnetic flux is a measure of how much magnetic field passes through a given area — you can picture it as the number of magnetic field lines threading through a loop. It depends on the strength of the field, on the size of the area, and on the tilt of the area with respect to the field. Its SI unit is the weber (Wb). Flux is treated carefully and formally in Section 12.4; for the moment this working picture is enough.
The phenomenon itself has a name that will be used throughout the chapter.
What Faraday Actually Observed
The experimental observations of Faraday are summarised below. Read them as a set: each one changes just one thing in the experiment, so that the effect of that one change can be seen clearly. This is exactly how good experimental physics is done.
Consider a closed circuit consisting of a coil connected to a galvanometer (G), a sensitive instrument that shows the presence and direction of a small current, as shown in Fig. 12.1.
Diagram
Fig. 12.1 – A bar magnet approaching a closed circuit consisting of a coil and galvanometer (G).
- When a magnet approaches a closed circuit consisting of a coil (Fig. 12.1), it produces a current in it. This current is known as induced current.
When the magnet is taken away from the closed circuit, a current is again produced, but in the opposite direction with respect to that in experiment (i). So approaching and receding give opposite effects.
If, instead of the magnet, the coil is moved towards the magnet or away from it, an induced current is produced in the coil (that is, in the closed circuit) just the same. What matters is the relative motion between the two, not which of the two you actually push.
If the polarity of the approaching or receding magnet is changed — the south pole is presented instead of the north pole — the direction of induced current in the coil is also changed.
The magnitude of the induced current depends on the relative speed of the coil with respect to the magnet. It also depends upon the number of turns in the coil. A faster movement, or a coil with more turns, gives a larger deflection of the galvanometer.
The induced current exists only so long as there is relative motion between the coil and the magnet. Stop the motion, and however close the magnet is to the coil, the galvanometer needle returns to zero.
Observation (vi) is the decisive one. A strong magnet held motionless inside the coil produces a large flux through the coil, but no current at all. So it is not the flux that produces the current — it is the changing of the flux.
The Same Effect Without Any Magnet
- Instead of a magnet and a closed circuit, two coils with their planes facing each other, as shown in Fig. 12.2, also produce similar effects to those described in experiments (i) to (vi).
Diagram
Fig. 12.2 – Two coils with their planes facing each other; the left coil connected in series with a battery, rheostat and key, the right coil connected to a galvanometer (G).
One coil is connected in series with a battery, a rheostat (a variable resistance, used here to change the current smoothly) and a key. The ends of the other coil are connected to a galvanometer (G). The coil which contains a source of emf (the battery) is termed the primary coil, while the other is called the secondary coil.
Notice what has been removed from the experiment: there is now no magnet, and nothing needs to move at all. The primary coil carrying a current is itself a source of magnetic field, and that is all the secondary coil requires.
With these two coils, the following observations are made.
When the circuit in the primary coil is closed or broken, a momentary deflection is produced in the galvanometer at the time of make or break. When the circuit is closed and when it is broken, the directions of deflection in the galvanometer are opposite to each other.
When there is a relative motion between the two coils (with their circuits closed), an induced current is produced in the secondary coil, but it exists only so long as there is relative motion between the coils.
Whenever the current in the primary coil is changed — either increased or decreased — by sliding the rheostat-jockey, a deflection is produced in the galvanometer. This indicates the presence of induced current. The induced current exists only so long as there is a change of current in the primary coil.
The Common Thread
The above observations, taken together, indicate one single conclusion: so long as there is a change of magnetic flux inside a coil — produced either by means of a magnet or by a current-carrying coil — an induced emf is produced. The source of the flux is irrelevant. The way in which the flux is made to change is irrelevant.
Further, the direction of the induced emf reverses if, instead of increasing the flux, the flux is decreased, or vice versa. This is why every pair of observations above — approach and retreat, make and break — gave deflections in opposite senses.
In exam questions based on this section, you will most often be asked to decide whether an emf is induced at all in a described situation. The test is always the same: ask whether the flux linked with the circuit is changing at that instant. A magnet at rest inside a coil, a coil moving parallel to a uniform field without changing the area it presents, or a steady current in a nearby primary coil — none of these induces an emf, however strong the field may be.
The next section puts this conclusion into a quantitative form, as Faraday’s two laws of electromagnetic induction.
12.2 Faraday’s Laws of Electromagnetic Induction
On the basis of the experimental evidence described in the previous section, Faraday enunciated the following laws concerning electromagnetic induction. The first tells us when an emf appears; the second tells us how large it is.
The First Law
Whenever there is a change of magnetic flux in a closed circuit, an induced emf is produced in the circuit. Also, if a conductor cuts the lines of magnetic field, an emf is induced between its ends.
Notice that the law is stated in two parts, because there are two situations that look different to the eye. In the first, a complete circuit sits somewhere and the flux through it changes. In the second, a single conductor — not necessarily part of a closed loop — sweeps across magnetic field lines, and an emf appears between its two ends even though no current can flow.
This law is a qualitative law, as it only indicates the characteristics of the induced emf. It tells us that an emf exists and what causes it, but it gives us no number.
The Second Law
The magnitude of induced emf produced in the circuit is directly proportional to the rate of change of magnetic flux linked with the circuit. This law is known as the quantitative law, as it gives the magnitude of the induced emf.
The phrase “rate of change” is the heart of it. It is not the amount by which the flux changes that decides the emf, nor how long the change lasts, but the ratio of the two — how fast the change happens. A small change squeezed into a very short time produces a large emf; the same change spread over a long time produces a small one.
If \(\Phi\) is the magnetic flux linked with the coil at any instant \(t\), then the induced emf
\[e \propto \frac{d\Phi}{dt}\]
— (12.1)
\[e = K \frac{d\Phi}{dt}\]
Here \(K\) is a constant of proportionality. If \(e\), \(\Phi\) and \(t\) are measured in the same system of units, then \(K = 1\).
\[e = \frac{d\Phi}{dt}\]
— (12.2)
This is a point worth pausing over, because students often wonder where the constant disappeared to. It has not been ignored; the SI units of emf, flux and time have been defined in such a way that \(K\) works out to exactly 1. In other words, one weber of flux vanishing uniformly in one second produces exactly one volt of emf.
If we combine this expression with Lenz’s law — which is the subject of the next section and which fixes the direction of the induced emf — we get
\[e = -\frac{d\Phi}{dt}\]
— (12.3)
The minus sign carries no information about magnitude at all. Its entire job is to record the fact that the induced emf opposes the change of flux that produced it. Section 12.3.3 examines this sign in detail.
Coils with Many Turns
A practical coil is not one loop but many. If \(\Phi'\) is the flux associated with a single turn, then the total magnetic flux \(\Phi\) for a coil consisting of \(n\) turns is
\[\Phi = n\Phi'\]
Each turn is threaded by the same changing flux, and the turns are joined end to end, so their individual emfs add up in series. Therefore
\[e = -n\frac{d\Phi_B}{dt}\]
— (12.4)
This is also known as the ‘flux rule’, according to which the emf is equal to the rate at which the magnetic flux through a conducting circuit is changing.
In the symbols above: - \(e\) is the induced emf, measured in volt (V) - \(\Phi\) (or \(\Phi_B\)) is the magnetic flux linked with the circuit, measured in weber (Wb) - \(n\) is the number of turns in the coil, a pure number with no unit - \(t\) is time, measured in second (s)
In SI units, \(e\) is measured in volt and \(\dfrac{d\Phi}{dt}\) is measured in weber/second. One volt is therefore the same thing as one weber per second.
A Symmetry Worth Noticing
We have already learnt, while studying the magnetic effect of current, that charges in motion — that is, a current — can exert a force or a torque on a stationary magnet, such as a compass needle. That was the discovery that opened up electromagnetism.
Now, in Faraday’s experiments, we have observed the mirror image of that statement. A bar magnet in motion, or more generally a time-varying magnetic field, can exert a force on the stationary charges inside a conductor. That force pushes the free electrons along the conductor.
The consequence depends on whether the conductor forms a complete path. If the conductor is an open circuit, the pushed charges pile up at the ends and an induced emf appears across those ends. If the conductor forms a closed circuit, the charges keep moving all the way round, and we observe an induced current.
Nature, it turns out, is symmetric here after all: currents act on magnets, and moving magnets act on charges. The two halves of electromagnetism are two views of one subject.
12.3 Lenz’s Law
H. F. E. Lenz (1804–1864), without any knowledge of the work of Michael Faraday and Joseph Henry, duplicated many of their discoveries independently and almost at the same time.
Faraday’s laws tell us how large an induced emf is, but they do not by themselves tell us which way round the induced current flows. For determining the direction of an induced current in a loop, Lenz devised a rule, which goes by his name as Lenz’s law.
According to this rule, the direction of the induced current in a circuit is such that the magnetic field produced by the induced current opposes the change in the magnetic flux that induces the current. The direction of the induced emf is the same as that of the induced current.
In short, the induced emf tends to set up a current the action of which opposes the change that causes it.
Read that statement carefully, because it is easy to misremember. The induced current does not oppose the magnetic flux, and it does not oppose the magnet. It opposes the change in the flux. If the flux through a loop is increasing, the induced current fights the increase; if the same flux is decreasing, the induced current fights the decrease and therefore tries to maintain the flux. The same loop, in the same field, will carry current in opposite directions in these two cases.
Applications of Lenz’s Law
Lenz’s law becomes much clearer when it is applied to concrete situations rather than argued in the abstract. In the sections that follow, we look at a magnet moving towards a loop, at the deep reason behind the law in the conservation of energy, and finally at how the law is folded into Faraday’s equation as a single minus sign.
12.3.1 Motion of a Magnet Toward a Loop
In order to get a feel for Lenz’s law, let us consider the north pole of a magnet moving toward a conducting loop, as shown in Fig. 12.3.
Diagram
Fig. 12.3 – A bar magnet with its north pole moving towards a conducting loop; the induced current in the loop shown with its equivalent magnetic dipole.
As the magnet is moved toward the loop, the magnetic flux through the loop increases, and a current is induced in the loop. This induced current, circulating around the loop, produces a magnetic dipole — that is, the current-carrying loop behaves exactly like a small bar magnet, with a north face on one side and a south face on the other.
The dipole is oriented in such a way that it opposes the motion of the magnet. Thus the loop’s north pole must face the approaching north pole of the magnet, so as to repel it.
There is a small chain of reasoning hidden in that sentence, and it is worth making explicit. The loop must oppose the increase of flux. To oppose an increase, it must produce a field inside itself pointing opposite to the magnet’s field. A loop whose field points back towards the magnet is a loop whose north face is turned towards the magnet — and two north poles facing each other repel. Every step follows from Lenz’s law.
Finding Which Way the Current Goes
Knowing which face of the loop is north is not yet the same as knowing which way the current circulates. For that we use the curled right-hand (RH) rule for a magnetic dipole or magnetic field: curl the fingers of the right hand in the sense of the current around the loop, and the extended thumb points along the loop’s magnetic field, that is, out of its north face.
Applied here, the rule works in reverse. We already know the north face must point towards the approaching magnet, so the thumb must point that way, and the curled fingers then give the sense of the current. As seen in Fig. 12.3, the induced current in the loop will be in the counter-clockwise direction.
Now consider the opposite experiment. If the magnet is pulled away from the loop, the flux through the loop is decreasing, and the loop opposes the decrease. A current will again be induced in the loop, in such a way that the loop will have a south pole facing the retreating north pole, and will oppose the retreat by attracting it. The induced current in the loop will now flow in the clockwise direction.
In numerical and reasoning questions on this subsection, you will usually be given a situation and asked for the direction of the induced current, often as “clockwise or anticlockwise as seen from the magnet”. Work through it in the same fixed order every time: first decide whether the flux is increasing or decreasing, then decide which face of the loop must be north, then apply the right-hand rule to read off the sense of the current.
12.3.2 Energy Conservation in Lenz’s Law
We have learnt that the cause of an induced current may be either
- the motion of a conductor (wire) in a magnetic field, or
- the change of magnetic flux through a stationary circuit.
These two look quite different, but Lenz’s law handles both, and it is worth stating separately what “opposition” means in each case.
Case 1: A Conductor Moving in a Field
In the first case, the direction of the induced current in the moving conductor (wire) is such that the direction of the thrust exerted on the conductor by the magnetic field is opposite to the direction of its motion, and thus opposes the motion of the conductor. This is illustrated in Fig. 12.3 (c).
Diagram
Fig. 12.3 (c) – A straight conductor moving to the right through a magnetic field, with the induced current marked and the magnetic thrust on it shown pointing to the left, opposite to the motion.
The chain here is a familiar one. The moving wire carries an induced current; a current-carrying wire in a magnetic field experiences a force; and Lenz’s law tells us that this force must point backwards along the motion. The wire therefore behaves as though it were moving through something sticky. If you want it to keep moving at a steady speed, you must keep pushing.
Case 2: A Stationary Circuit in a Changing Field
In the second case, the induced current sets up a magnetic field of its own, which within the area bounded by the circuit is
- opposite to the original magnetic field if this field is increasing, but
- in the same direction as the original field if the field is decreasing.
Thus it is the ‘change in flux’ through the circuit — not the flux itself — which is opposed by the induced current. Part (b) is the half that students most often get wrong: when the applied field is dying away, the induced current does not fight the field, it reinforces it, because what it is fighting is the loss.
Why the Law Has to Be This Way
Lenz’s law follows directly from the conservation of energy. It is not an extra experimental fact bolted onto Faraday’s law; it is forced on us.
Suppose, for a moment, that the law worked the other way round. If an induced current flowed in a circuit in such a direction that it helped the cause that produced it, then we would soon find that the induced current and the magnetic flux penetrating the loop would lead to an infinite growth. A little current would make a little extra flux, the extra flux would induce more current, and so on without limit.
The induced current, once started flowing in the loop, would keep increasing indefinitely, producing joule heating at no extra cost, and would thus be self-sustaining — a perpetual motion machine. This will clearly violate the law of conservation of energy, since heat would be pouring out of the circuit with nothing being supplied to it.
We thus see that Lenz’s law is a necessary consequence of the law of conservation of energy. The opposing sense of the induced current is one manifestation of a general statement of Lenz’s law:
Seen this way, the “opposition” is simply nature’s book-keeping. Whatever electrical energy appears in the circuit has to be paid for — by the person pushing the magnet, by the engine turning the generator, by whatever agency is causing the flux to change.
Open Circuits: A Useful Trick
In order to have an induced current, we must have a closed circuit. But an emf can appear across a conductor that is not part of any loop, and we still need a way to work out which end is positive.
If a conductor is not forming a closed circuit, we mentally construct a circuit between the two ends of the conductor or wire, and use Lenz’s law to determine the direction of induced current in this imagined loop. Then the polarity of the ends of the open-circuited conductor can be found easily: the end from which the imagined current would emerge into the external path is the positive terminal, exactly as with the terminals of a cell.
The imagined loop is only a scaffolding for the reasoning. Once the polarity is known, it can be taken away.
12.3.3 Lenz’s Law and Faraday’s Law
We now have two separate statements: Faraday’s law, which fixes the size of the induced emf, and Lenz’s law, which fixes its direction. In this subsection we see that the two are not really separate at all — the direction has already been written into Faraday’s law, in the form of a single minus sign.
Consider Faraday’s law with special attention to the negative sign:
\[e = -\frac{d\Phi}{dt}\]
To see what the sign is doing, we need a definite situation with definite directions.
Setting Up a Definite Case
Consider that the area vector \(\vec{A}\) of the loop — the vector drawn perpendicular to the plane of the loop, with magnitude equal to the area — is fixed in position, and is oriented parallel (\(\theta = 0\)) to the magnetic field \(\vec{B}\). Let the magnetic field \(\vec{B}\) increase with time. This arrangement is shown in Fig. 12.3 (d).
Diagram
Fig. 12.3 (d) – A flat loop with its area vector A drawn perpendicular to its plane and parallel to the magnetic field B, with the chosen positive sense of circulation marked around the loop.
Using the definition of flux, \(\Phi = \vec{B} \cdot \vec{A}\), Faraday’s law can be written as
\[e = -\frac{d}{dt}(BA) = -|\vec{A}|\frac{dB}{dt}\]
— (12.5)
The area has been taken outside the derivative because the loop is rigid and fixed, so only \(B\) is changing with time.
Now look at the right-hand side of Eq. (12.5). The quantity \(|\vec{A}|\) is positive, since it is the magnitude of a vector, and \(\dfrac{dB}{dt}\) is positive because \(B\) is increasing with time. With the minus sign in front, the right-hand side is a negative quantity.
Reading the Sign on the Left-Hand Side
For the equation to balance, the left-hand side must also be negative. But what does a “negative emf” mean physically? It means an emf directed opposite to whichever sense of circulation around the loop we have agreed to call positive — so we must first fix that agreement.
The screwdriver rule (the right-handed screw rule) fixes the positive sense of circulation around the loop: turning a right-handed screw in that sense should drive it forward along \(\vec{A}\). For the arrangement of Fig. 12.3 (d), this makes the clockwise direction the positive sense of circulation.
Meanwhile, Lenz’s law tells us independently what the loop actually does. The flux through the loop is increasing, so the induced current must oppose the increase, and its sense in this arrangement is counter-clockwise — that is, negative on the convention just fixed. Since the induced emf acts in the same sense as the induced current, the induced emf is negative too. The left-hand side of Eq. (12.5) is indeed a negative quantity, exactly as required to equal the right-hand side.
Thus the negative sign in the equation
\[e = -\frac{d\Phi}{dt}\]
incorporates Lenz’s law into Faraday’s law. One compact equation now carries both pieces of information: its magnitude gives the size of the induced emf, and its sign gives the direction.
In practice, when solving numerical problems you will almost always be asked only for the magnitude of the induced emf, and you may safely work with \(|e| = \left|\dfrac{d\Phi}{dt}\right|\). Bring the minus sign back whenever the question asks for a direction, a polarity, or an argument about energy.
12.4 Flux of the Field
The concept of flux of the magnetic field is vital to our understanding of Faraday’s law. So far we have used the word loosely, as “the amount of field passing through a loop”. In this section we make it precise, because every quantitative statement in the rest of the chapter depends on it.
Giving a Direction to an Area
An area is a flat patch; it has size, but at first sight it has no direction. To speak of the “tilt” of an area we must first agree on how to give it one.
As shown in Fig. 12.4 (a), consider a small element of area \(d\vec{a}\). A direction is assigned to this element of area such that, if the curve bounding the area is traversed in the direction of the arrow, then the normal comes out of the plane of the paper towards the reader. In other words, it is the direction in which a right-handed screw will move if rotated in the sense of the arrow on the curve.
Diagram
Fig. 12.4 (a) – A small element of area d(a) bounded by a curve traversed in the anticlockwise sense, with the normal vector coming out of the plane of the paper (right-handed screw rule).
This is the same right-handed screw rule already used in Section 12.3.3 to fix the positive sense of circulation around a loop.
Flux Through a Small Element
Suppose the element of area \(d\vec{a}\) is situated in a magnetic field \(\vec{B}\). Then the scalar quantity
\[d\Phi = \vec{B} \cdot d\vec{a} = |\vec{B}|\,|d\vec{a}|\cos\theta\]
— (12.6)
is called the flux of \(\vec{B}\) through the area \(d\vec{a}\), where \(\theta\) is the angle between the direction of the magnetic field \(\vec{B}\) and the direction assigned to the area \(d\vec{a}\).
In this equation: - \(d\Phi\) is the magnetic flux through the small area, measured in weber (Wb) - \(|\vec{B}|\) is the magnitude of the magnetic field, measured in tesla (T) - \(|d\vec{a}|\) is the magnitude of the small area, measured in square metre (m²) - \(\theta\) is the angle between \(\vec{B}\) and the normal to the area
Note carefully that the flux is a scalar, even though it is built out of two vectors. It has magnitude and sign but no direction of its own. Its sign depends on the sense chosen for the area vector, which is why the convention above had to come first.
Flux Through a Finite Surface
This result can be generalised to define the flux over a finite area \(\vec{S}\). It should be remembered that the magnetic field \(\vec{B}\) will not be the same at different points within the finite area — it may be stronger at one corner than another, and it may point in slightly different directions at different points.
Therefore the area is divided into small sections of area \(d\vec{a}\), small enough that \(\vec{B}\) can be treated as constant over each one. The flux over each section is calculated using Eq. (12.6), and these contributions are then added up by integrating over the entire area, as shown in Fig. 12.4 (b).
Diagram
Fig. 12.4 (b) – A finite curved surface S bounded by a contour C, divided into small elements of area d(a), each with its own normal and its own value of the magnetic field.
Thus, the flux passing through \(S\) is
\[\Phi = \int_S \vec{B} \cdot d\vec{a}\]
— (12.7)
We cannot take \(\vec{B}\) out of the integral in Eq. (12.7) unless \(\vec{B}\) is the same everywhere in \(\vec{S}\). When the field is uniform, however, this is exactly what we do, and the integral collapses to the familiar \(\Phi = BA\cos\theta\) used in most school-level problems.
If the magnetic field at every point changes with time as well, then the flux \(\Phi\) will also change with time:
\[\Phi(t) = \int_S \vec{B}(t) \cdot d\vec{a}\]
— (12.8)
Faraday’s Law in Its Full Form
Faraday’s discovery was that the rate of change of flux, \(\left(\dfrac{d\Phi}{dt}\right)\), is related to the work done to take a unit positive charge around the contour \(C\) shown in Fig. 12.4 (b) in the ‘reverse’ direction. This work done per unit charge is just the emf.
Accordingly, Faraday’s law states that the induced emf can be written as
\[e = -\frac{d\Phi}{dt} = -\frac{d}{dt}\int_S \vec{B}(t) \cdot d\vec{a}\]
— (12.9)
In SI units, the emf \(e\) will be in volt, the flux \(\Phi\) in weber, and time \(t\) in second.
The Two Ways Flux Can Change
From Eq. (12.9) we can see that even if \(\vec{B}\) does not change with time, the flux may still vary if the area \(S\) changes with time. Stretching a loop, sliding one side of it, rotating it in a steady field — all of these change the integral without any change in the field itself.
So there are two independent routes to an induced emf: change the field while holding the circuit still, or change the circuit while holding the field steady. Faraday’s experiments in Section 12.1 used both, and the next section takes up the second route in detail, under the name motional emf.
12.5 Motional Electromotive Force
In the previous section we saw that flux can change in two ways: the field can change, or the circuit can change. This section deals entirely with the second way. The magnetic field will be perfectly steady throughout; everything happens because a conductor moves.
Translational Motion of a Conductor
As shown in Fig. 12.5, a C-shaped frame of wires ABCD of area \((l \times x)\) is situated in a constant magnetic field \(\vec{B}\). BC is a conducting wire that slides on the frame parallel to AD.
Diagram
Fig. 12.5 – A C-shaped frame of wire ABCD in a magnetic field \(\vec{B}\) directed into the plane, with the wire BC of length \(l\) sliding along the x-axis with velocity \(v\).
As the wire BC of length \(l\) is moved out with velocity \(\vec{v}\) so as to increase \(x\), the area of the loop ABCD increases. Since the field is the same everywhere and perpendicular to the loop, more field lines are now enclosed, and thus the flux of \(\vec{B}\) through the loop increases with time.
According to the ‘flux rule’, the induced emf will be equal to the rate at which the magnetic flux through a conducting circuit is changing, as stated in Eq. (12.9). The induced emf will cause a current in the loop.
It is assumed that there is enough resistance in the wire so that the induced currents are very small, producing a negligible magnetic field of their own. This assumption keeps the problem simple: the only magnetic field we need to consider is the external one.
Working Out the Emf from the Flux Rule
As the flux \(\Phi\) through the frame ABCD is \(Blx\) — the product of the field and the enclosed area \(lx\) — the magnitude of the induced emf can be written as
\[|e| = \frac{d\Phi}{dt} = \frac{d}{dt}(Blx) = Bl\frac{dx}{dt} = Blv\]
— (12.10)
where \(v\) is the velocity of the wire BC increasing the length \(x\) of the wires AB and CD. Here \(B\) is in tesla (T), \(l\) in metre (m), \(v\) in metre per second (m s⁻¹), and \(e\) comes out in volt (V).
Note which quantities were treated as constants when differentiating. The field \(B\) is constant, and the length \(l\) of the sliding wire is fixed; only \(x\) grows with time, and its rate of growth is exactly the speed of the wire.
The Same Result from the Force on the Charges
Equation (12.10) came from a rule about flux. But flux is a mathematical construction, and a rule alone does not explain what physically pushes the charges. We can now understand the above result from the magnetic forces on the charges in the moving wire BC.
A charge \(q\) which is carried along by the moving wire BC experiences the Lorentz force
\[\vec{F} = q(\vec{v} \times \vec{B})\]
This force is perpendicular to both \(\vec{v}\) and \(\vec{B}\), and hence is parallel to the wire BC. That is the crucial geometrical fact: the force on the charges points along the wire, so it can drive them from one end of BC to the other.
The force \(\vec{F}\) is constant along the length \(l\) of the wire BC, since \(v\) and \(B\) are both constant, and it is zero elsewhere, because \(v = 0\) for the stationary part CDAB of the wire frame. The moving wire alone acts as the seat of the emf; the rest of the loop is merely a conducting path, exactly like the wires connecting a battery to a bulb.
When the charge \(q\) moves a distance \(l\) along the wire, the work done by the Lorentz force is
\[W = F \cdot l = qvB\sin\theta \cdot l\]
where \(\theta\) is the angle between \(\vec{B}\) and \(\vec{v}\).
The emf generated is the work done per unit charge, that is,
\[e = \frac{W}{q} = vB l \sin\theta\]
— (12.11)
For maximum induced emf, \(\sin\theta = 1\), which happens when the velocity is perpendicular to the magnetic field:
\[e_{max} = Blv\]
— (12.12)
which is the same result as obtained in Eq. (12.10), derived from the rate of change of flux. Two completely different lines of argument — one about flux through an area, the other about forces on individual charges — have led to the same expression. This agreement is not an accident.
In general, it can be proved that for any circuit whose parts move in a fixed magnetic field, the induced emf is the time derivative of flux (\(\Phi\)), regardless of the shape of the circuit.
Why the Flux Rule Also Works for a Stationary Circuit
The flux rule is also applicable in the case of a wire loop that is kept stationary while the magnetic field is changed. But the explanation just given cannot be the reason in that case, because if the wire does not move, then \(v = 0\) and the magnetic part of the Lorentz force vanishes.
The full Lorentz force on the electrical charges is given by
\[\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})\]
There are no new special forces due to changing magnetic fields. Any force on charges at rest in a stationary wire comes from the \(\vec{E}\)-term. If the charges in a stationary loop are being pushed around the loop, an electric field must be present to push them.
Faraday’s observations led to the discovery that electric and magnetic fields are related by a new law: in a region where the magnetic field is changing with time, electric fields are generated. It is this electric field which drives the electrons around the conductor circuit, and as such is responsible for the induced emf in a stationary circuit whenever there is a changing magnetic flux.
This is a striking result, and worth stating plainly. The electric field produced in this way does not need any charges to create it. A changing magnetic field alone is enough, and the electric field it produces exists whether or not a wire happens to be there to reveal it.
Motional Emf in a Rotating Bar
A rotating bar is different in nature from the sliding bar. In the sliding bar, every point moves with the same velocity. In a rotating bar, the end near the pivot barely moves at all, while the far end sweeps round quickly, so no single value of \(v\) can be substituted into \(e = Blv\).
The way out is to divide the bar into pieces small enough that each piece has a definite speed. As shown in Fig. 12.6, consider a small segment \(dr\) of the bar at a distance \(r\) from the pivot. It is a short length \(dr\) of the conductor which is moving with velocity \(\vec{v}\) in the magnetic field \(\vec{B}\), and it has an induced emf generated in it just like a sliding bar.
Diagram
Fig. 12.6 – A conducting bar rotating about a pivot at one end in a uniform magnetic field perpendicular to the plane of rotation, with a small segment \(dr\) marked at a distance \(r\) from the pivot.
By imagining all such segments as sources of emf, we can see that all these segments are joined end to end, that is, they are in series, and therefore the emfs of the individual segments will be added.
Now we know that the induced emf \(de\) in the small segment \(dr\) of the rotating conductor is
\[de = Bv\,dr\]
The total induced emf in the rotating rod is obtained by integrating this from the pivot to the far end. Using \(v = \omega r\) for a point at distance \(r\) from the pivot, where \(\omega\) is the angular velocity of the bar in radian per second:
\[e = \int de = \int B v \, dr = \int_0^l B\omega r \, dr = B\omega \frac{l^2}{2}\]
\[e = \frac{1}{2}B\omega l^2\]
— (12.13)
Compare the above result with the induced emf in a sliding bar, \(e = Blv\).
In examination problems, the commonest error is to use the tip speed \(\omega l\) in place of the average speed, which gives an answer exactly twice too large. Whenever a rod, spoke, disc radius or aircraft propeller blade rotates about one end, the factor of one-half belongs in the answer.
12.6 Induced emf in a Stationary Coil in a Changing Magnetic Field
In the previous section the field stood still and the conductor moved. We now turn the arrangement around: the coil stays exactly where it is, and the magnetic field at the coil is made to change. The experiment described here is one that can actually be assembled in a school or college laboratory, and it shows the whole of Faraday’s law at work in a single swinging motion.
The Magnet–Coil System
As shown in Fig. 12.7 (a), in a magnet-coil system a permanent bar magnet is mounted on an arc of a semicircle of radius 50 cm. The arc is a part of a rigid frame of aluminium and is suspended at the centre of the arc, so that the whole system can oscillate freely in its plane, like a pendulum.
A coil of about 10,000 turns of copper wire loops the arc, so that the bar magnet can pass through the coil freely as the frame swings. The coil itself never moves.
Diagram
Fig. 12.7 (a) – Magnet–coil system: a bar magnet mounted on a semicircular aluminium arc suspended at its centre, oscillating through a fixed multi-turn coil placed at the mean position.
When the magnet moves through the coil, the magnetic flux through the coil changes. That is the whole source of the emf: a stationary coil sitting in a field that grows and shrinks as the magnet sweeps past.
Aluminium is chosen for the frame for a good reason — it is non-magnetic, so the frame itself does not disturb the field of the magnet.
Measuring the Induced Emf
The emf produced lasts only for the brief moment when the magnet is near the coil, so an ordinary voltmeter would never have time to respond to it. A different method is needed.
In order to measure the induced emf, a capacitor (C) and a diode (D) are connected across the coil, as shown in Fig. 12.7 (b). The induced emf produced in the coil is used for charging the capacitor through the diode. Then the voltage developed across the capacitor is measured.
Diagram
Fig. 12.7 (b) – Measurement of induced emf: the coil connected through a diode D to a capacitor C, with a milliammeter (mA) in the charging circuit.
The diode is a one-way device: it allows current through in one direction only. Its role here is to let the capacitor collect charge from pulses of one polarity while ignoring those of the other, so that the capacitor voltage builds up instead of being cancelled on the return swing.
The capacitor may not get charged up to the peak value in a single swing, as the time constant (RC) may be larger than the time during which the emf in the coil is generated. This may take about a few oscillations to charge the capacitor to the peak value, and this is indicated by the ammeter (mA), which will tell us when the charging current ceases to flow.
How the Field at the Coil Varies with Time
As the magnet, kept in the middle of the arc as in Fig. 12.7 (a), starts far away from the coil, moves through it, and recedes, the magnetic field and hence the flux through the coil changes from a small value, increases to its maximum, and becomes small again, thus inducing an emf. Actually, there is a substantial magnetic field at the coil only when the coil is very near the magnet, because the field of a bar magnet falls off sharply with distance.
There is a second effect at work as well. The speed of the magnet is largest when it approaches the coil, since the coil is placed at the mean position of the oscillation, where a pendulum always moves fastest. Thus the magnetic field changes quite slowly with time when the magnet is far away, and changes rapidly when it approaches the coil. The variation of the magnetic field \(\vec{B}\) at the coil in the mean position with time is shown in Fig. 12.7 (c).
Diagram
Fig. 12.7 (c) – Graph of magnetic field B at the coil against time t, showing a hump bcde with a flat top cd, and steep portions at times \(t_1\) and \(t_2\).
The centre of the hump ‘bcde’ refers to the time when the magnet is inside the coil. The flat portion (cd) at the top corresponds to the finite length of the magnet — while the middle of a long magnet is passing through, the field at the coil hardly alters, because the coil is simply seeing the uniform middle region of the magnet.
The magnetic flux (\(\Phi\)) is related to the magnetic field (\(B\)) through a constant, namely the effective area, which is the number of turns multiplied by the area of the coil. Since this constant does not change during the experiment, the graph of flux against time has exactly the same shape as the graph of \(B\) against time.
Reading the Emf Off the Graph
Now, the induced emf is proportional to \(\dfrac{d\Phi}{dt}\), that is, to the slope of the curve in Fig. 12.7 (c). This is a useful skill in itself: given a flux-time graph, the emf at any instant is the steepness of the curve at that instant, and nothing else.
As the slope of the curve is largest at the times \(t_1\) and \(t_2\), the magnitude of the induced emf will be largest at these times. At the flat top cd, where the flux is greatest, the slope is nearly zero and so the emf is nearly zero — a vivid reminder that maximum flux and maximum emf occur at quite different moments.
But Lenz’s law gives the minus sign in Eq. (12.3),
\[e = -\left(\frac{d\Phi}{dt}\right)\]
which means that the emf \(e\) is ‘negative’ when \(\Phi\) is increasing at \(t_1\), and ‘positive’ when \(\Phi\) is decreasing at time \(t_2\). This is shown in Fig. 12.7 (d), relating the induced emf (\(e\)) with time (\(t\)).
Diagram
Fig. 12.7 (d) – Graph of induced emf e against time t, showing a negative pulse at \(t_1\) followed by a positive pulse at \(t_2\).
Remember the sequence of two pulses: one ‘negative’ and one ‘positive’ occurs during just half a cycle of motion of the magnet. On the return swing of the magnet, they will be repeated.
Effect of the Pulses on the Charging Circuit
Now we consider the effect of these pulses on the charging circuit of Fig. 12.7 (b). The diode will conduct only during the ‘positive’ pulse.
At the first half swing, the capacitor will charge up to a potential, say \(e_1\). During the next half swing, the diode will be cut off until a ‘positive’ pulse is produced, and then the capacitor will charge up to a slightly higher potential, say \(e_2\).
This will continue for a few oscillations, till the capacitor charges up to its peak value \(e_0\) set by the voltage or emf pulse. At this stage, the ammeter will show no kick, that is, no further increase in the current of the circuit. The reading has then settled, and \(e_0\) is the peak induced emf we set out to measure.
Estimating the Peak Emf
In order to have an estimate of \(e_0\), the equation for the induced emf can be written using the chain rule as
\[e = \frac{d\Phi}{dt} = \frac{d\Phi}{d\theta} \cdot \frac{d\theta}{dt}\]
— (12.14)
Here \(\theta\) is the angular displacement of the oscillating frame from its mean position. The chain rule has split the problem into two independent pieces: how the flux depends on where the magnet is, and how fast the magnet is moving at that place.
The first term, \(\dfrac{d\Phi}{d\theta}\), depends on the geometry of the magnet and the coil. At \(\theta = 0\), the mean position, we have maximum \(\Phi\). But we are interested in \(\dfrac{d\Phi}{d\theta}\), which is actually zero at \(\theta = 0\), since the flux is passing through a maximum there and the graph is momentarily flat.
The second term, \(\dfrac{d\theta}{dt}\), can be deduced from the oscillation equation. For an oscillating system,
\[\theta = \theta_0 \sin 2\pi \nu t\]
where \(\theta_0\) is the amplitude of the oscillating magnet and \(\nu\) is the frequency. Since frequency is the reciprocal of the time period,
\[\nu = \frac{1}{T}\]
so that
\[\theta = \theta_0 \sin \frac{2\pi t}{T}\]
Differentiating with respect to time,
\[\frac{d\theta}{dt} = \theta_0 \left(\frac{2\pi}{T}\right)\cos\frac{2\pi t}{T}\]
\[\left(\frac{d\theta}{dt}\right)_{max} = \frac{2\pi \theta_0}{T}\]
— (12.15)
The peak voltage (emf) \(e_0\) in the induced emf pulse corresponds to \(\left(\dfrac{d\Phi}{dt}\right)_{max}\).
We can see from Fig. 12.7 (c) that \(\left(\dfrac{d\Phi}{d\theta}\right)_{max}\) occurs at positions near the mean position. In Eq. (12.15), the cosine term does not differ much from unity for very small angles, that is, close to zero. So very near the mean position, both factors in Eq. (12.14) are close to their largest values at the same time, and we may multiply the two maxima together.
Hence we conclude that
\[e_0 = \left(\frac{d\Phi}{dt}\right)_{max} = \left(\frac{2\pi \theta_0}{T}\right)\left(\frac{d\Phi}{d\theta}\right)_{max}\]
— (12.16)
For a given magnet-coil system, the peak induced emf \(e_0\) is directly proportional to the angular amplitude (\(\theta_0\)) and inversely proportional to the time period (\(T\)).
12.7 Generators
In Chapter 10 you have learnt the principle of electric motors. The basic construction of an electric generator is the same as that of a motor. The difference lies entirely in what is supplied and what is taken out: a motor is given electrical energy and delivers rotation, while a generator is given rotation and delivers electrical energy.
In a generator, the armature is turned by some external agency or torque, as shown in Fig. 12.8 (a). The armature is the rotating assembly of coils; the external agency may be a steam turbine, a falling column of water, a diesel engine or even a hand crank.
Diagram
Fig. 12.8 (a) – Schematic of a generator: an armature coil rotating between the poles of a magnet, with its ends connected through a commutator and brushes to an external circuit.
As the conductor wires cut across the magnetic lines of force, an induced emf (\(e = Blv\)) is produced across the terminals of the commutator. This is precisely the motional emf of Section 12.5, now being produced continuously rather than in a single sweep. The induced emf is found to be proportional to the speed of rotation (\(\omega\)) of the armature — turn the machine faster and it delivers a larger emf.
Following One Conductor Round
Let us focus our attention on one conductor of the armature, as shown in Fig. 12.8 (b). Watching a single wire through one full turn explains the shape of everything that follows.
Diagram
Fig. 12.8 (b) – Waveform generation: positions (i), (ii) and (iii) of one armature conductor during rotation, and (iv) the resulting sinusoidal current-time graph for the lamp.
In position (i), the conductor is moving upward across the lines of force, inducing maximum emf. Its velocity is perpendicular to the field, so \(\sin\theta = 1\) in Eq. (12.11) and the emf is as large as it can be.
When the armature reaches position (ii), the conductor is moving parallel to the field, and there is no induced emf (\(e = 0\)). It is sliding along the field lines rather than cutting them, so no lines are being crossed.
At position (iii), the same conductor moves down across the lines of force, and the induced emf and current are directed opposite to that in case (i). Half a rotation later, the same wire is doing the opposite job.
The graph plotted between the current flowing in the lamp as a function of time (\(t\)) shows a sinusoidally varying current, as is shown in part (iv) of Fig. 12.8 (b).
The Emf of a Rotating Coil
We can now derive the result rather than merely describe it. When a coil is rotating with a constant angular velocity \(\omega\), the angle between the magnetic field \(\vec{B}\) and the area vector \(\vec{A}\) of the coil at any instant \(t\) is
\[\theta = \omega t\]
assuming \(\theta = 0\) at \(t = 0\). As the effective area of the coil — that is, the area it presents to the field — is changing due to rotation in the magnetic field \(B\), the flux \(\Phi_B\) at any time can be written as
\[\Phi_B = B \cdot A \cos\theta = B \cdot A \cos\omega t\]
From Faraday’s law, the induced emf \(e\) generated by a rotating coil of \(N\) turns is
\[e = -\frac{N d\Phi_B}{dt} = -N \cdot \frac{d}{dt}\left(BA\cos\omega t\right)\]
\[e = NBA\omega \sin\omega t\]
The minus sign has disappeared because differentiating \(\cos\omega t\) itself produces a factor of \(-\omega \sin\omega t\), and the two negatives cancel.
For \(\sin\omega t = \pm 1\),
\[e = \pm NBA\omega = \pm e_0\]
\[\therefore e = e_0 \sin\omega t\]
\[e = e_0 \sin 2\pi f t\]
— (12.17)
where \(f\) is the frequency of revolution of the coil.
In these expressions: - \(e\) is the instantaneous induced emf, in volt (V) - \(e_0 = NBA\omega\) is the peak emf, in volt (V) - \(N\) is the number of turns, a pure number - \(B\) is the magnetic field, in tesla (T) - \(A\) is the area of the coil, in square metre (m²) - \(\omega = 2\pi f\) is the angular velocity, in radian per second (rad s⁻¹) - \(f\) is the frequency of revolution, in hertz (Hz) or revolutions per second
Alternating Current
Since the value of \(\sin\omega t\) varies between \(+1\) and \(-1\), the polarity of the emf changes with time. The emf has its extremum value at \(\theta = 90°\) and \(270°\), as the change in flux is greatest at these points.
As the direction of the induced current changes periodically, it is called alternating current (AC), shown in Fig. 12.8 (c).
Diagram
Fig. 12.8 (c) – Graph of alternating current against time, a full sine wave with equal positive and negative halves.
The frequency of AC is equal to the number of times per second the current changes from positive (+ve) to negative (−ve) and back again. The domestic electrical current varies at a frequency of 50 cycles per second.
Getting Direct Current Instead
For the purpose of charging a storage battery, it is necessary to generate a steady or direct current (DC), since a battery can only be charged by current flowing in one fixed direction.
The reversing action of the commutator can be used to generate pulsating DC, as depicted in Fig. 12.8 (d). The commutator acts like a rapid switch, which reverses the connections to the armature at just the right times to match with the reversals in the current.
Diagram
Fig. 12.8 (d) – Graph of pulsating direct current against time, showing successive humps all on the positive side of the axis.
The result is not a smooth steady current but a series of humps, all on the same side of the axis, as the figure shows. The negative half of each cycle has been flipped over rather than removed, so no energy is thrown away.
Modern AC motors are more compact and rugged than DC motors, which is one reason why alternating current dominates in industry and in household supply.
12.8 Back emf and Back Torque
We know that emf can be generated in a circuit in different ways. It is worth listing them together, because it shows that induction is one member of a family rather than a special case standing alone.
In a battery, it is the chemical force which gives rise to emf. In piezoelectric crystals, mechanical pressure generates the emf. In a thermocouple, it is the temperature gradient which is responsible for producing emf in a circuit containing the junctions of two metallic wires. In a photoelectric cell, the incident light above a certain frequency is responsible for producing the emf. In a Van de Graaff generator, the electrons are literally loaded onto a conveyor belt and swept along to create a potential difference.
A generator utilises the movement of a wire through a magnetic field to produce motional emf and current through a circuit. That is the member of the family this chapter is concerned with.
A Motor Is Always a Generator As Well
We have seen that the physical construction of a DC generator and a motor is practically the same. If a DC generator is connected to a battery, it will run as a motor. If a motor is turned by any external means, it will behave as a generator.
This has an unavoidable consequence. So whenever a motor is running, its generator action cannot be turned off. The armature coils of a running motor are rotating in a magnetic field, and rotating coils in a magnetic field generate an emf whether or not we want them to.
By Lenz’s law, the induced emf will tend to oppose the change which causes it. In the present case, the ‘cause’ is the current through the armature. Therefore, the induced emf will tend to reduce the armature current.
The same argument applied to a running generator gives the mirror-image effect. The current drawn from a generator flows through its own armature coils, which sit in a magnetic field, and so the armature experiences a torque opposing its rotation — the back torque. It is this back torque that the turbine or engine must work against, and it is the reason a generator becomes physically harder to turn the moment a load is connected to it.
Why a Motor Draws a Large Current at Starting
Initially, when a motor is just starting up, its armature is not turning, and hence it is not producing any back emf. At that instant the only thing limiting the current is the small resistance of the armature windings, so a large current flows.
As the motor starts speeding up, the back emf increases and the armature current decreases. The net emf driving current through the armature is the applied emf minus the back emf, and this difference shrinks as the speed builds up.
This explains the reason why the current through a motor is larger in the beginning than when the motor is running at full speed.
12.9 Induction and Energy Transfer
Lenz’s law told us in Section 12.3.2 that induction must obey the conservation of energy. In this section we do the book-keeping in full, for one definite arrangement, and check that the mechanical energy put in exactly equals the electrical energy that comes out.
The Arrangement
Consider a loop ABCD taken out with constant velocity \(\vec{v}\) through a uniform magnetic field \(\vec{B}\), as shown in Fig. 12.9 (a). A current \(i\) is induced in the loop in the clockwise direction, and the loop segments, being still in the magnetic field, experience forces \(F_1\), \(F_2\) and \(F_3\). The dashed lines show the limits of the magnetic field, that is, the boundary beyond which there is no field.
Diagram
Fig. 12.9 (a) – A rectangular loop ABCD of width L being pulled with velocity v out of a region of uniform magnetic field directed into the page, with the induced current and the forces \(F_1\), \(F_2\) and \(F_3\) marked.
To maintain the velocity \(\vec{v}\) constant, it is required to apply an external force \(\vec{F}\) on the loop, so as to overcome the magnetic force of equal magnitude but acting in the opposite direction. If we stopped pulling, the loop would slow down and stop.
Therefore the rate of doing work on the loop is
\[P = \frac{\text{Work }(W)}{\text{time }(t)} = \frac{\text{Force }(F) \times \text{displacement }(d)}{\text{time }(t)}\]
\[P = \text{Force }(F) \times \text{velocity }(v) = \vec{F} \cdot \vec{v}\]
— (12.18)
We would like to find the expression for \(P\) in terms of \(B\) and the characteristics of the loop, i.e., its resistance (\(R\)), width (\(L\)) and area (\(A\)). Only then can we compare it with the heat produced.
The Induced Current
As the loop is moved to the right, the area lying within the magnetic field decreases, thus causing a decrease in the magnetic flux linked with the moving loop. The decreasing magnetic flux induces a current in the loop, as dictated by Lenz’s law. The induced current in the loop gives rise to a force that opposes the pulling of the loop out of the magnetic field.
We know that the magnitude of magnetic flux through the loop is
\[\Phi_B = B \cdot A = B \cdot L \cdot x\]
— (12.19)
where \(x\) is the length of the loop still inside the field region, and \(L\) is its width.
As \(x\) decreases, the flux decreases. According to Faraday’s law, the magnitude of the induced emf is
\[e = \frac{d\Phi}{dt} = \frac{d(BLx)}{dt} = BL\frac{dx}{dt} = BLv\]
— (12.20)
The induced emf \(e\) is represented on the left and the collective resistance \(R\) of the loop on the right in Fig. 12.9 (b), so that the moving loop is drawn as an ordinary circuit — a source of emf feeding a resistance. The direction of the induced current \(i\) is obtained by the right-hand (RH) rule.
Diagram
Fig. 12.9 (b) – Equivalent circuit of the moving loop: induced emf e on the left driving a current i through the collective resistance R of the loop on the right.
The magnitude of the induced current \(i\) can be written using Eq. (12.20) as
\[i = \frac{|e|}{R} = \frac{BLv}{R}\]
— (12.21)
The Force Opposing the Pull
The three segments of the current-carrying loop that lie inside the field experience the deflecting forces \(\vec{F}_1\), \(\vec{F}_2\) and \(\vec{F}_3\) in the magnetic field \(\vec{B}\), in accordance with \(\vec{F} = i\vec{L} \times \vec{B}\).
From the symmetry, the forces \(\vec{F}_2\) and \(\vec{F}_3\), being equal and opposite, cancel each other. These act on the two long sides of the loop, in which the induced current runs in opposite senses, so their effects on the motion cancel exactly.
The remaining force \(\vec{F}_1\) is directed opposite to the external force \(\vec{F}\) on the loop. So
\[\vec{F} = -\vec{F}_1\]
The magnitude of \(|\vec{F}_1|\) can be written as
\[|\vec{F}_1| = iLB\sin 90° = iLB = |\vec{F}|\]
— (12.22)
From Eq. (12.21) and Eq. (12.22),
\[|\vec{F}| = |\vec{F}_1| = iLB = \left(\frac{BLv}{R}\right)LB = \frac{B^2L^2v}{R}\]
— (12.23)
Mechanical Power Supplied
From Eq. (12.18) and Eq. (12.23), the rate of doing mechanical work, that is, the power, is
\[P = \vec{F} \cdot \vec{v} = \frac{B^2L^2v}{R} \times v = \frac{B^2L^2v^2}{R}\]
— (12.24)
This is the energy the person pulling the loop must supply every second, simply to keep it moving at a steady speed. Note that no part of it is going into kinetic energy, since the speed is constant.
Electrical Power Dissipated
If current \(i\) is flowing in the closed circuit with collective resistance \(R\), then the rate of production of heat energy in the loop, as we pull it along at constant speed \(v\), can be written as
Rate of production of heat energy
\[P = i^2R\]
— (12.25)
From Eq. (12.21) and Eq. (12.25),
\[P = \left(\frac{BLv}{R}\right)^2 R = \frac{B^2L^2v^2}{R}\]
— (12.26)
Comparing Eq. (12.24) and Eq. (12.26), we find that the rate of doing mechanical work is exactly the same as the rate of production of heat energy in the circuit or loop. The two expressions are identical, term for term.
Thus the work done in pulling the loop through the magnetic field appears as heat energy in the loop. Nothing is lost and nothing is created; the muscular effort of the person pulling ends up warming the wire.
12.10 Eddy Currents
In the last section the induced current was confined to a thin wire loop, so it had only one path to follow. What happens if the conductor is not a wire at all, but a solid block of metal?
Suppose the conducting loop of Fig. 12.9 (a) is replaced by a solid conducting plate. The relative motion between the conductor and the magnetic field induces a current in the conductor plate, as shown in Fig. 12.10 (a). In this case again, we encounter an opposing force, so we must do work while moving the conductor with uniform velocity \(v\).
Diagram
Fig. 12.10 (a) – Eddy currents induced in a solid conducting plate being pulled out of a magnetic field, shown as a swirling closed loop of current within the plate.
The difference lies in the paths available to the charges. The conduction electrons making up the induced current do not follow one path, as they do with the loop, but swirl about within the plate as if they were caught in an eddy of water. Such a current is called an eddy current. An eddy current can be represented by a single path, as shown in Fig. 12.10 (a), although in reality the swirls fill the whole body of the metal.
Eddy Currents Convert Motion into Heat
The induced current in the conductor plate is responsible for the transfer of mechanical energy into heat energy. The energy accounting is exactly the one worked out in Section 12.9; only the geometry of the current path has changed.
The dissipation of energy as heat energy is more apparent in the arrangement shown in Fig. 12.10 (b), where a conducting plate, free to rotate about a pivot, is allowed to swing down like a pendulum through a magnetic field.
Diagram
Fig. 12.10 (b) – A solid metal plate pivoted at the top, swinging like a pendulum between the poles of a magnet and coming rapidly to rest.
In each swing, when the plate enters and leaves the field, a portion of its mechanical energy is transformed into heat energy. After several such swings, there is no mechanical energy left with the pendulum, and the converted heat energy is dissipated in the solid plate, making it warm.
The pendulum stops far sooner than an identical plate swinging outside the field, and the difference has nothing to do with friction or air resistance. It is Lenz’s law taking its payment.
Reducing Eddy Currents
Eddy currents can be reduced by introducing a discontinuity in the structure of the conductor plate, as depicted in Fig. 12.10 (c).
Diagram
Fig. 12.10 (c) – A metal plate cut with parallel slots, showing how the slots break up the large eddy current loops into small ones.
The reason is straightforward. Slots cut into the plate break the wide circulating paths into narrow ones. The induced emf around each small path is smaller, and the resistance of the thin strip is higher, so far less current flows and far less energy is wasted as heat.
Eddy Currents Put to Work
Eddy currents are not only a nuisance to be suppressed. Where heating or braking is what we actually want, they are extremely useful.
12.11 Self-Inductance
So far, every induced emf in this chapter has appeared in one circuit because of something happening outside it — a magnet moving nearby, a neighbouring coil switching on, or the circuit itself moving through a field. We now come to a case where a circuit induces an emf in itself.
Consider a circuit (coil) in which the current is changing, as shown in Fig. 12.11. The changing current will vary the magnitude of the magnetic flux linked with the coil (circuit) itself, and consequently an emf will be induced in the circuit.
Diagram
Fig. 12.11 – A coil connected to a battery through a key K, with the magnetic field lines produced by its own current threading through the coil itself.
The logic is a short chain, and every link is something already established. A current produces a magnetic field; that field passes through the very coil producing it; if the current changes, this flux changes; and a changing flux induces an emf.
The Coefficient of Self-Induction
Let, at any instant, the value of magnetic flux linked with the circuit itself be \(\Phi\), corresponding to a current \(i\) in it (Fig. 12.11). It is obvious that \(\Phi\) will be proportional to the current \(i\), since doubling the current everywhere in the coil doubles the field it produces and hence doubles the flux.
\[\Phi \propto i\]
\[\Phi = Li \qquad \text{or} \qquad L = \frac{\Phi}{i}\]
— (12.27)
where \(L\) is a constant of proportionality and is termed as the self-inductance (or coefficient of self induction) of the coil.
For a closely wound coil of \(N\) turns, the same magnetic flux will be linked with all the turns. When the flux through the coil changes, each turn of the coil contributes towards the induced emf. Therefore, a term flux linkage is used for a closely wound coil. The flux linkage for a coil with \(N\) turns corresponding to current \(i\) will be written as
\[N\Phi_B \propto i\]
\[N\Phi_B = Li\]
\[L = \frac{N\Phi_B}{i}\]
— (12.28)
The inductance (\(L\)) depends only on the geometry and the material properties of the core of the coil. This is an important practical point: the shape, size, number of turns and core material fix \(L\) once and for all, and it does not change when the current changes. In this respect \(L\) is like the capacitance of a capacitor or the resistance of a resistor.
Unit of Inductance
According to Faraday’s law, the induced emf \(e\) is given by
\[e = -\frac{d\Phi}{dt}\]
Using Eq. (12.27),
\[e = -\frac{d}{dt}(Li) = -L\frac{di}{dt}\]
— (12.29)
This is one of the most useful equations of the chapter. It says that the self-induced emf depends not on the current, but on how fast the current is changing, and the minus sign records that this emf always opposes the change.
\[\text{Unit of } L = \frac{|e|}{|di/dt|} = \frac{\text{volt}}{\text{A/s}} = \text{henry}\]
The unit is named after Joseph Henry, whose work was described in Section 12.1. One henry is a large unit for ordinary circuits; the coils met in laboratory work are usually measured in millihenry (mH) or microhenry (μH).
Definition of L
Self-inductance \(L\) may be defined in the following ways. All three describe the same physical quantity, and which one is most convenient depends on what information a problem gives you.
(i) From Eq. (12.27), \(\Phi = Li\), or \(L = \dfrac{\Phi}{i}\).
Hence, the self-inductance of a circuit is the ratio of the magnetic flux — produced due to the current in the circuit — linked with the circuit, to the current flowing in it.
(ii) Using Eq. (12.29), \(L = \left|\dfrac{e}{di/dt}\right|\).
Hence, self-inductance of a circuit is the ratio of the induced emf, caused by the changing current in the circuit, produced around the circuit, to the rate of change of current in it.
(iii) When a current increases in the circuit, an induced emf acts opposite to it. Consequently, work will have to be done in order to establish the magnetic flux associated with a steady current \(i_0\) in the circuit.
The work done in time \(dt\) is
\[dW = e \cdot i \, dt = \left(L\frac{di}{dt}\right) i \, dt = Li \, di\]
Here \(e\) has been taken as the magnitude of the emf that the source must work against, and \(i\,dt\) is the charge that flows in the time \(dt\).
Therefore, the total work done in raising the current from zero to \(i_0\) is
\[W = \int dW = \int_0^{i_0} Li \, di = \frac{Li_0^2}{2}\]
— (12.30)
\[\text{or} \qquad W = \frac{1}{2}Li_0^2 \quad \text{(magnitude)}\]
Now if \(i_0 = 1\),
then
\[W = \frac{1}{2}L \qquad \text{or} \qquad L = 2W \ \text{(numerically)}\]
— (12.31)
Electrical Inertia
This work done, \(W\), will represent the energy of the circuit. It has not been lost; it has been stored in the magnetic field, and it comes back out of the circuit when the current is switched off.
\[\therefore \text{Energy of the circuit} = \frac{1}{2}Li_0^2\]
— (12.32)
We know that mechanical energy is expressed in terms of kinetic energy as
\[KE = \frac{1}{2}mv^2\]
— (12.33)
Comparing Eq. (12.32) and Eq. (12.33), we find that the self-inductance (\(L\)) of an electrical circuit plays the same role — that of electrical inertia — as is played by mass (inertia) in mechanical motion.
The comparison is worth taking seriously, because it makes the behaviour of inductive circuits easy to predict. Current corresponds to velocity, and inductance to mass. Just as a heavy body is hard to set moving and hard to stop, a large inductance makes the current in a circuit hard to start and hard to stop.
Inductance of a Solenoid
If a current \(i\) is established in the windings (turns) of a long solenoid, the current produces a magnetic flux \(\Phi_B\) through the central region. The inductance of the solenoid is given by \(L = \dfrac{N\Phi_B}{i}\), where \(N\) is the number of turns, and \(N\Phi_B\) is called the magnetic flux linkage.
For a length \(l\) near the middle of the solenoid, the flux linkage is
\[N\Phi_B = (nl)(\vec{B} \cdot \vec{A}) = nlBA \qquad \text{(for } \theta = 0°)\]
where \(n\) is the number of turns per unit length, \(B\) is the magnetic field inside, and \(A\) is the cross-sectional area of the solenoid. The number of turns in the length \(l\) is \(nl\), and the angle is zero because the field runs straight along the axis, perpendicular to the cross-section.
We know that the magnetic field inside the solenoid is given by Eq. (10.65) as
\[B = \mu_0 n i\]
Hence
\[L = \frac{N\Phi_B}{i} = \frac{(nl)(BA)}{i} = \frac{(nl)(\mu_0 n i)A}{i} = \mu_0 n^2 lA\]
where \(Al\) is the interior volume of the solenoid. Notice that the current \(i\) has cancelled out completely — as it must, since \(L\) cannot depend on the current.
Therefore, the inductance per unit length near the middle of a long solenoid is
\[\frac{L}{l} = \mu_0 n^2 A = \mu_0 n^2 \left(\frac{\pi d^2}{4}\right)\]
— (12.34)
\(d\) being the diameter of the solenoid.
This implies that the inductance of a solenoid follows \(L \propto n^2\) and \(L \propto d^2\). The square on \(n\) has a clear physical origin: doubling the turns per unit length doubles the field produced by a given current, and also doubles the number of turns available to link that field, so the flux linkage rises fourfold.
As \(n\) is a number per unit length, inductance can be written as a product of the permeability constant \(\mu_0\) and a quantity with the dimension of length. This implies that \(\mu_0\) can be expressed in henry/metre (H/m).
Inductances in Series or Parallel
If several inductances are connected in series or in parallel, then the total inductance is determined by using the following relations:
\[L_{Total} = L_1 + L_2 + L_3 + \ldots \qquad \text{(Series Combination)}\]
\[\frac{1}{L_{Total}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + \ldots \qquad \text{(Parallel Combination)}\]
These are the same rules as for resistances in series and parallel, and they are easy to remember for that reason. In series the total inductance is larger than any single one; in parallel it is smaller than the smallest.
12.12 Energy Stored in a Magnetic Field
We have seen that the changing magnetic flux in a coil causes an induced emf. The induced emf so produced opposes the change, and hence energy has to be spent to overcome it in order to build up the magnetic field.
This energy may be recovered as heat in a resistance of the circuit — it is not destroyed when the current is switched off. That fact gives the logical concept of the energy being stored in the magnetic field while the current flows.
The Electrostatic Parallel
We have dealt with a similar problem in electrostatics, where the total electrostatic energy \(U_E\) stored in the medium between the plates of a capacitor with capacitance \(C\) and charge \(q\), held at potential \(V\), is
\[U_E = \frac{q^2}{2C} = \frac{CV^2}{2} \qquad [\because q = CV]\]
Recalling this result is useful, because the magnetic case that follows has exactly the same mathematical shape, and the same physical interpretation: work must be done to set up a field, and that work is stored in the field.
Energy Stored in an Inductor
Now we can estimate the energy spent to build up a current \(I\) in a circuit having an inductance \(L\).
From Eq. (12.29), the induced emf is
\[e = -L\frac{di}{dt}\]
The work done in moving a charge \(dq\) against this emf is
\[dw = e\,dq = L\frac{di}{dt}\,dq = L\,di\,\frac{dq}{dt} = Li\,di \qquad \left[\because \frac{dq}{dt} = i\right]\]
The step that does the work here is the rearrangement in the middle: \(dq\) has been divided by \(dt\) to give the current, leaving the neat product \(Li\,di\), which can be integrated at once.
Therefore the total work done in raising the current from zero to its final value \(I\) is
\[W = \int dw = \int_0^I Li\,di = \frac{1}{2}LI^2 = U_B\]
— (12.35)
This is the energy stored (\(U_B\)) in the magnetic field, and it is analogous to the energy stored (\(U_E\)) in the electric field in a capacitor, given above.
Note the square on the current. Doubling the current in a coil stores four times as much energy in its field, which is why the switching-off of a heavy-current electromagnet has to be handled carefully — all of that stored energy has to go somewhere in a very short time.
Energy Density of the Field
It can be shown that this energy stored up in the magnetic field per unit volume (\(u_B\)) comes out to be
\[u_B = \frac{B^2}{2\mu_0}\ \text{joules per cubic metre}\]
which parallels \(\dfrac{1}{2}\varepsilon_0 E^2\), the energy density (\(u_E\)) in an electric field \(E\), with \(\mu_0\) and \(\varepsilon_0\) being the permeability and the permittivity of free space respectively.
This is a significant shift in viewpoint, and it deserves to be stated plainly. Equation (12.35) locates the energy in the coil, as a property of the circuit; the energy density locates it in the space where the field exists, whether or not any coil is nearby. The second view is the deeper one, and Section 12.13 derives the result properly.
12.13 Energy Density of a Magnetic Field
In the previous section the result \(u_B = \dfrac{B^2}{2\mu_0}\) was quoted without proof. We now derive it, using a solenoid because it is the one arrangement in which the field is uniform, confined and easy to calculate.
Consider a long solenoid having length \(l\) near the middle, cross-sectional area \(\vec{A}\), and carrying a current \(i\) through it, as shown in Fig. 12.12. The volume associated with the length \(l\) will be \(A \cdot l\).
Diagram
Fig. 12.12 – A current-carrying solenoid producing a uniform magnetic field in its interior region, with a length \(l\) near the middle and cross-sectional area A marked.
The energy \(U_B\) stored by the length \(l\) of the solenoid must lie entirely within the volume \(Al\), because the magnetic field outside the solenoid is almost zero. Since there is no field outside, there can be no energy outside either.
Moreover, the energy stored will be uniformly distributed within the volume, as the magnetic field \(\vec{B}\) is uniform everywhere inside the solenoid. These two facts together are what make the solenoid the right choice for this derivation: we can simply divide the total energy by the total volume, without any integration.
Deriving the Energy Density
Thus, the energy stored per unit volume in the magnetic field is
\[u_B = \frac{U_B}{Al}\]
— (12.36)
From Eq. (12.35), we know that \(U_B = \dfrac{1}{2}LI^2\). Therefore
\[u_B = \frac{\frac{1}{2}LI^2}{Al} = \frac{1}{2}\left(\frac{L}{l}\right)\frac{I^2}{A}\]
— (12.37)
The rearrangement here is deliberate. It has produced the group \(\dfrac{L}{l}\), the inductance per unit length, which we have already worked out for a solenoid.
For a long solenoid, we know that the inductance (\(L\)) per unit length is given by Eq. (12.34) as
\[\frac{L}{l} = \mu_0 n^2 A\]
where \(L\) is the inductance of a long solenoid having length \(l\) in the middle, \(n\) is the number of turns per unit length, \(A\) is the cross-sectional area of the solenoid, and \(\mu_0\) is the permeability constant for air, \(4\pi \times 10^{-7}\) T·m/A or \(4\pi \times 10^{-7}\) H/m \([\because 1\ \text{H (henry)} = 1\ \text{T·m}^2\text{/A}]\).
Substituting the value of \(\left(\dfrac{L}{l}\right)\) in Eq. (12.37), we get
\[u_B = \frac{1}{2}\frac{\mu_0 n^2 A I^2}{A}\]
\[u_B = \frac{1}{2}\mu_0 n^2 I^2\]
— (12.38)
The area \(A\) has cancelled, which is a good sign — an energy density should not depend on how wide the solenoid happens to be.
For a solenoid, the magnetic field at interior points is given by Eq. (10.65) as
\[B = \mu_0 I n\]
Squaring this gives \(B^2 = \mu_0^2 n^2 I^2\), so that \(\mu_0 n^2 I^2 = \dfrac{B^2}{\mu_0}\). Therefore, the expression for the energy density (\(u_B\)) stored in the magnetic field can be written as
\[u_B = \frac{B^2}{2\mu_0}\]
— (12.39)
This equation gives the density of stored energy at any point where the magnetic field is \(B\). Although it has been derived here for a solenoid, this equation holds good for all magnetic fields, no matter how they are produced.
That last sentence is the whole point of the section. The solenoid was only scaffolding. Once the result is written in terms of \(B\) alone, every reference to turns, current and geometry has vanished, and what is left is a statement about the field itself — near a bar magnet, inside a transformer core, or in the space between the Sun and the Earth.
12.14 Mutual Inductance (M)
Self-inductance described a coil acting on itself. Mutual inductance describes one coil acting on another — which is exactly the two-coil experiment that opened this chapter, now put into quantitative form.
Let us consider a case of two coils placed side by side, as shown in Fig. 12.13. Suppose a fixed current \(I_1\) is flowing through coil 1. Due to this current, a magnetic field \(\vec{B}_1(x, y, z)\) will be produced in the nearby region surrounding coil 1.
Diagram
Fig. 12.13 – Mutual inductance of two coils: coil 1 connected to a battery, rheostat and key K, coil 2 connected to a galvanometer, with the field lines of coil 1 shown passing through coil 2.
Let \(\Phi_{21}\) be the magnetic flux linked with the surface area \(s_2\) of coil 2 due to the magnetic field \(\vec{B}_1\). It can be written as
\[\phi_{21} = \int_{s_2} \vec{B}_1 \cdot \vec{\delta a}\]
— (12.40)
where \(s_2\) represents the effective surface (or area) enclosed by coil 2.
Note the order of the subscripts, which is a convention worth fixing in your mind now: the first number is the coil where the flux arrives, and the second is the coil that produced it.
The Coefficient of Mutual Induction
If the positions of the coils are fixed in space,
Then \(\phi_{21} \propto I_1\)
\(\phi_{21} = \text{constant} \cdot I_1\)
or
\[\phi_{21} = M_{21} I_1\]
— (12.41)
where \(M_{21}\) is a constant of proportionality and is termed as the mutual inductance or coefficient of mutual induction of coil 2 (or circuit \(c_2\)) with respect to coil 1 (or circuit \(c_1\)).
Suppose \(I_1\) changes slowly with time. Then the magnetic field \(B_1\) in the vicinity of coil 2 is related to the current \(I_1\) in coil 1 in the same way as it would be related for a steady current. The magnetic flux \(\phi_{21}\) will therefore change in proportion as \(I_1\) changes.
The condition “slowly” matters. It allows us to use the ordinary steady-current formula for the field at each instant, treating the field as though it keeps up with the current without any delay.
The induced emf in coil 2 will be written as
\[e_{21} = -\frac{d\phi_{21}}{dt}\]
\[e_{21} = -M_{21}\frac{dI_1}{dt}\]
Now we allow current \(I_2\) to flow through coil 2. On account of this current, the magnetic flux \(\phi_{12}\) linked with coil 1 is obviously proportional to \(I_2\).
That is
\[\phi_{12} \propto I_2 \qquad \text{or} \qquad \phi_{12} = M_{12} I_2\]
— (12.42)
\[\text{or} \qquad M_{12} = \frac{\phi_{12}}{I_2}\]
— (12.43)
\(M_{12}\) is known as the mutual inductance of coil 1 with respect to coil 2. The induced emf in coil 1 will be
\[e_{12} = -M_{12}\frac{dI_2}{dt}\]
— (12.44)
It may be noted that, by symmetry,
\[M_{12} = M_{21} = M\]
This is a genuinely surprising result. The two coils may be of quite different sizes, with different numbers of turns, yet the coupling is the same in both directions: a given rate of change of current in the small coil induces exactly the same emf in the large one as the same rate of change in the large coil would induce in the small one. Because of this, we simply speak of the mutual inductance \(M\) of the pair.
Alternative Definitions of Mutual Inductance
It is evident from Eq. (12.41) and Eq. (12.42) that
\[\phi_{21} = MI_1 \quad \text{and} \quad \phi_{12} = MI_2\]
\[\text{or} \qquad M = \frac{\phi_{21}}{I_1} = \frac{\phi_{12}}{I_2}\]
— (12.45)
The circuit in which current is provided by an external source is usually referred to as the primary circuit, while the other is called the secondary.
Therefore, the mutual inductance \(M\) of two circuits (or coils) is the magnetic flux (\(\phi_s\)) linked with the secondary circuit per unit current (\(I_p\)) of the primary circuit.
\[\therefore M = \frac{\phi_s}{I_p} \qquad \text{or} \qquad \phi_s = MI_p\]
— (12.46)
Also from Faraday’s law,
\[e_s = -\frac{d\phi_s}{dt} = -\frac{d}{dt}(MI_p) = -M\frac{dI_p}{dt}\]
\[\text{or} \qquad M = \left|\frac{e_s}{(dI_p/dt)}\right|\]
— (12.47)
The unit of mutual inductance is the henry (H), the same unit as for self-inductance.
\[\text{henry} = \frac{\text{volt}}{\text{A/s}} = \text{ohm} \cdot \text{s}\]
\[1\ \text{henry} = 1\ \text{ohm} \cdot \text{s}\]
If, corresponding to a 1 A/s rate of change of current in the primary circuit, the induced emf produced in the secondary circuit is 1 volt, then the mutual inductance (\(M\)) of the two circuits is 1 H.
Coefficient of Coupling Between Two Circuits
In practice, not all of the flux produced by one coil reaches the other. Some of it leaks away into the surrounding space. The coefficient of coupling (K) is a measure of the portion of flux that reaches coil 2, which is in the vicinity of coil 1. The greater the coefficient of coupling, the greater will be the mutual inductance (\(M\)).
The inductance of any circuit is proportional to the induced voltage it can develop. This is equally true for mutual inductance.
\[M \propto e_{21}\]
where \(e_{21}\) is the induced emf developed in coil 2 due to the portion of the flux from coil 1 reaching coil 2, which is \(K\Phi_1\).
But the induced emf is also proportional to the number of turns in the coil,
So, \(e_{21} \propto N_2 K \Phi_1\)
But \(\Phi_1 \propto N_1\)
\[\therefore e_{21} \propto N_2 K N_1\]
Also \(L \propto N^2\), or \(N \propto \sqrt{L}\)
\[\therefore N_1 N_2 \propto \sqrt{L_1}\sqrt{L_2} = \sqrt{L_1 L_2}\]
Replacing \(e_{21}\) with \(M\), we now have
\[M = K\sqrt{L_1 L_2}\]
— (12.48)
\(K\) is usually less than unity. If \(K = 1\), the two coils will be perfectly coupled, and \(M = \sqrt{L_1 L_2}\). This is the largest value \(M\) can possibly take for a given pair of coils.
- If \(K > 0.5\), the two coils are tightly coupled.
- If \(K < 0.5\), the coils are loosely coupled.
- If \(L_1 = L_2\), then a coil with self-inductance \(L\) is coupled to itself with mutual inductance \(M = \sqrt{L_1 L_2} = \sqrt{L^2} = L\).
When Coupling Is Not Wanted
It may not always be desirable to have a large value of mutual inductance (\(M\)). A large value of \(M\) is desirable for a transformer, but a higher \(M\) is not desirable for home appliances such as an electric clothes dryer.
A dangerous emf can be induced on the metallic case of the dryer if the mutual inductance between its heating coils and the case is large. In order to minimise \(M\), the heating coils are counter wound, so that their magnetic fields cancel one another and reduce \(M\) with the case of the dryer. Counter winding means the wire is taken along the former and then brought back alongside itself, so that the two adjacent currents run in opposite directions and their fields very nearly cancel outside the winding.
Theoretically, the coupling between two coils is never perfect. If two coils are wound on a common iron core, the coefficient of coupling (\(K\)) can be considered as unity, because the iron confines almost the whole of the flux to the core and guides it from one winding to the other.
For two air-core coils, or two coils on separate iron cores, the coefficient of coupling depends on the distance between the two coils and the angle between the axes of the two coils, as illustrated in Fig. 12.13 (b).
Diagram
Fig. 12.13 (b) – Two coils shown in three relative orientations: axes parallel and in line (maximum K), axes at an intermediate angle, and axes at right angles (minimum K).
When the coils are parallel (and in line), the coefficient \(K\) is maximum. If the axes of the coils are at right angles (and in line), \(K\) is minimum. If we want to prevent interaction between the coils, the coils should be oriented at right angles to each other and be kept as far apart as possible.
The \(K\)-value for radio coils (radio frequency and intermediate frequency transformers) lies between 0.001 to 0.05.
12.15 Transformer
Mutual inductance is the basis of all types of transformers. A transformer is a device used for changing the voltage of alternating current from a low value to a high value, or vice versa. We can see transformers by the road sides in villages and cities.
Whenever the magnetic flux linked with a coil changes, an emf is induced in the neighbouring coil. In a transformer there are two coils, primary (p) and secondary (s), insulated from each other and wound on a soft iron core, as shown in Fig. 12.14. The primary and secondary coils are called input and output coils respectively.
Diagram
Fig. 12.14 – Transformer consisting of primary and secondary coils wound on a soft iron core.
The soft iron core has a definite job. It confines almost the entire flux produced by the primary and guides it through the secondary, so the coefficient of coupling approaches unity, as noted in Section 12.14. Soft iron is used rather than steel because it magnetises and demagnetises easily, following the alternating current without much loss.
When an AC voltage is applied to the primary coil, the current through the coil changes sinusoidally, causing similar changes in the magnetic flux through the core. As the changing magnetic flux is linked with both the primary and secondary coils, an emf is induced in each coil. The magnetic flux linked with a coil depends upon the number of turns in that coil.
The Transformer Equation
Let \(\phi\) be the magnetic flux linked per turn with both the coils at an instant \(t\). Let \(N_p\) and \(N_s\) be the number of turns in the primary and secondary coil respectively.
Then at the instant \(t\), the magnetic flux linked with the primary coil is \(\phi_p = N_p\phi\), and with the secondary coil is \(\phi_s = N_s\phi\).
The crucial assumption here is that the same flux per turn threads both windings, which is what the closed iron core makes possible.
The induced emf in the primary and secondary coil will be
\[e_p = -\frac{d\phi_p}{dt} = -N_p\frac{d\phi}{dt}\]
and
\[e_s = -N_s\frac{d\phi}{dt}\]
Dividing one by the other, the common factor \(\dfrac{d\phi}{dt}\) cancels, leaving
\[\therefore \frac{e_s}{e_p} = \frac{N_s}{N_p}\]
— (12.49)
The ratio \(\dfrac{N_s}{N_p}\) is called the turn ratio (transformer ratio) of the transformer. Equation (12.49) is known as the equation for the transformer.
Currents in an Ideal Transformer
A transformer cannot create energy. For an ideal transformer, in which no energy is lost in the core or the windings,
input power = output power
\[e_p i_p = e_s i_s\]
\[\frac{e_s}{e_p} = \frac{i_p}{i_s}\]
— (12.50)
Combining Eqs. (12.49) and (12.50),
\[\frac{e_s}{e_p} = \frac{N_s}{N_p} = \frac{i_p}{i_s}\]
— (12.51)
Read the last equality carefully: the current ratio is inverted with respect to the turn ratio. Whatever a transformer gives in voltage, it takes back in current.
Case 1: When \(N_s > N_p\),
then \(e_s > e_p\) (step up transformer)
and \(i_p > i_s\). The current in the primary coil is more than that in the secondary coil.
Case 2: When \(N_s < N_p\),
then \(e_s < e_p\) (step down transformer)
and \(i_p < i_s\). The current in the primary coil is less than that in the secondary coil.
12.16 Summary
Electromagnetic Induction
- Electromagnetic induction is the production of an emf, and hence a current in a closed circuit, whenever the magnetic flux linked with the circuit changes.
- Faraday (1831) and Joseph Henry (1799–1878) independently established that a time-varying magnetic field produces a current in a closed circuit.
- An induced current appears only while the flux is changing. A steady flux, however large, induces nothing.
- Faraday’s experiments showed that: the induced current reverses when the magnet’s motion or polarity is reversed; only the relative motion of coil and magnet matters; and the magnitude of the induced current depends on the relative speed and on the number of turns in the coil.
- Two coils facing each other show the same effects. The coil carrying current from an external source is the primary, the other is the secondary. A deflection appears at make and break of the primary circuit, during relative motion of the coils, and whenever the primary current is changed by a rheostat.
Faraday’s Laws
- First law (qualitative): whenever the magnetic flux linked with a closed circuit changes, an induced emf is produced in it; also, when a conductor cuts magnetic field lines, an emf is induced between its ends.
- Second law (quantitative): the magnitude of the induced emf is directly proportional to the rate of change of magnetic flux linked with the circuit.
\[e \propto \frac{d\Phi}{dt}, \qquad e = K\frac{d\Phi}{dt}, \qquad K = 1 \text{ in SI units}\]
\[e = \frac{d\Phi}{dt} \qquad \text{--- (12.2)}\]
- Combined with Lenz’s law:
\[e = -\frac{d\Phi}{dt} \qquad \text{--- (12.3)}\]
- For a coil of \(n\) turns, with \(\Phi = n\Phi'\):
\[e = -n\frac{d\Phi_B}{dt} \qquad \text{--- (12.4)}\]
- This is the flux rule: the emf equals the rate at which the magnetic flux through a conducting circuit changes.
- In SI units, \(e\) is in volt and \(\dfrac{d\Phi}{dt}\) is in weber/second.
Lenz’s Law
- The direction of the induced current is such that the magnetic field produced by it opposes the change in magnetic flux that causes it. The induced emf has the same sense as the induced current.
- General statement: “Every effect of induction acts in opposition to the cause that produces it.”
- For a magnet approaching a loop, the loop presents a north pole to an approaching north pole and repels it (induced current counter-clockwise as seen in Fig. 12.3); for a receding north pole the loop presents a south pole and attracts it (current clockwise).
- When a conductor moves in a field, the magnetic thrust on it opposes its motion. When the flux through a stationary circuit changes, the induced current’s own field is opposite to the original field if that field is increasing, and in the same direction if it is decreasing.
- Lenz’s law is a necessary consequence of the conservation of energy; the opposite rule would give a self-sustaining perpetual motion machine.
- The negative sign in \(e = -\dfrac{d\Phi}{dt}\) is what incorporates Lenz’s law into Faraday’s law.
- For an open-circuited conductor, imagine a closed circuit between its ends, apply Lenz’s law, and read off the polarity of the ends.
Flux of the Magnetic Field
- The direction assigned to an area element follows the right-handed screw rule.
- Flux through a small area element:
\[d\Phi = \vec{B}\cdot d\vec{a} = |\vec{B}||d\vec{a}|\cos\theta \qquad \text{--- (12.6)}\]
- Flux through a finite surface:
\[\Phi = \int_S \vec{B}\cdot d\vec{a} \qquad \text{--- (12.7)}\]
- \(\vec{B}\) can be taken out of the integral only if it is the same everywhere over \(S\).
- If the field varies with time, \(\Phi(t) = \displaystyle\int_S \vec{B}(t)\cdot d\vec{a}\), and
\[e = -\frac{d\Phi}{dt} = -\frac{d}{dt}\int_S \vec{B}(t)\cdot d\vec{a} \qquad \text{--- (12.9)}\]
- SI units: \(e\) in volt, \(\Phi\) in weber, \(t\) in second.
- Even if \(\vec{B}\) is constant in time, the flux changes if the area \(S\) changes with time.
Motional emf
- For a wire of length \(l\) sliding with velocity \(v\) on a frame in a field \(B\), with \(\Phi = Blx\):
\[|e| = \frac{d}{dt}(Blx) = Blv \qquad \text{--- (12.10)}\]
- From the Lorentz force \(\vec{F} = q(\vec{v}\times\vec{B})\) on the charges in the moving wire, with \(W = qvB\sin\theta \cdot l\):
\[e = \frac{W}{q} = vBl\sin\theta \qquad \text{--- (12.11)}\]
\[e_{max} = Blv \qquad (\sin\theta = 1) \qquad \text{--- (12.12)}\]
- For a stationary circuit in a changing field, the force comes from the \(\vec{E}\)-term of \(\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})\): in a region where the magnetic field changes with time, electric fields are generated.
- The flux rule holds so long as the flux changes due to a change in the field, or the motion of the circuit, or both.
- For a bar of length \(l\) rotating about one end with angular velocity \(\omega\), adding the emfs \(de = Bv\,dr\) of all segments in series:
\[e = \frac{1}{2}B\omega l^2 \qquad \text{--- (12.13)}\]
Induced emf in a Stationary Coil in a Changing Field
- In the magnet–coil system, the emf is proportional to the slope of the flux–time graph; it is largest where the flux changes fastest, and is negative while the flux increases and positive while it decreases.
- Using the chain rule:
\[e = \frac{d\Phi}{dt} = \frac{d\Phi}{d\theta}\cdot\frac{d\theta}{dt} \qquad \text{--- (12.14)}\]
- For an oscillating magnet, \(\theta = \theta_0\sin\dfrac{2\pi t}{T}\), so
\[\left(\frac{d\theta}{dt}\right)_{max} = \frac{2\pi\theta_0}{T} \qquad \text{--- (12.15)}\]
\[e_0 = \left(\frac{d\Phi}{dt}\right)_{max} = \left(\frac{2\pi\theta_0}{T}\right)\left(\frac{d\Phi}{d\theta}\right)_{max} \qquad \text{--- (12.16)}\]
- For a given magnet–coil system, the peak emf \(e_0\) is directly proportional to the angular amplitude \(\theta_0\) and inversely proportional to the time period \(T\).
Generators, Back emf and Back Torque
- A generator has the same construction as a motor, with the armature turned by an external torque; the induced emf is proportional to the speed of rotation.
- For a coil of \(N\) turns rotating with angular velocity \(\omega\), with \(\Phi_B = BA\cos\omega t\):
\[e = NBA\omega\sin\omega t = e_0\sin\omega t = e_0\sin 2\pi f t \qquad \text{--- (12.17)}\]
where \(f\) is the frequency of revolution and \(e_0 = NBA\omega\).
- The emf is extremum at \(\theta = 90°\) and \(270°\), where the change of flux is greatest. Domestic electrical current varies at 50 cycles per second.
- The commutator reverses the connections to the armature at the right instants and converts the output into pulsating DC.
- A running motor is always also a generator; the induced emf that opposes the armature current is the back emf. At starting there is no back emf, so the current is large; as the speed rises, the back emf rises and the current falls.
Induction and Energy Transfer
- For a loop of width \(L\) and resistance \(R\) pulled with velocity \(v\) out of a field \(B\):
\[\Phi_B = BLx \qquad \text{--- (12.19)}\]
\[e = BLv \qquad \text{--- (12.20)}\]
\[i = \frac{BLv}{R} \qquad \text{--- (12.21)}\]
\[|F| = iLB = \frac{B^2L^2v}{R} \qquad \text{--- (12.23)}\]
\[P = F\cdot v = \frac{B^2L^2v^2}{R} \qquad \text{--- (12.24)}\]
- The rate of heat production, \(P = i^2R = \dfrac{B^2L^2v^2}{R}\) — (12.26), is exactly equal to the rate of doing mechanical work. The work done in pulling the loop appears as heat in the loop.
Eddy Currents
- Eddy currents are the circulating induced currents set up in the body of a solid conductor when the flux linked with it changes; the electrons swirl within the plate instead of following a single path.
- They convert mechanical energy into heat, which is why a metal plate swinging as a pendulum through a magnetic field is quickly brought to rest and grows warm.
- They are reduced by making the conductor discontinuous — slots in a plate, or lamination of a core.
Self-Inductance
- Self-inductance is the production of an induced emf in a circuit due to a change of current in that same circuit.
\[\Phi = Li, \qquad L = \frac{\Phi}{i} \qquad \text{--- (12.27)}\]
\[N\Phi_B = Li, \qquad L = \frac{N\Phi_B}{i} \qquad \text{--- (12.28)}\]
- \(L\) depends only on the geometry and the material properties of the core.
\[e = -\frac{d}{dt}(Li) = -L\frac{di}{dt} \qquad \text{--- (12.29)}\]
- Unit: \(\dfrac{\text{volt}}{\text{A/s}} = \text{henry (H)}\).
- \(L\) may be defined as the flux linked per unit current, as the induced emf per unit rate of change of current, or as twice the work done in establishing the flux associated with unit current.
\[W = \int_0^{i_0} Li\,di = \frac{1}{2}Li_0^2 \qquad \text{--- (12.30)}\]
\[L = 2W \ \text{(numerically, for } i_0 = 1) \qquad \text{--- (12.31)}\]
\[\text{Energy of the circuit} = \frac{1}{2}Li_0^2 \qquad \text{--- (12.32)}\]
- Comparison with \(KE = \dfrac{1}{2}mv^2\) shows that \(L\) is the electrical inertia of a circuit, playing the role that mass plays in mechanics.
- For a long solenoid, using \(B = \mu_0 ni\) and \(N\Phi_B = nlBA\):
\[L = \mu_0 n^2 lA, \qquad \frac{L}{l} = \mu_0 n^2 A = \mu_0 n^2\left(\frac{\pi d^2}{4}\right) \qquad \text{--- (12.34)}\]
so that \(L \propto n^2\) and \(L \propto d^2\); \(\mu_0\) may be expressed in henry/metre (H/m).
- Series and parallel combinations:
\[L_{Total} = L_1 + L_2 + L_3 + \ldots \qquad \text{(Series)}\]
\[\frac{1}{L_{Total}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + \ldots \qquad \text{(Parallel)}\]
Energy Stored in a Magnetic Field
- Work must be done against the self-induced emf to build up a current, and this work is stored in the magnetic field:
\[dW = Li\,di, \qquad W = \int_0^I Li\,di = \frac{1}{2}LI^2 = U_B \qquad \text{--- (12.35)}\]
- This is the magnetic analogue of the electrostatic energy \(U_E = \dfrac{q^2}{2C} = \dfrac{CV^2}{2}\) of a capacitor.
- Energy stored per unit volume, derived for a solenoid using \(\dfrac{L}{l} = \mu_0 n^2 A\) and \(B = \mu_0 In\):
\[u_B = \frac{U_B}{Al} \qquad \text{--- (12.36)}\]
\[u_B = \frac{1}{2}\mu_0 n^2 I^2 \qquad \text{--- (12.38)}\]
\[u_B = \frac{B^2}{2\mu_0} \qquad \text{--- (12.39)}\]
- This holds for all magnetic fields, however they are produced, and parallels the electric energy density \(u_E = \dfrac{1}{2}\varepsilon_0 E^2\).
Mutual Inductance
- For two fixed coils, the flux linked with coil 2 due to current \(I_1\) in coil 1 is
\[\phi_{21} = \int_{s_2}\vec{B}_1\cdot\vec{\delta a} \qquad \text{--- (12.40)}\]
\[\phi_{21} = M_{21}I_1 \qquad \text{--- (12.41)}\]
\[\phi_{12} = M_{12}I_2, \qquad M_{12} = \frac{\phi_{12}}{I_2} \qquad \text{--- (12.42), (12.43)}\]
\[e_{21} = -M_{21}\frac{dI_1}{dt}, \qquad e_{12} = -M_{12}\frac{dI_2}{dt} \qquad \text{--- (12.44)}\]
- By symmetry, \(M_{12} = M_{21} = M\).
\[M = \frac{\phi_{21}}{I_1} = \frac{\phi_{12}}{I_2} \qquad \text{--- (12.45)}\]
\[M = \frac{\phi_s}{I_p}, \qquad \phi_s = MI_p \qquad \text{--- (12.46)}\]
\[M = \left|\frac{e_s}{(dI_p/dt)}\right| \qquad \text{--- (12.47)}\]
- \(M\) is the flux linked with one circuit per unit current in the other, or the emf induced in the secondary per unit rate of change of current in the primary.
- Unit: henry; \(1\ \text{henry} = 1\ \text{ohm}\cdot\text{s}\). \(M = 1\) H if a rate of change of 1 A/s in the primary induces 1 volt in the secondary.
- Mutual potential energy of two circuits: \(W = MI_1I_2\), so \(M = \dfrac{W}{I_1I_2}\).
- Coefficient of coupling \(K\):
\[M = K\sqrt{L_1L_2} \qquad \text{--- (12.48)}\]
- \(K\) is usually less than unity; \(K = 1\) means perfect coupling. \(K > 0.5\): tightly coupled; \(K < 0.5\): loosely coupled. If \(L_1 = L_2 = L\), then \(M = L\).
- \(K\) is maximum when the coil axes are parallel and in line, and minimum when they are at right angles. For two coils on a common iron core \(K\) may be taken as unity; for radio-frequency and intermediate-frequency transformer coils, \(K\) lies between 0.001 and 0.05.
- A large \(M\) is wanted in a transformer but not in appliances such as a clothes dryer, where the heating coils are counter wound to minimise \(M\) with the metal case.
Transformer
- A transformer changes the voltage of an alternating supply, using two coils — primary and secondary — insulated from each other and wound on a soft iron core.
- With \(\phi\) the flux linked per turn, \(\phi_p = N_p\phi\) and \(\phi_s = N_s\phi\), giving
\[e_p = -N_p\frac{d\phi}{dt}, \qquad e_s = -N_s\frac{d\phi}{dt}\]
\[\frac{e_s}{e_p} = \frac{N_s}{N_p} \qquad \text{--- (12.49)}\]
- \(N_s/N_p\) is the turn ratio. For an ideal transformer, input power = output power:
\[e_pi_p = e_si_s, \qquad \frac{e_s}{e_p} = \frac{i_p}{i_s} \qquad \text{--- (12.50)}\]
\[\frac{e_s}{e_p} = \frac{N_s}{N_p} = \frac{i_p}{i_s} \qquad \text{--- (12.51)}\]
- Step up (\(N_s > N_p\)): \(e_s > e_p\) and \(i_p > i_s\). Step down (\(N_s < N_p\)): \(e_s < e_p\) and \(i_p < i_s\).
12.17 Objective Questions
- A circular coil of 100 turns with a cross-sectional area (A) of 1 m² is kept with its plane perpendicular to the magnetic field (B) of 1 T. What is the magnetic flux linkage with the coil?
- 1 Wb
- 100 Wb
- 50 Wb
- 200 Wb
- A conductor rod of length (l) is moving with velocity (v) in a direction normal to a uniform magnetic field (B). What will be the magnitude of induced emf produced between the ends of the moving conductor?
- \(BLv\)
- \(BLv^2\)
- \(\dfrac{1}{2}Blv\)
- \(\dfrac{2Bl}{v}\)
- Two inductor coils with inductance 10 mH and 20 mH are connected in series. What is the resultant inductance of the combination of the two coils?
- 20 mH
- 30 mH
- 10 mH
- \(\dfrac{20}{3}\) mH
- A current through a coil of self inductance 10 mH increases from 0 to 1 A in 0.1 s. What is the induced emf in the coil?
- 0.1 V
- 1 V
- 10 V
- 0.01 V
- What is the energy required to build up a current of 1 A in an inductor of 20 mH?
- 10 mJ
- 20 mJ
- 20 J
- 10 J
12.18 Check Your Concepts
A strong bar magnet is held completely still inside a closed coil connected to a galvanometer. The flux linked with the coil is large, yet the galvanometer reads zero. Explain why, and state what quantity actually determines the induced emf.
A coil is connected to a battery through a key. Explain why the galvanometer connected to a second, nearby coil deflects only at the instants of closing and opening the key, and why the two deflections are in opposite directions.
The north pole of a magnet is moved towards a conducting loop, and then withdrawn. In each case, state which face of the loop behaves as a north pole and which way the induced current flows, giving your reasoning step by step.
Suppose the induced current in a loop flowed in the direction that assisted the change producing it. Describe what would happen to the current and the flux, and identify the physical law that would be violated.
What is the role of the negative sign in \(e = -\dfrac{d\Phi}{dt}\)? Would the equation still give correct magnitudes without it? Explain.
A rectangular coil moves with constant velocity through a region of uniform magnetic field that is much wider than the coil. Explain why an emf is induced while the coil is entering and while it is leaving the field, but not while it is completely inside.
A conducting rod slides on rails in a magnetic field. Explain the origin of the induced emf in terms of the Lorentz force on the free charges in the rod, and state why the stationary parts of the circuit contribute no emf.
A rod rotating about one end in a magnetic field develops an emf \(\dfrac{1}{2}B\omega l^2\), whereas a rod sliding with speed v develops \(Blv\). Explain the physical origin of the factor \(\dfrac{1}{2}\).
A wire loop is held stationary while the magnetic field through it is increased. Since the loop does not move, the magnetic part of the Lorentz force is zero. What then drives the charges around the loop? State the general conclusion this leads to.
In a rotating-coil generator, at which position of the coil is the flux maximum, and at which position is the induced emf maximum? Explain why these two positions are different.
Why is the current drawn by an electric motor much larger at the instant of starting than when it is running at full speed? Explain using the idea of back emf, and state one practical consequence.
A loop is pulled out of a magnetic field at constant velocity. Show, in words, why the mechanical power supplied by the person pulling must be exactly equal to the electrical power dissipated in the loop.
Why are the iron cores of transformers and motors built up from thin, insulated laminations instead of a single solid block? What would happen if a solid core were used?
Self-inductance is often described as “electrical inertia”. Explain this description by comparing the roles of L and i in an electrical circuit with those of m and v in mechanics.
The self-inductance of a solenoid depends on \(n^2\), where n is the number of turns per unit length. Explain why the dependence is on the square of n and not simply on n.
Coil 1 has very few turns and coil 2 has very many turns. Explain why the mutual inductance of the pair is the same whichever coil is used as the primary.
Two coils are to be placed near each other in an instrument, but their interaction must be kept as small as possible. How should they be oriented and positioned, and why? Also explain why the heating coils of a clothes dryer are counter wound.
A transformer works perfectly on an AC supply but does nothing when connected to a battery of the same voltage. Explain this observation, and state what the soft iron core contributes to its working.
A step-up transformer increases the voltage of a supply. Does it therefore increase the power delivered? Explain, and state the relation between the turn ratio and the current ratio for an ideal transformer.
Electrical energy is transmitted over long distances at very high voltage and then stepped down near the consumer. Explain the physical reason for this practice.
What do you mean by electromagnetic induction? State Faraday’s law of induction.
State and explain Lenz’s law in the light of the principle of conservation of energy.
What are eddy currents? State applications of eddy currents.
If the copper disc of a pendulum swings between the poles of a magnet, the pendulum comes to rest very quickly. Explain the reason. What happens to the mechanical energy of the pendulum?
Explain why the inductance of two coils connected in parallel is less than the inductance of either coil.
12.19 Practice with Numericals
The magnetic flux linked with each turn of a 120-turn coil changes steadily from \(6.0\times10^{-3}\) Wb to \(1.0\times10^{-3}\) Wb in 0.040 s. Calculate the magnitude of the emf induced in the coil.
A flat circular coil of radius 8.0 cm is placed in a uniform magnetic field of 0.25 T, with the normal to its plane making an angle of 60° with the field. Calculate the magnetic flux linked with the coil.
A square coil of side 15 cm and 300 turns lies with its plane perpendicular to a uniform magnetic field. The field increases steadily from 0.10 T to 0.70 T in 0.25 s. Calculate the magnitude of the induced emf in the coil.
A straight conducting rod of length 0.60 m slides on horizontal rails at a constant speed of 5.0 m s⁻¹ in a uniform magnetic field of 0.35 T directed perpendicular to the plane of the rails. Calculate the emf induced across the ends of the rod. If the total resistance of the circuit is 1.5 Ω, find the induced current.
A conducting rod of length 0.50 m rotates about one of its ends with an angular velocity of 40 rad s⁻¹ in a uniform magnetic field of 0.20 T, directed perpendicular to the plane of rotation. Calculate the emf induced between the ends of the rod.
A rectangular loop of width 0.30 m and total resistance 0.60 Ω is pulled out of a uniform magnetic field of 0.50 T at a constant speed of 3.0 m s⁻¹, the plane of the loop being perpendicular to the field. Calculate (a) the induced emf, (b) the induced current, (c) the external force required, and (d) the rate of production of heat in the loop.
A coil of 400 turns and area \(2.5\times10^{-2}\) m² is rotated at 30 revolutions per second about an axis perpendicular to a uniform magnetic field of 0.15 T. Calculate the peak value of the induced emf and the emf at the instant when the plane of the coil is perpendicular to the field.
A DC motor working on a 240 V supply has an armature resistance of 3.0 Ω. When it runs at full speed, the back emf developed is 216 V. Calculate the armature current at full speed and the current at the instant of starting.
A coil of self-inductance 180 mH carries a current that increases uniformly from 0.50 A to 2.50 A in 0.010 s. Calculate the magnitude of the self-induced emf in the coil.
An air-cored solenoid of length 25 cm and cross-sectional area \(6.0\times10^{-4}\) m² has 500 turns. Calculate its self-inductance. (\(\mu_0 = 4\pi\times10^{-7}\) H/m)
A closely wound coil of 150 turns has a self-inductance of 25 mH. Calculate the total flux linkage and the flux through the cross-section of the coil when it carries a current of 8.0 mA.
Three inductors of 12 mH, 18 mH and 36 mH are available. Calculate the resultant inductance when all three are connected (a) in series, and (b) in parallel.
An inductor of 0.75 H carries a steady current of 4.0 A. Calculate the energy stored in its magnetic field. What current would store the same energy in an inductor of 3.0 H?
The magnetic field inside a long air-cored solenoid is 0.080 T. Calculate the energy density of the field, and the total energy stored in a 0.50 m length of the solenoid whose cross-sectional area is \(1.2\times10^{-3}\) m². (\(\mu_0 = 4\pi\times10^{-7}\) H/m)
Two coils have a mutual inductance of 60 mH. Calculate the emf induced in the secondary coil when the current in the primary falls uniformly from 6.0 A to zero in 0.25 s. Also calculate the change in flux linkage with the secondary.
Two coils of self-inductances 90 mH and 40 mH are coupled with a coefficient of coupling 0.60. Calculate their mutual inductance. What would the mutual inductance be if the coupling were perfect?
A long solenoid of length 40 cm and cross-sectional area \(5.0\times10^{-4}\) m² has 1000 turns. A secondary coil of 80 turns is wound tightly around its centre. Calculate the mutual inductance of the arrangement. (\(\mu_0 = 4\pi\times10^{-7}\) H/m)
An ideal transformer has 1200 turns in the primary and 150 turns in the secondary, and its primary is connected to a 240 V AC supply. Calculate the secondary voltage. If the secondary current is 8.0 A, calculate the primary current and the power delivered.
An ideal step-up transformer is required to convert 220 V to 6600 V. If the primary has 500 turns, calculate the number of turns needed in the secondary, and the primary current when the secondary supplies 0.40 A.
A horizontal metal rod 1.20 m long is dropped from rest and falls freely, remaining horizontal and perpendicular to the horizontal component of the Earth’s magnetic field of \(2.0\times10^{-5}\) T. Calculate the emf induced between its ends 2.0 s after it is released. (Take \(g = 9.8\) m s⁻²)
In a Faraday disc dynamo, a metal disc of radius R rotates with an angular velocity \(\omega\) about an axis perpendicular to the plane of the disc and passing through its centre. The disc is placed in a magnetic field B acting perpendicular to the plane of the disc. Determine the induced emf between the rim and the axis of the disc.
A horizontal wire 20 m long extending from east to west is falling with a velocity of 10 m/s normal to the Earth’s magnetic field of \(0.5\times10^{-4}\) T. What is the value of induced emf in the wire?
A metal disc is made to spin at 20 revolutions per second about an axis passing through its centre and normal to its plane. The disc has a radius of 30 cm and spins in a uniform magnetic field of 0.20 T, which is parallel to the axis of rotation. Calculate (a) the area swept out per second by the radius of the disc, (b) the flux cut per second by a radius of the disc, (c) the induced emf in the disc.
A pair of adjacent coils has a mutual inductance of 1.5 H. If the current in one coil changes from 0 to 10 A in 0.2 s, what is the change of flux linkage with the other coil?
A long solenoid has 1500 turns/m. A coil C having cross sectional area 25 cm² and 150 turns (\(N_c\)) is wound tightly around the centre of the solenoid. If a current of 3.0 A flows through the solenoid, calculate: (a) the magnetic flux density at the centre of the solenoid, (b) the flux linkage in the coil C, (c) the average emf induced in coil C if the direction of the current in the solenoid is reversed in a time of 0.5 s. (\(\mu_0 = 4\pi\times10^{-7}\) T·m/A)
A search coil having 2000 turns with area 1.5 cm² is placed in a magnetic field of 0.60 T. The coil is moved rapidly out of the field in a time of 0.2 second. Calculate the induced emf across the search coil.
An aircraft of wing span of 50 m flies horizontally in earth’s magnetic field of \(6\times10^{-5}\) T at a speed of 400 m/s. Calculate the emf generated between the tips of the wings of the aircraft.
A stiff semi-circular wire of radius R is rotated in a uniform magnetic field B about an axis passing through its ends. If the frequency of rotation of the wire is f, calculate the amplitude of the alternating emf induced in the wire.
Calculate the value of induced emf between the ends of an axle of a railway carriage 1.75 m long travelling on level ground with a uniform velocity of 50 km per hour. The vertical component of Earth’s magnetic field (\(B_v\)) is given to be \(5\times10^{-5}\) T.
The value of mutual inductance of two coils is 10 mH. If the current in one of the coils changes from 5 A to 1 A in 0.2 s, calculate the value of emf induced in the other coil.
An emf of 96.0 mV is induced in the windings of a coil when the current in a nearby coil is increasing at the rate of 1.20 A/s. What is the mutual inductance (M) of the two coils?
A long solenoid of length l, cross-sectional area A and having \(N_1\) turns (primary coil), has a small coil of \(N_2\) turns (secondary coil) wound about its centre. Determine the mutual inductance (M) of the two coils.
The primary and secondary coil of a transformer each have an inductance of \(200\times10^{-6}\) H. The mutual inductance (M) between the windings is \(4\times10^{-6}\) H. What percentage of the flux from one coil reaches the other?
A toroidal ring, having 100 turns per cm of a thin wire is wound on a non-magnetic metal rod of length 1 m and diameter 1 cm. If the permeability of the bar is equal to that of free space (\(\mu_0\)), calculate the magnetic field inside the bar (B) when the current (i) circulating through the turns is 1 A. Also determine the self-inductance (L) of the coil.
A uniform magnetic field B(t), pointing upward fills a circular region of radius s in a horizontal plane. If B is changing with time, find the induced electric field. [Hint: Part of Maxwell’s equation, applied to a time varying magnetic flux, leads us to the equation \(\oint \vec{E}\cdot d\vec{l} = -\dfrac{d\Phi_m}{dt}\), where \(\vec{E}\) is the electric field induced when the magnetic flux changes at the rate of \(\dfrac{d\Phi_m}{dt}\).]