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Chapter 12: Magnetism

For centuries, sailors trusted a small piece of stone that always turned towards the north. Nobody knew why. Some believed a giant magnetic mountain stood at the North Pole and pulled the needle towards itself. Others thought the pole star reached down and tugged at it. The compass worked, but the reason behind it was a mystery.

Around the year 1600, the English physician and natural philosopher William Gilbert decided to stop guessing and start testing. He carved a sphere out of lodestone — naturally magnetised rock — and called it a terrella, meaning “little Earth”. Then he moved a tiny compass needle over the surface of this stone ball and carefully noted its direction at every position.

What he saw changed everything. Near the middle of the sphere, the needle lay flat. As he moved it towards the top or bottom of the ball, the needle tilted, and at the ends it stood almost upright. This was exactly the behaviour that navigators had reported at different places on Earth.

Gilbert drew the bold conclusion that no mountain and no star was responsible. The Earth itself is a magnet, and the compass needle simply obeys the Earth’s own magnetic field.

Diagram

Fig. 12.1 – William Gilbert holding a small compass needle at different points over a spherical lodestone (terrella), the needle lying flat near the middle and tilting steeply near the ends.

NoteCuriosity Corner

Q1. If you break a bar magnet into two pieces, can you separate its north pole from its south pole? Q2. Electric field lines start on a positive charge and end on a negative charge. Why do magnetic field lines instead form closed loops? Q3. Why is the total magnetic flux passing through any closed surface always exactly zero, no matter where you draw the surface? Q4. At the same distance from a bar magnet, the field on its axis is exactly twice the field on its equator. Why twice? Q5. A compass in Mumbai does not point to the true geographic north. How large is the error, and what causes it? Q6. Why does a freely pivoted magnetic needle stay horizontal near the equator but tilt steeply downwards as you carry it towards the poles?

This chapter will help you answer each of these questions.

12.1 Introduction

  1. What is a bar magnet?
  2. What are magnetic lines of force?
  3. What rules do the lines of force follow?
  4. If you hang a bar magnet horizontally so that it can turn freely, which direction does it finally settle in?
A Very Old Observation, A Very New Understanding

Magnetism is not a modern discovery. People knew about it earlier than 600 B.C., long before anyone had a theory of atoms, electrons or fields. What they knew was purely observational: certain naturally occurring stones could attract small pieces of iron, and a sliver of such stone, when allowed to swing freely, would always come to rest pointing roughly north and south.

For more than two thousand years, that was more or less where matters stood. The effect was real, useful and completely unexplained. It is only in the twentieth century that scientists began to genuinely understand magnetism, and only then could technologies be built on that understanding — the electric motor, the generator, magnetic storage, the MRI scanner.

The Scientists Who Built the Subject

Three names mark the turning points in this story.

William Gilbert (1544–1603) was the first person to investigate magnetism systematically, using the scientific method — that is, by doing controlled experiments and drawing conclusions from what he actually observed rather than from what he expected. It was he who discovered that the Earth itself is a weak magnet.

The Danish physicist Hans Oersted (1777–1851) took the subject in a new direction. He suggested that electricity and magnetism are linked — that these are not two unrelated phenomena but two connected ones.

Around 1820, while demonstrating an electric circuit during a lecture, Hans Oersted noticed something he had not planned for. A compass needle lying near the wire twitched and turned the moment current flowed through it, and returned when the current stopped. A wire carrying current was behaving like a magnet. This single accidental observation opened the door to the entire subject of electromagnetism.

James Clerk Maxwell (1831–1879) completed the picture. He proved that electricity and magnetism are not merely linked — they are different aspects of one and the same fundamental force field. What we call an “electric” effect and what we call a “magnetic” effect are two faces of a single electromagnetic reality.

Borrowing the Language of Electrostatics

You have already met, in electrostatics, the idea of an electric field and the force it exerts on charges and on electric dipoles. A great deal of that language carries over directly into magnetism, which is what makes this chapter easier than it first looks. Wherever you see a magnetic quantity in this chapter, it will usually have an electrostatic cousin that you already know.

But there is one deep difference, and it must be stated at the very start. Electric charges exist in isolation: a lone positive charge is a perfectly ordinary object. Magnetic poles do not. A north pole never occurs on its own, separated from a south pole. What we always have instead is a magnetic dipole — a north and a south pole together — or a more complicated arrangement such as a quadrupole.

Q. If a north pole always comes attached to a south pole, why bother talking about “poles” at all?

A. Because the pair of poles is exactly what gives a magnet its behaviour. The two poles together define a direction (south to north) and a strength, and from these we build the magnetic dipole moment — the single quantity that controls the field of a bar magnet and the torque it feels. The poles are a useful bookkeeping device, even though neither one can be taken away on its own.

This is why so much of magnetism is written in the language of dipoles rather than in the language of single poles, as you will see from Section 12.3 onwards.

Facts Worth Fixing in Your Mind

Some commonly known facts about magnetism: (i) Every magnet, whatever its size or shape, has two poles — a north pole and a south pole. (ii) If a magnet is broken into two or more pieces, each piece behaves as an independent magnet with its own north and south poles, though with a somewhat weaker magnetic field. Isolated magnetic monopoles therefore do not exist. The search for magnetic monopoles is still going on. (iii) Like magnetic poles repel each other; unlike poles attract each other. (iv) A bar magnet or a magnetic needle, when suspended freely or pivoted, aligns itself along the geographic North–South direction.

Point (ii) deserves a second look, because students often expect the opposite. If you saw a bar magnet in Fig. 12.2 and cut it exactly at the middle, you might think you would be left with a piece carrying only a north pole and a piece carrying only a south pole. You would not. A fresh south pole appears at the cut face of the first piece and a fresh north pole at the cut face of the second. Each fragment is a complete, smaller magnet.

Diagram

Fig. 12.2 – A bar magnet cut into two, then into four pieces, each fragment showing its own N and S poles at the cut faces.

Q1. If you break a bar magnet into two pieces, can you separate its north pole from its south pole?

A. No. The moment you break it, new poles appear at the broken faces, and each piece turns out to be a complete magnet with both a north and a south pole. You can keep cutting into smaller and smaller fragments and you will never obtain a lone pole. This is because an isolated magnetic pole — a magnetic monopole — has never been found. Magnetism, unlike electric charge, comes only in dipoles.

What This Chapter Will Cover

Magnetism is a large subject, and this chapter deliberately keeps to two parts of it: the elementary aspects of magnetism — magnetic field, lines of force, the bar magnet as a dipole, and Gauss’ law of magnetism — and terrestrial magnetism, which is the study of the Earth’s own magnetic field and the parameters used to describe it.

12.2 Magnetic Lines of Force and Magnetic Field

Borrowing a Picture from Electrostatics

You have already studied the properties of electric lines of force in the chapter on electrostatics. A line of force is a drawing device: an imaginary line drawn in the region around a source such that the field at any point on it points along the line. It lets us see, on paper, something that is otherwise invisible.

Magnetic lines of force work in exactly the same spirit. Outside a bar magnet, they originate from the north pole and end at the south pole. If you sprinkle iron filings on a card held over a bar magnet, the filings arrange themselves along these lines, and the pattern becomes visible.

Take a bar magnet and a small compass needle. Fix the bar magnet at one position on a sheet of paper, then place the compass needle at various points around it. At each point, note the direction in which the needle settles. Joining these directions traces out the magnetic field direction at various locations — you will have mapped the field of the magnet with nothing but a needle.

The Four Properties of Magnetic Lines of Force

The magnetic lines of force of a magnet have the following properties.

(i) They form closed loops. The magnetic lines of force of a magnet, or of a solenoid, form closed loops. Outside the magnet they run from the north pole to the south pole; inside the magnet they continue from the south pole back to the north pole, closing the loop, as shown in Fig. 12.3.

Diagram

Fig. 12.3 – Field lines of a bar magnet drawn as complete closed loops, running N to S outside the magnet and S to N inside it.

This is in sharp contrast to the case of an electric dipole. There, the electric lines of force originate from the positive charge and end on the negative charge, without forming a complete loop (see Fig. 12.8). The difference traces straight back to the fact you met in the previous section: electric charges can exist alone, magnetic poles cannot.

Q2. Electric field lines start on a positive charge and end on a negative charge. Why do magnetic field lines instead form closed loops?

A. An electric field line can start somewhere and stop somewhere because there is an isolated positive charge to start on and an isolated negative charge to end on. A magnetic field line has nowhere to stop, because an isolated magnetic pole does not exist. So the line simply keeps going — out of the north pole, around through space, in at the south pole, and then straight on through the body of the magnet back to the north pole. Having no place to begin or end, it closes on itself.

(ii) The tangent gives the direction of the field. The direction of the net magnetic field \(\vec{B}\) at a point is given by the tangent to the magnetic line of force at that point, taken in the direction of the line of force. So a line of force is not just decoration — reading off its tangent at any point tells you exactly which way \(\vec{B}\) points there.

(iii) The crowding of the lines gives the strength of the field. The number of lines of force crossing per unit area decides the magnitude of the magnetic field \(\vec{B}\). Where the lines are packed close together, the field is strong; where they spread apart, it is weak. This is why the lines bunch up near the poles of a bar magnet — that is where its field is strongest.

(iv) They never intersect. The magnetic lines of force do not intersect. The reason follows from property (ii). If two lines crossed at a point, you could draw two different tangents there, and the magnetic field would have two different directions at the same point. The direction of the field would not be unique at that point, which is impossible.

Q. A field is a vector. Can’t a vector at a point have two directions at once?

A. No. At any given point in space, the magnetic field has one definite magnitude and one definite direction — that is what it means to be a vector field. Two lines crossing would demand two directions at the same point, so crossing lines are ruled out. This is the identical argument you used for electric lines of force, and it is worth remembering as a one-line reason in an exam.

From a Picture to a Number: Flux and Field

So far the lines have given us a picture. Now we turn the picture into a measurable quantity.

The density of lines of force — that is, the number of lines of force per unit area normal to the surface around a particular point — determines the strength of the magnetic field at that point. The word normal matters here: the area must be held perpendicular to the lines, otherwise you are not counting all the lines that actually pass through it.

The number of lines of force itself is called the magnetic flux, written \(\phi\). Its SI unit is the weber (Wb).

NoteDefinition

Magnetic flux (\(\phi\)) is the number of magnetic lines of force crossing a given surface. Its SI unit is the weber (Wb).

Consider now the specific case of a uniform magnetic field which is normal to a finite area \(A\), as in Fig. 12.4. “Uniform” means the field has the same magnitude and direction everywhere over that area, so the lines are evenly spaced and parallel; “normal” means they strike the area at right angles. Under these two conditions, and only under these two, the magnitude of the magnetic field strength \(B\) at a point in the area \(A\) is simply the flux divided by the area:

\[\text{Magnetic Field} = \frac{\text{magnetic flux}}{\text{area}}\]

\[B = \frac{\phi}{A} \qquad \text{--- (12.1)}\]

Diagram

Fig. 12.4 – Uniform, evenly spaced parallel field lines striking a flat area A at right angles, illustrating B = φ/A.

Here \(B\) is the magnitude of the magnetic field, \(\phi\) is the magnetic flux through the area in weber (Wb), and \(A\) is the area in square metre (m²).

NoteDefinition

For a uniform magnetic field normal to a finite area \(A\), the magnitude of the magnetic field \(B\) is the magnetic flux \(\phi\) passing through that area divided by the area, \(B = \phi/A\).

Units of Magnetic Field

From Eq. (12.1), the SI unit of magnetic field (\(B\)) is expressed as weber/m², which is given the name tesla.

There is also an older, smaller unit still in common use, the gauss:

\[1\ \text{Tesla} = 10^{4}\ \text{Gauss}\]

Because one tesla equals ten thousand gauss, a field of one tesla is an extremely strong field. This conversion is a favourite trap in numerical problems — a value quoted in gauss must be converted to tesla before it is put into any SI formula.

The Earth’s magnetic field at its surface is of the order of a few times \(10^{-5}\) T, which is under a gauss — feeble, yet enough to steer every compass needle on the planet. A small refrigerator magnet is a few times \(10^{-3}\) T at its surface. The superconducting magnet inside a hospital MRI scanner runs at roughly 1.5 T to 3 T, which is why loose iron objects must never be carried into an MRI room: at those field strengths, a steel oxygen cylinder becomes a projectile.

A uniform magnetic field of 0.25 T is directed normally through a flat rectangular surface of dimensions 20 cm × 15 cm. Find the magnetic flux through the surface in weber. Also express the given magnetic field in gauss.

An Honest Warning About Lines of Force

Useful as they are, magnetic lines are only a crude way of representing the magnetic field. They are a pictorial representation of the strength of the magnetic field (\(B\)) — a drawing, not a definition. Nobody can point to a real line in space; what exists is the field, at every point, with a magnitude and a direction.

The magnetic field is better defined in terms of the Lorentz force law, which you will learn later: the field is defined by the force it exerts on a moving charge. In this chapter, however, the picture of lines of force will serve us well, and every result we obtain will be consistent with the fuller definition that follows later.

12.3 The Bar Magnet

Pole Strength and the Magnetic Dipole

A bar magnet is said to have magnetic pole strength \(+q_m\) at its north pole and \(-q_m\) at its south pole. Pole strength is a measure of how “strongly magnetic” a pole is, in the same way that charge measures how strongly electric a charged body is. The north pole is taken as positive and the south pole as negative, exactly as we take charges positive and negative.

The separation of the magnetic poles inside the magnet is \(2l\). Notice the phrase inside the magnet — the poles are not two beads stuck on the ends, they are effective positions lying within the body of the magnet, and \(2l\) is the distance between them.

Since the bar magnet has two poles, with equal and opposite pole strength, separated by a distance, it is called a magnetic dipole. This is precisely analogous to an electric dipole, which is two equal and opposite charges separated by a small distance.

NoteDefinition

A magnetic dipole is a pair of equal and opposite magnetic poles, \(+q_m\) and \(-q_m\), separated by a distance \(2l\). A bar magnet is a magnetic dipole.

The Magnetic Dipole Moment

For an electric dipole you defined a dipole moment: charge multiplied by the separation vector, pointing from the negative charge to the positive charge. The magnetic dipole moment is built in exactly the same way. It therefore becomes

\[\vec{m} = q_m \cdot 2\vec{l}\]

where \(\vec{m}\) is the magnetic dipole moment of the magnet, \(q_m\) is the pole strength, and \(2\vec{l}\) is a vector from the south pole to the north pole — that is, it has magnitude \(2l\) and points from S to N, in analogy with the electric dipole moment which points from the negative to the positive charge.

Getting this direction right matters. Every result in the rest of this chapter — the axial field, the equatorial field, the field at an arbitrary point — is written relative to the direction of \(\vec{m}\), so a reversed \(\vec{m}\) reverses all of them.

NoteDefinition

The magnetic dipole moment of a bar magnet is \(\vec{m} = q_m \cdot 2\vec{l}\), the product of its pole strength and the vector \(2\vec{l}\) drawn from its south pole to its north pole.

The SI unit of pole strength (\(q_m\)) is A m (ampere metre).

The SI unit of magnetic dipole moment (\(m\)) is A m² (ampere metre squared).

These units follow directly from the definition: multiplying a pole strength in A m by a length in m gives A m². They look unusual at first because there is no “magnetic charge” in the SI system to hang them on — the ampere in them is a reminder that, at bottom, all magnetism comes from moving charge.

Q. Pole strength is measured in A m and dipole moment in A m². Why do units of current appear in a quantity that describes a piece of iron with no current flowing through it?

A. Because a magnet’s magnetism ultimately arises from currents at the atomic scale — the circulating and spinning electrons inside its atoms. The SI system therefore builds every magnetic unit out of the ampere rather than inventing a separate unit of “magnetic charge”. A bar magnet of moment 0.5 A m² produces the same kind of external field as a suitably chosen tiny current loop of the same moment.

Axis, Equator and the Geometry of a Bar Magnet

Three geometrical terms are used constantly in this chapter, and they are best fixed once and for all with the help of Fig. 12.5.

Diagram

Fig. 12.5 – A bar magnet showing its N and S poles, the axis through both poles, the equator perpendicular to the axis through the centre O, the geometrical length, and the magnetic length 2l.

Axis: It is the line passing through both the poles of a bar magnet. Obviously, there is only one axis for a given bar magnet, since two points determine exactly one straight line.

NoteDefinition

The axis of a bar magnet is the line passing through both of its poles.

Equator: A line passing through the centre of a magnet and perpendicular to its axis is called the magnetic equator. There are many such lines, not one — the axis has an entire perpendicular plane of them through the centre. The plane containing all equators is called the equatorial plane.

The locus of points, on the equatorial plane, which are equidistant from the centre of the magnet is called the equatorial circle. The popularly known ‘equator’ in Geography is actually an ‘equatorial circle’. Any diameter of such a circle is an equator.

NoteDefinition

The magnetic equator of a bar magnet is a line through its centre, perpendicular to its axis. The plane containing all such lines is the equatorial plane, and the circle of points on that plane equidistant from the centre is the equatorial circle.

Magnetic Length Is Not Geometric Length

Magnetic length (2l): It is the distance between the two poles of a magnet.

This is not the same as the length of the bar you can measure with a ruler, which is called the geometric length. The effective poles sit a little inside the two ends of the bar, so the magnetic length is always the shorter of the two. For a normal bar magnet the two are related by

\[\text{Magnetic length } (2l) = \frac{5}{6} \times \text{Geometric length} \qquad \text{--- (12.2)}\]

NoteDefinition

The magnetic length \(2l\) of a magnet is the distance between its two poles, and is given by \(2l = \dfrac{5}{6} \times\) geometric length.

In every formula in this chapter, the length that appears is the magnetic length. If a problem hands you the physical length of the bar, Eq. (12.2) must be applied first. Skipping this step is one of the commonest slips in numericals on bar magnets, and it changes the dipole moment by nearly 17%.

In a school laboratory, when a bar magnet is used in a deflection magnetometer to compare magnetic moments, the bar is first measured with a ruler and then Eq. (12.2) is applied to obtain \(2l\). Using the ruler value directly would place the effective poles too far apart and give a magnetic moment that is consistently too large — a systematic error, one that no amount of repeating the reading will remove.

A bar magnet has a geometric length of 6.0 cm and a pole strength of 4.0 A m. Calculate (i) its magnetic length and (ii) its magnetic dipole moment, stating the SI unit in each case.

12.3.1 Magnetic field due to a bar magnet at a point along its axis and at a point along its equator

Setting Up the Two Standard Positions

There are two positions around a bar magnet that we treat as standard, because every other position can later be built out of them. The first is a point on the axis of the magnet — the axial position, shown in Fig. 12.6 (a). The second is a point on the equator of the magnet — the equatorial position, shown in Fig. 12.6 (b).

Diagram

Fig. 12.6 (a) – Bar magnet with poles S and N, centre O, dipole moment m along the axis, and a point P on the axis at distance r from O, with B_axis marked pointing along m.

Consider a bar magnet of dipole length \(2l\) and magnetic dipole moment \(\vec{m}\) as shown in Fig. 12.6 (a). We will now find the magnetic field at a point P along the axis of the bar magnet.

Let \(r\) be the distance of the point P from the centre O of the magnetic dipole. Since the poles are placed symmetrically about the centre,

\[OS = ON = l\]

For a point P lying on the equator of the magnet, as in Fig. 12.6 (b), P is equally far from both poles, and by Pythagoras’ theorem in the right triangle formed by O, N and P,

\[NP = SP = \sqrt{\left(r^{2} + l^{2}\right)}\]

Diagram

Fig. 12.6 (b) – Bar magnet with centre O, dipole moment m along the axis, and a point P on the equator at distance r from O, with B_eq shown pointing opposite to m.

Using the Electrostatic Analogy

We could now work out the field pole by pole, but there is a much shorter route. We use the electrostatic analogy to obtain the magnetic field due to a bar magnet at a large distance, \(r \gg l\). To do that, we first recall the electric field due to an electric dipole with a dipole moment \(p\), and then simply translate the result into magnetic language.

The Electrostatic Analogue

As suggested by Maxwell, electricity and magnetism could be studied analogously. The pole strength (\(q_m\)) in magnetism can be compared with charge \(q\) in electrostatics. Accordingly, we can write the equivalent physical quantities in electrostatics and magnetism as shown in Table 12.1.

Table 12.1: The Electrostatic Analogue

Quantity Electrostatics Magnetism
Basic physical quantity Electrostatic charge Magnetic pole
Field Electric Field \(\vec{E}\) Magnetic Field \(\vec{B}\)
Constant \(\dfrac{1}{4\pi\varepsilon_0}\) \(\dfrac{\mu_0}{4\pi}\)
Dipole moment \(\vec{p} = q\,(2\vec{l}\,)\), along (−ve) \(\rightarrow\) (+ve) charge \(\vec{m} = q_m (2\vec{l}\,)\) (bar magnet), along S \(\rightarrow\) N pole
Force \(\vec{F} = q\vec{E}\) \(\vec{F} = q_m \vec{B}\)
Energy (in external field) of a dipole \(U = -\,\vec{p}\cdot\vec{E}\) \(U = -\,\vec{m}\cdot\vec{B}\)
Coulomb’s law \(F = \left(\dfrac{1}{4\pi\varepsilon_0}\right)\dfrac{q_1 q_2}{r^{2}}\) No analogous law, as magnetic monopoles do not exist
Axial field for a short dipole \(\dfrac{2p}{4\pi\varepsilon_0 r^{3}}\), along \(\vec{p}\) \(\dfrac{\mu_0\,2\vec{m}}{4\pi r^{3}}\)
Equatorial field for a short dipole \(\dfrac{p}{4\pi\varepsilon_0 r^{3}}\), opposite to \(\vec{p}\) \(\dfrac{-\mu_0 \vec{m}}{4\pi r^{3}}\)

Read the table one row at a time. Each row says: whatever role the electrostatic quantity plays in the left-hand column, the magnetic quantity plays in the right-hand column. The single row that has no partner is Coulomb’s law — and that gap is not an oversight, it is the physics. There is no law of force between two isolated magnetic poles, because isolated magnetic poles do not exist.

Translating the Two Electric Results

You have studied the electric field due to an electric dipole of length \(2l\) (with \(p = 2ql\)) at a distance \(r\) along the dipolar axis (Eq. 10.24), which is given by

\[\left|\vec{E}_a\right| = \frac{1}{4\pi\varepsilon_0}\,\frac{2p}{r^{3}}, \qquad r \gg l\]

The electric field on the equator (Eq. 10.28) is antiparallel to \(\vec{p}\) and is given by

\[\left|\vec{E}_{eq}\right| = \frac{1}{4\pi\varepsilon_0}\,\frac{p}{r^{3}}, \qquad r \gg l\]

Now apply the dictionary in Table 12.1. Replace \(\vec{p}\) by \(\vec{m}\), and replace the electrostatic constant \(\dfrac{1}{4\pi\varepsilon_0}\) by the magnetic constant \(\dfrac{\mu_0}{4\pi}\). Nothing else changes — not the powers of \(r\), not the factor of 2, not the directions.

Using the analogy given in Table 12.1, we can thus write the axial magnetic field of a bar magnet at a distance \(r\), with \(r \gg l\), where \(2l\) is the length of the bar magnet:

\[\vec{B}_a = +\,\frac{\mu_0}{4\pi}\,\frac{2\vec{m}}{r^{3}} \qquad \text{--- (12.3)}\]

Similarly, the equatorial magnetic field is

\[\vec{B}_{eq} = -\,\frac{\mu_0}{4\pi}\,\frac{\vec{m}}{r^{3}} \qquad \text{--- (12.4)}\]

In both equations, \(\vec{m}\) is the magnetic dipole moment in A m², \(r\) is the distance of the point from the centre O of the magnet in metre (m), \(\mu_0\) is the permeability of free space, and \(B\) comes out in tesla (T), or equivalently Wb/m².

The negative sign in Eq. (12.4) shows that the direction of \(\vec{B}_{eq}\) is opposite to \(\vec{m}\). On the axis, the field points along the dipole moment — that is, in the S-to-N direction. On the equator, the field points against the dipole moment. This sign is not decoration; it is the answer to any question that asks for the direction of the field at an equatorial point, and it must not be dropped.

Q. Both Eq. (12.3) and Eq. (12.4) carry the condition \(r \gg l\). What breaks if the point P is close to the magnet?

A. These are short-dipole results — they are obtained by treating the magnet as a point dipole, which is only legitimate when the observation distance is far larger than the magnet’s own length. Close to the magnet, the two poles are “seen” separately, the exact expressions contain terms in \((r^2 - l^2)\) and \((r^2 + l^2)\), and the simple \(1/r^{3}\) dependence no longer holds. In exam problems, the phrase “short magnetic dipole” is precisely the signal that Eqs. (12.3) and (12.4) may be used.

The Factor of Two

Compare the magnitudes of Eq. (12.3) and Eq. (12.4). At the same distance \(r\) from the centre O of a bar magnet, the axial field carries a factor of 2 that the equatorial field does not. Hence

\[B_{axis} = 2B_{eq} \qquad \text{--- (12.5)}\]

NotePrinciple / Law

For a short bar magnet, at the same distance \(r\) from its centre (\(r \gg l\)), the magnitude of the magnetic field at an axial point is twice that at an equatorial point: \(B_{axis} = 2B_{eq}\). The axial field is directed along \(\vec{m}\), the equatorial field opposite to \(\vec{m}\).

Q4. At the same distance from a bar magnet, the field on its axis is exactly twice the field on its equator. Why twice?

A. On the axis, the point is nearer to one pole and further from the other, so the two poles’ contributions point the same way and add, and the difference in their distances is what survives. On the equator, the point is equally far from both poles, so their contributions are equal in size but partly cancel, leaving only the components along the magnet. Working the geometry out for \(r \gg l\), the surviving axial contribution comes to \(\mu_0 2m / 4\pi r^{3}\) and the equatorial one to \(\mu_0 m / 4\pi r^{3}\). The 2 is a piece of geometry, not a coincidence, and it appears in the electric dipole case for exactly the same reason.

A short bar magnet has a magnetic dipole moment of 0.8 A m². Calculate the magnitude of its magnetic field at a point 40 cm from its centre (i) on its axis and (ii) on its equator. Verify that your two answers are in the ratio 2 : 1, and state the direction of the field in each case relative to \(\vec{m}\). (Take \(\mu_0 = 4\pi \times 10^{-7}\) SI units.)

12.3.2 Magnetic field due to a bar magnet at an arbitrary point

The Trick: Split the Dipole, Not the Point

So far we can find the field only at two very special places — straight out along the axis, or straight out along the equator. But a point in space is usually neither. It sits somewhere off to one side, at some angle to the magnet.

Fig. 12.7 shows a bar magnet of magnetic moment \(\vec{m}\) with centre at O. P is any point in its magnetic field, at a distance \(r\) from O, and the line OP makes an angle \(\theta\) with \(\vec{m}\).

Diagram

Fig. 12.7 – Bar magnet with centre O and moment m; a point P at distance r making angle θ with m; m resolved into m cos θ along r and m sin θ perpendicular to r; the resultant field B at P making angle α with r.

The trick is to leave the point alone and split the magnet instead. The magnetic moment \(\vec{m}\) is resolved (about the centre of the magnet) into two components: one along \(\vec{r}\) and one perpendicular to \(\vec{r}\). The words in italics matter — both components are imagined to sit at the same centre O, so both are at the same distance \(r\) from P.

Now look at what P has become. For the component \(m\cos\theta\) along \(\vec{r}\), the point P lies straight out along that component’s own direction — so with respect to this component, P is an axial point. And for the component \(m\sin\theta\) perpendicular to \(\vec{r}\), the point P lies at right angles to that component — so with respect to it, P is an equatorial point, at the same distance \(\vec{r}\).

Q. How can one point P be an axial point and an equatorial point at the same time?

A. It cannot be both for the same dipole — but it can be axial for one component and equatorial for the other. We have replaced a single dipole \(\vec{m}\) by two perpendicular dipoles \(m\cos\theta\) and \(m\sin\theta\) placed at the same centre, which is legitimate because the magnetic moment is a vector and adds like one. P happens to be axial for the first and equatorial for the second. This is the standard way of extending a result derived for two special positions to every position.

Adding the Two Fields

Using the results of axial and equatorial fields — Eqs. (12.3) and (12.4) — we get, for the component along \(\vec{r}\),

\[B_a = \frac{\mu_o}{4\pi}\,\frac{2m\cos\theta}{r^{3}} \qquad \text{--- (12.6)}\]

directed along \(m\cos\theta\), and for the perpendicular component,

\[B_{eq} = \frac{\mu_o}{4\pi}\,\frac{m\sin\theta}{r^{3}} \qquad \text{--- (12.7)}\]

directed opposite to \(m\sin\theta\).

Here \(m\) is the magnetic dipole moment in A m², \(r\) is the distance OP in metre, \(\theta\) is the angle between \(\vec{m}\) and \(\vec{r}\), and both fields are in tesla.

These two fields are at right angles to each other, because the two components of \(\vec{m}\) that produced them are at right angles. Perpendicular vectors are combined by Pythagoras’ theorem, not by simple addition. Thus, the magnitude of the resultant magnetic field \(B\) at point P is given by

\[B = \sqrt{B_a^{\,2} + B_{eq}^{\,2}}\]

\[\therefore\ B = \frac{\mu_o}{4\pi}\,\frac{m}{r^{3}}\,\sqrt{\left[2\cos\theta\right]^{2} + \left[\sin\theta\right]^{2}}\]

The common factor \(\dfrac{\mu_o}{4\pi}\dfrac{m}{r^{3}}\) has simply been taken outside the square root, leaving the angle-dependent part inside. That part now simplifies. Since \(4\cos^{2}\theta + \sin^{2}\theta = 3\cos^{2}\theta + \left(\cos^{2}\theta + \sin^{2}\theta\right)\), and \(\cos^{2}\theta + \sin^{2}\theta = 1\), we obtain

\[\therefore\ B = \frac{\mu_o}{4\pi}\,\frac{m}{r^{3}}\,\sqrt{3\cos^{2}\theta + 1} \qquad \text{--- (12.8)}\]

Equation (12.8) is the general result, and the two special cases fall straight out of it. Put \(\theta = 0^\circ\) (an axial point): \(\sqrt{3 + 1} = 2\), giving the axial field with its factor of 2. Put \(\theta = 90^\circ\) (an equatorial point): \(\sqrt{0 + 1} = 1\), giving the equatorial field. A good way to check any answer is to see that it collapses correctly at these two angles.

The Direction of the Resultant Field

Knowing the magnitude is only half the answer; we also need to know which way \(\vec{B}\) points at P.

Let \(\alpha\) be the angle made by the direction of \(\vec{B}\) with \(\vec{r}\). Since \(B_a\) lies along \(\vec{r}\) and \(B_{eq}\) lies perpendicular to it, the tangent of the angle between the resultant and \(\vec{r}\) is the ratio of the perpendicular component to the along-\(\vec{r}\) component. Then, by using Eq. (12.6) and Eq. (12.7),

\[\tan\alpha = \frac{B_{eq}}{B_a} = \frac{1}{2}\left(\tan\theta\right) \qquad \text{--- (12.9)}\]

because the factor \(\dfrac{\mu_o}{4\pi}\dfrac{m}{r^{3}}\) cancels between numerator and denominator, and \(\dfrac{\sin\theta}{2\cos\theta} = \dfrac{1}{2}\tan\theta\).

Notice what Eq. (12.9) is telling you: except at \(\theta = 0^\circ\) and \(\theta = 90^\circ\), the magnetic field at P is not along OP, and it is not along \(\vec{m}\) either. It leans somewhere in between. The angle between the directions of \(\vec{B}\) and \(\vec{m}\) is then \(\left(\theta + \alpha\right)\).

This general formula is what makes magnetic surveying possible. When archaeologists or engineers sweep a magnetometer over the ground to locate buried iron objects or old pipelines, the buried object behaves as a small magnetic dipole, and the instrument reads a field whose strength and direction depend on the angle \(\theta\) at which the sensor happens to sit relative to that dipole. Interpreting the reading — turning a wiggle on a screen into “there is something metallic 1.5 m below and slightly to your left” — is an exercise in Eqs. (12.8) and (12.9).

A short magnetic dipole has a magnetic moment of 0.5 A m². Calculate its magnetic field at a distance of 20 cm from the centre of the magnetic dipole, on (i) the axis and (ii) the equatorial line. (Given \(\mu_0 = 4\pi \times 10^{-7}\) SI units)

Solution:

\[m = 0.5\ \text{A m}^{2}, \quad r = 20\ \text{cm} = 0.2\ \text{m}\]

The point lies on the axis, so we use the axial field of a short dipole, Eq. (12.3). Note that \(\dfrac{\mu_0}{4\pi} = 10^{-7}\) in SI units, which is why \(10^{-7}\) appears in the working.

\[B_a = \frac{\mu_0}{4\pi}\,\frac{2m}{r^{3}} = \frac{10^{-7} \times 2 \times 0.5}{(0.2)^{3}} = \frac{1 \times 10^{-7}}{8 \times 10^{-3}}\]

\[= \frac{1}{8} \times 10^{-4} = 1.25 \times 10^{-5}\ \text{Wb/m}^{2}\]

For the equatorial point at the same distance we use Eq. (12.4), which is identical except that the factor 2 is absent.

\[B_{eq} = \frac{\mu_0}{4\pi}\,\frac{m}{r^{3}} = \frac{10^{-7} \times 0.5}{(0.2)^{3}} = \frac{5 \times 10^{-8}}{8 \times 10^{-3}} = 0.625 \times 10^{-5}\ \text{Wb/m}^{2}\]

The two answers are in the ratio 2 : 1, exactly as Eq. (12.5) requires — a quick self-check you should perform whenever a problem asks for both.

A short bar magnet of magnetic moment 1.2 A m² has its centre at O. A point P lies at a distance of 30 cm from O, such that OP makes an angle of \(60^\circ\) with the direction of \(\vec{m}\). Calculate (i) the magnitude of the magnetic field at P, and (ii) the angle \(\alpha\) that this field makes with OP. (Take \(\mu_0 = 4\pi \times 10^{-7}\) SI units.)

12.4 Gauss’ Law of Magnetism

The Electric Law You Already Know

The Gauss’ law for the electric field is known to you. It states that the net electric flux through a closed Gaussian surface is proportional to the net electric charge enclosed by the surface (Eq. 10.18):

\[\Phi_E = \int \vec{E} \cdot \vec{dS} = \frac{q}{\varepsilon_0}\]

where \(q\) is the net charge enclosed by the surface, \(\Phi_E\) is the electric flux, and \(\varepsilon_0\) is the permittivity of free space. A Gaussian surface is simply an imaginary closed surface — a bag drawn in space — through which we count field lines. Lines going out count positive, lines coming in count negative.

The law says something with a clear physical meaning: flux comes out of a closed surface only if there is a source of field lines sitting inside it. Enclose a positive charge, and lines stream outward. Enclose nothing, and every line that enters must also leave.

The Magnetic Law

The Gauss’ law for magnetic fields states that the net magnetic flux \(\Phi_B\) through a closed Gaussian surface is zero, i.e.,

\[\Phi_B = \int \vec{B} \cdot \vec{dS} = 0 \qquad \text{(Gauss' law for magnetic fields)}\]

Here \(\vec{B}\) is the magnetic field in tesla, \(\vec{dS}\) is a vector element of area on the closed surface directed along the outward normal, in m², and \(\Phi_B\) is the magnetic flux in weber. Note the word closed — the law is about a surface that completely encloses a region of space, not about an open sheet.

NotePrinciple / Law

Gauss’ law of magnetism: the net magnetic flux through any closed Gaussian surface is always zero, \(\Phi_B = \displaystyle\int \vec{B}\cdot\vec{dS} = 0\).

The magnitude of this statement is easy to miss, because it is a statement about nothing. It does not say the flux is small, or usually zero, or zero on average. It says that for every closed surface you can possibly draw — around a magnet, inside a magnet, around a whole galaxy — the answer is exactly zero.

Why It Is Zero: Reading the Three Figures

The magnetic force lines of (a) a bar magnet, (b) a current carrying finite solenoid, and (c) an electric dipole are shown in Fig. 12.8 (a), 12.4 (b) and 12.4 (c) respectively. The curves labelled (i) and (ii) are cross sections of three-dimensional closed Gaussian surfaces.

Diagram

Fig. 12.8 (a) – Bar magnet with its closed field lines, and two closed Gaussian surfaces marked (i) and (ii) in cross-section, surface (ii) enclosing the N pole region.

Diagram

Fig. 12.8 (b) – Current (I) carrying solenoid with its field lines, and closed Gaussian surfaces (i) and (ii) shown in cross-section.

Diagram

Fig. 12.8 (c) – Electric dipole with field lines running from the positive to the negative charge, and closed Gaussian surfaces (i) and (ii), surface (ii) enclosing only the positive charge.

First take surface (i). If we compare the number of lines of force entering in and leaving out of the surface (i), it is clearly seen that they are equal. The Gaussian surface does not include poles. It means that the flux associated with any closed surface is equal to zero. Every line that dives into the bag comes out somewhere else, so the ins and the outs cancel exactly.

Now take surface (ii) in Fig. 12.8 (b), which is more interesting, because here we are deliberately enclosing the North pole. Surely now we have caught a source of field lines, and the flux should be non-zero?

It is not. As even a thin slice of a bar magnet will have North and South poles associated with it, the closed Gaussian surface will also include a South pole. You cannot bag a north pole by itself. Cut the surface as finely as you like around the end of the magnet: whatever piece of magnetic material you have enclosed is itself a complete little magnet, with a south pole inside your bag supplying exactly as many lines as the north pole removes. The flux is zero again.

Contrast this with Fig. 12.8 (c), for an electric dipole. There the field lines begin from the positive charge and end on the negative charge. For the closed surface (ii), there is a net outward flux, since it does include a net (positive) charge. According to the Gauss’ law of electrostatics as studied earlier,

\[\Phi_E = \int \vec{E} \cdot \vec{dS} = \frac{q}{\varepsilon_0}\]

where \(q\) is the positive charge enclosed. Here the bag can be drawn around one charge alone, because a single charge is a thing that exists on its own.

Thus, the situation is entirely different from magnetic lines of force, which are shown in Fig. 12.8 (a) and Fig. 12.8 (b). Thus, Gauss’ law of magnetism can be written as

\[\Phi_B = \int \vec{B}\cdot\vec{dS} = 0\]

Q3. Why is the total magnetic flux passing through any closed surface always exactly zero, no matter where you draw the surface?

A. Because magnetic field lines have no beginning and no end — they are closed loops. A closed loop that enters a closed surface must, sooner or later, come back out of it. So the inward and outward flux always cancel completely. The only way to get a non-zero answer would be to trap a lone source of field lines inside the surface, that is, a magnetic monopole. Since no monopole exists, no closed surface anywhere in the universe has ever been found with a net magnetic flux through it.

What the Law Really Tells Us

From the above we conclude that, for electrostatics, an isolated electric charge exists but an isolated magnetic pole does not exist. In short, only dipoles exist in the case of magnetism.

This is the whole content of Gauss’ law of magnetism, compressed into one line. The law is not really a statement about flux at all — it is a statement about monopoles, written in the language of flux. The day a magnetic monopole of pole strength \(q_m\) is discovered, the right-hand side of the law will stop being zero, and every physics textbook will be rewritten.

Q. Both surfaces (i) and (ii) give zero flux. Does that mean the magnetic field itself is zero on those surfaces?

A. Certainly not. The field can be strong at every point of the surface — near the pole of the magnet it is very strong indeed. Zero net flux only means the lines that go in are exactly matched by the lines that come out. Flux is a signed total over the whole closed surface, not a measure of how big the field is at any one place. Confusing “zero flux” with “zero field” is a classic error in reasoning questions.

In 1931, Paul Dirac showed that if even a single magnetic monopole existed anywhere in the universe, it would neatly explain why electric charge always comes in whole multiples of a fixed amount. That was a powerful motive to go looking. Physicists have searched ever since — in rocks, in moon dust, in cosmic rays, and today in the debris of collisions at particle accelerators. Not one monopole has ever been confirmed. Gauss’ law of magnetism has held perfectly, every time it has been tested.

Engineers who design transformers and electromagnets rely on this law in a very practical way. Because no magnetic flux can be created or destroyed inside a closed surface, all the flux that leaves one part of an iron core must return through another. Magnetic circuits are therefore analysed like closed plumbing loops: whatever flux flows out must flow back. Any flux that appears to “go missing” from the core has not vanished — it has leaked into the surrounding air, and the designer must account for it.

A closed cylindrical Gaussian surface of radius 5.0 cm and length 20 cm is placed in a uniform magnetic field of 0.30 T, with the axis of the cylinder parallel to the field. Calculate the magnetic flux through (i) the flat face at which the field enters, (ii) the flat face at which the field leaves, (iii) the curved surface, and (iv) the entire closed surface. State whether your answer to (iv) agrees with Gauss’ law of magnetism.

12.5 Earth’s Magnetism

The Evidence That the Earth Is Magnetic

It is common experience that a bar magnet or a magnetic needle suspended freely in air always aligns itself along the geographic N–S direction. That alone is a clue: something must be exerting a torque on the needle, turning it until it settles.

There is a second, less familiar observation. If the needle also has the freedom to rotate about a horizontal axis, it does not stay level. It inclines at some angle with the horizontal, tilting in the vertical N–S plane.

This fact clearly indicates that there is some magnetic field present everywhere on the Earth. This is called Terrestrial Magnetism. It is extremely useful during navigation — for centuries, it was the only direction-finder available to a sailor out of sight of land.

The magnetic parameters of the Earth are described below. The magnetic lines of force enter the Earth’s surface at the north pole and emerge from the south pole.

One convention must be fixed before we go further, because almost every mistake in this section comes from ignoring it. Unless and otherwise stated, the directions mentioned (South, North, etc.) are always Geographic. When a magnetic direction is meant, it will be said so explicitly.

Diagram

Fig. 12.9 – The Earth shown as a huge tilted bar magnet, marking the geographic North and South poles (Ng, Sg), the geomagnetic North and South poles (Nm, Sm), the magnetic axis MM′, the rotation axis RR′, and the magnetic equatorial circle AA′.

The Earth as a Huge Magnet

Magnetic Axis: The Earth is considered to be a huge magnet. The magnetic north pole (N) of the Earth is located below Antarctica, while the south pole (S) is below north Canada. The straight line NS joining these two poles is called the magnetic axis, MM′, as shown in Fig. 12.9.

Read that sentence twice, because it sounds wrong the first time. The magnet’s north pole is down south, and its south pole is up north. But this is exactly what it must be. The north pole of a compass needle is attracted towards the geographic north — and unlike poles attract. So whatever lies beneath northern Canada has to be the south pole of the Earth’s internal magnet. The place we casually call the “magnetic north pole” of the Earth, on maps, is magnetically a south pole.

Q. If the Earth’s magnetic south pole is up in the north, why does everyone call that spot the “north magnetic pole”?

A. Out of habit and convenience — it is named after the direction it lies in, not after its magnetic polarity. Navigators needed a name for the place their compass pointed towards, and they called it magnetic north. Physics, however, must be consistent: since the needle’s N pole is attracted there, the Earth’s own pole buried there behaves as an S pole. Keep the two usages apart, and Fig. 12.9 will never confuse you.

Magnetic equator: A great circle in the plane perpendicular to the magnetic axis is the magnetic equatorial circle, AA′. It happens to pass through India, near Thiruvananthapuram. A compass carried to Thiruvananthapuram therefore sits almost perfectly flat — a fact we will explain in a moment.

Meridians: Two Vertical Planes That Do Not Quite Coincide

The next two definitions are planes, not lines, and both are vertical planes standing on the Earth’s surface.

Geographic Meridian: A plane perpendicular to the surface of the Earth (a vertical plane), perpendicular to the geographic axis, is the geographic meridian (Fig. 12.10). This is the plane that contains true, geographic North at your location.

Magnetic Meridian: A plane perpendicular to the surface of the Earth (a vertical plane) and passing through the magnetic axis is the magnetic meridian. The direction of the resultant magnetic field of the Earth is always along or parallel to the magnetic meridian (Fig. 12.10).

That last sentence is what makes the magnetic meridian worth defining at all. The Earth’s field at your location has no component sticking out sideways from this plane — the whole of \(\vec{B}\) lies in it. So the plane fixes the vertical sheet within which everything about the Earth’s field at that place happens.

NoteDefinition

The magnetic meridian at a place is the vertical plane passing through the magnetic axis of the Earth. The Earth’s resultant magnetic field at that place lies along or parallel to this plane.

Magnetic Declination

Since the magnetic axis and the geographic axis are not the same line, the two meridians at a place do not generally coincide. The angle between them measures the mismatch.

Magnetic declination: The angle between the geographic and the magnetic meridian at a place is called ‘magnetic declination’ (\(\alpha\)), shown in Fig. 12.10.

Diagram

Fig. 12.10 – The geographic meridian and the magnetic meridian shown as two vertical planes with the angle of declination α between them; the Earth’s field B at the place resolved into its horizontal component B_H and vertical component B_V, with the angle of dip φ marked.

NoteDefinition

Magnetic declination (\(\alpha\)) at a place is the angle between the geographic meridian and the magnetic meridian at that place.

The declination is small in India. It is 0° 58′ west at Mumbai and 0° 41′ east at Delhi. Thus, at both these places, the magnetic needle shows true North accurately (Fig. 12.10). Fifty-eight minutes of arc is less than one degree — an error too small to matter for anyone finding their way on land.

Q5. A compass in Mumbai does not point to the true geographic north. How large is the error, and what causes it?

A. The error is the magnetic declination, and at Mumbai it is 0° 58′ west — under one degree. It arises because the compass needle aligns itself with the magnetic meridian, which is set by the Earth’s magnetic axis, while true north is set by the Earth’s geographic (rotational) axis. The two axes do not coincide, so the two meridians make a small angle with each other. At Delhi the declination is 0° 41′, but east rather than west. At both places the needle is close enough to true north to be trusted for ordinary purposes.

Magnetic Inclination, or Angle of Dip

Magnetic inclination or angle of dip (\(\phi\)): The angle made by the direction of the resultant magnetic field with the horizontal at a place is the inclination or angle of dip at that place (Fig. 12.11).

Diagram

Fig. 12.11 – A dip circle: a compass needle free to rotate in a vertical plane, with the horizontal marked 0° and the needle tilted below it through the angle of dip φ, pointing along the Earth’s magnetic field B.

NoteDefinition

The angle of dip (\(\phi\)) at a place is the angle between the direction of the Earth’s resultant magnetic field at that place and the horizontal.

Declination and dip together describe where the Earth’s field points at your location: declination is measured in the horizontal plane, dip is measured out of it. One is a left-right error, the other an up-down tilt.

Resolving the Earth’s Field

Earth’s magnetic field: The magnetic force experienced per unit pole strength is the magnetic field \(\vec{B}\) at that place. It can be resolved into components along the horizontal, \(\vec{B}_H\), and along the vertical, \(\vec{B}_V\). The vertical component can be conveniently determined.

Since \(\phi\) is the angle that \(\vec{B}\) makes with the horizontal, the two components follow from simple trigonometry of a right-angled triangle whose hypotenuse is \(B\):

\[B_H = B\cos\phi, \qquad B_V = B\sin\phi\]

Dividing the second by the first, \(B\) cancels and we are left with a relation between the components and the dip alone:

\[\frac{B_V}{B_H} = \tan\phi \qquad \text{--- (12.10)}\]

Squaring and adding the two components instead, and using \(\cos^2\phi + \sin^2\phi = 1\):

\[B^{2} = B_V^{\,2} + B_H^{\,2}\]

\[\therefore\ B = \sqrt{B_V^{\,2} + B_H^{\,2}} \qquad \text{--- (12.11)}\]

Here \(B\) is the magnitude of the Earth’s resultant magnetic field at the place, \(B_H\) its horizontal component and \(B_V\) its vertical component, all in tesla (T), and \(\phi\) is the angle of dip.

Equations (12.10) and (12.11) are the working tools of this section. Between them, they let you find any one of \(B\), \(B_H\), \(B_V\), \(\phi\) from any two of the others — which is exactly what numerical problems on terrestrial magnetism ask you to do. The three quantities \(B_H\), \(\alpha\) and \(\phi\) are called the magnetic elements of the Earth at a place, and together they fix the field completely.

Special Cases

1) At the magnetic North pole, \(\vec{B} = \vec{B}_V\), directed upward, \(\vec{B}_H = 0\) and \(\phi = 90^{0}\).

2) At the magnetic south pole, \(\vec{B} = \vec{B}_V\), directed downward, \(\vec{B}_H = 0\) and \(\phi = 270^{0}\).

3) Anywhere on the magnetic great circle (magnetic equator), \(B = B_H\) along South to North, \(B_V = 0\) and \(\phi = 0\).

Look at what these three cases say physically. At the magnetic poles the field is purely vertical, so a dip needle stands on end and an ordinary horizontal compass is useless — it has no horizontal field left to align with. On the magnetic equator the field is purely horizontal, so the dip needle lies perfectly flat. Everywhere in between, the field tilts, and the dip grows steadily as you move from the magnetic equator towards a magnetic pole.

Q6. Why does a freely pivoted magnetic needle stay horizontal near the equator but tilt steeply downwards as you carry it towards the poles?

A. Because the needle aligns itself with the Earth’s resultant field, and the direction of that field changes with latitude. On the magnetic equator the field runs horizontally, from South to North, so \(B_V = 0\) and the dip is \(\phi = 0\) — the needle lies flat. As you move towards a magnetic pole, the field lines curve into the Earth, the vertical component \(B_V\) grows and the horizontal component \(B_H\) shrinks. Since \(\tan\phi = B_V/B_H\), the angle of dip increases. At the magnetic pole itself \(B_H = 0\), the field is entirely vertical, and the needle stands upright with \(\phi = 90^\circ\).

Magnetic Maps of the Earth

Magnetic maps of the Earth: The magnetic elements of the Earth (\(B_H\), \(\alpha\) and \(\phi\)) vary from place to place and also with time. The maps providing these values at different locations are called magnetic maps. These are extremely useful for navigation. Magnetic maps drawn by joining places with the same value of a particular element are called Iso-magnetic charts.

Three such families of lines are named:

  • Lines joining the places of equal horizontal components (\(B_H\)) are known as Isodynamic lines.
  • Lines joining the places of equal declination (\(\alpha\)) are called Isogonic lines.
  • Lines joining the places of equal inclination or dip (\(\phi\)) are called Aclinic lines.

The phrase “also with time” in the definition above is not a formality. The Earth’s magnetic north pole has been drifting steadily out of the Canadian Arctic towards Siberia, and in recent decades it sped up markedly. The drift became fast enough that the World Magnetic Model — the reference used by ships, aircraft and smartphone compasses — had to be updated ahead of its normal five-year schedule so that navigation systems would not accumulate errors. The Earth’s magnetism is a living, shifting thing, not a fixed set of numbers.

Aircraft runways are numbered by their magnetic heading, rounded to the nearest ten degrees: a runway pointing along a magnetic bearing of 268° is painted “27”. Because declination changes slowly over the years, that painted number can drift out of date, and airports around the world have had to repaint their runways and update their charts when the magnetic heading crosses the rounding boundary. An abstract-sounding quantity, \(\alpha\), ends up as a job for a painting crew.

The Earth’s magnetic field at the equator is approximately \(4 \times 10^{-5}\) T. Calculate the Earth’s dipole moment. (Radius of Earth \(= 6.4 \times 10^{6}\) m, \(\mu_0 = 4\pi \times 10^{-7}\) SI units)

Solution: Given

\[B_{eq} = 4 \times 10^{-5}\ \text{T}\]

\[r = 6.4 \times 10^{6}\ \text{m}\]

Assume that the Earth is a bar magnet with N and S poles being the geographical South and North poles, respectively. A point on the Earth’s magnetic equator is an equatorial point of this dipole, at a distance equal to the radius of the Earth from its centre. So the equatorial magnetic field due to the Earth’s dipole can be written using Eq. (12.4):

\[B_{eq} = \frac{\mu_0 m}{4\pi r^{3}}\]

Rearranging to make \(m\) the subject:

\[m = 4\pi B_{eq} \times r^{3} / \mu_0\]

\[= 4 \times 10^{-5} \times (6.4 \times 10^{6})^{3} \times 10^{7}\]

\[= 1.05 \times 10^{20}\ \text{A m}^{2}\]

At a given place on the Earth, a bar magnet of magnetic moment \(\vec{m}\) is kept horizontal in the East–West direction. P and Q are the two neutral points due to the magnetic field of this magnet, and \(\vec{B}_H\) is the horizontal component of the Earth’s magnetic field.

(A) Calculate the angles between the position vectors of P and Q with the direction of \(\vec{m}\).

(B) Points P and Q are 1 m from the centre of the bar magnet and \(B_H = 3.5 \times 10^{-5}\) T. Calculate the magnetic dipole moment of the bar magnet.

A neutral point** is that point where the resultant magnetic field is zero.**

Solution: (A) As seen from the figure, the direction of the magnetic field \(\vec{B}\) due to the bar magnet is opposite to \(\vec{B}_H\) at the points P and Q. That is the only way the two fields can cancel and leave zero. Also, \(\left(\theta + \alpha\right) = 90^{0}\) at P and it is \(270^{0}\) at Q.

Recall from Eq. (12.9) that \(\alpha\), the angle of the magnet’s field with the position vector, satisfies

\[\tan\alpha = \frac{1}{2}\tan\theta\]

\[\therefore\ \tan\theta = 2\tan\alpha\]

\[= 2\tan\left(90 - \theta\right) \text{ and } 2\tan\left(270 - \theta\right)\]

\[\therefore\ \tan\theta = \pm\,2\cot\theta\]

\[\therefore\ \tan^{2}\theta = 2\]

\[\therefore\ \tan\theta = \pm\sqrt{2}\]

\[\therefore\ \theta = \tan^{-1}\left(\pm\sqrt{2}\right)\]

\[\therefore\ \theta = 54^{0}44' \text{ and } 180^{0} - 54^{0}44' = 116^{0}4'\]

Diagram

Fig. 12.12 – Bar magnet placed horizontally along East–West with centre O and moment m; the two neutral points P and Q shown with their position vectors, the angles θ and α marked at each, and the direction of B_H indicated.

  1. From part (A), \(\tan^{2}\theta = 2\). Using the identity \(\sec^{2}\theta = 1 + \tan^{2}\theta\):

\[\tan^{2}\theta = 2\]

\[\therefore\ \sec^{2}\theta = 1 + \tan^{2}\theta = 1 + 2 = 3\]

\[\therefore\ \cos^{2}\theta = \frac{1}{3}\]

\[r = 1\,\text{m and } B = B_H = 3.5 \times 10^{-5}\ \text{T} \qquad \text{(Given)}\]

At a neutral point the magnet’s own field has magnitude equal to \(B_H\), so we may put \(B = B_H\) into the general arbitrary-point formula, Eq. (12.8):

\[B = \frac{\mu_0}{4\pi}\,\frac{m}{r^{3}}\,\sqrt{3\cos^{2}\theta + 1}\]

\[\therefore\ m = \frac{B_H \times r^{3}}{\left(\dfrac{\mu_0}{4\pi}\right)\sqrt{3\cos^{2}\theta + 1}}\]

\[= \frac{3.5 \times 10^{-5} \times 1^{3}}{10^{-7} \times \sqrt{3 \cdot \dfrac{1}{3} + 1}}\]

\[\therefore\ m = \frac{350}{\sqrt{2}} = 247.5\ \text{A m}^{2}\]

Always remember: In this Chapter we have used \(B\) as a symbol for the magnetic field. Calling it magnetic induction is unreasonable. We have used the words magnetic field, which are the words used in spoken language.

At a certain place, the horizontal component of the Earth’s magnetic field is \(3.2 \times 10^{-5}\) T and the angle of dip is \(60^\circ\). Calculate (i) the vertical component of the Earth’s magnetic field and (ii) the magnitude of the Earth’s resultant magnetic field at that place.

12.6 Summary

The Nature of Magnetism

  1. Magnetism has been known since before 600 B.C., but it was understood, and turned into technology, only in the twentieth century. William Gilbert first studied it systematically and found that the Earth is a weak magnet; Hans Oersted suggested a link between electricity and magnetism; James Clerk Maxwell proved that electricity and magnetism are different aspects of the same fundamental force field.

  2. Every magnet, whatever its size or shape, has two poles. Like poles repel, unlike poles attract. A freely suspended bar magnet or magnetic needle aligns itself along the geographic North–South direction.

  3. Breaking a magnet does not separate its poles. Each piece becomes a complete, somewhat weaker magnet with its own N and S poles. Isolated magnetic monopoles do not exist — magnetism always presents itself as a dipole (or a quadrupole).

Magnetic Lines of Force
  1. Outside a magnet, magnetic lines of force run from the north pole to the south pole; inside, they run from S to N, so the lines form closed loops. Electric lines of force, by contrast, begin on a positive charge and end on a negative charge without closing.
  2. The tangent to a line of force at a point gives the direction of the net magnetic field \(\vec{B}\) there.
  3. The number of lines per unit area decides the magnitude of \(\vec{B}\): crowded lines mean a strong field.
  4. Lines of force never intersect, since two tangents at one point would give the field two directions there.
  5. Lines of force are only a crude, pictorial representation. The magnetic field is better defined through the Lorentz force law.
Magnetic Flux and Magnetic Field
  1. The number of lines of force is the magnetic flux \(\phi\). SI unit: weber (Wb).
  2. For a uniform magnetic field normal to a finite area \(A\):

\[B = \frac{\phi}{A} \qquad \text{--- (12.1)}\]

  1. SI unit of magnetic field \(B\): weber/m², called the tesla (T).

\[1\ \text{Tesla} = 10^{4}\ \text{Gauss}\]

The Bar Magnet as a Magnetic Dipole
  1. A bar magnet has pole strength \(+q_m\) at its N pole and \(-q_m\) at its S pole, separated inside the magnet by a distance \(2l\). Being two equal and opposite poles, it is a magnetic dipole.
  2. Magnetic dipole moment:

\[\vec{m} = q_m \cdot 2\vec{l}\]

where \(2\vec{l}\) is directed from the south pole to the north pole.

  1. SI unit of pole strength \(q_m\): A m. SI unit of magnetic dipole moment \(m\): A m².
  2. Axis: the line passing through both poles — there is only one.
  3. Equator: a line through the centre, perpendicular to the axis. All such lines form the equatorial plane; points on it equidistant from the centre form the equatorial circle.
  4. Magnetic length is the distance between the two poles, and is shorter than the geometric (ruler) length:

\[\text{Magnetic length } (2l) = \frac{5}{6} \times \text{Geometric length} \qquad \text{--- (12.2)}\]

Field of a Short Bar Magnet (\(r \gg l\))

  1. The electrostatic analogue: magnetism copies electrostatics with \(q \rightarrow q_m\), \(\vec{p} \rightarrow \vec{m}\), \(\vec{E} \rightarrow \vec{B}\) and \(\dfrac{1}{4\pi\varepsilon_0} \rightarrow \dfrac{\mu_0}{4\pi}\). The one electrostatic law with no magnetic counterpart is Coulomb’s law, because magnetic monopoles do not exist. Energy of a dipole in an external field: \(U = -\vec{m}\cdot\vec{B}\); force on a pole: \(\vec{F} = q_m\vec{B}\).

  2. Axial field:

\[\vec{B}_a = +\,\frac{\mu_0}{4\pi}\,\frac{2\vec{m}}{r^{3}} \qquad \text{--- (12.3)}\]

directed along \(\vec{m}\).

  1. Equatorial field:

\[\vec{B}_{eq} = -\,\frac{\mu_0}{4\pi}\,\frac{\vec{m}}{r^{3}} \qquad \text{--- (12.4)}\]

the negative sign showing that \(\vec{B}_{eq}\) is opposite to \(\vec{m}\).

  1. At the same distance \(r\) from the centre:

\[B_{axis} = 2B_{eq} \qquad \text{--- (12.5)}\]

Field at an Arbitrary Point

For a point P at distance \(r\) from the centre, with OP making an angle \(\theta\) with \(\vec{m}\), the moment is resolved about the centre into \(m\cos\theta\) along \(\vec{r}\) (for which P is axial) and \(m\sin\theta\) perpendicular to \(\vec{r}\) (for which P is equatorial):

\[B_a = \frac{\mu_o}{4\pi}\,\frac{2m\cos\theta}{r^{3}} \qquad \text{--- (12.6)}\]

directed along \(m\cos\theta\), and

\[B_{eq} = \frac{\mu_o}{4\pi}\,\frac{m\sin\theta}{r^{3}} \qquad \text{--- (12.7)}\]

directed opposite to \(m\sin\theta\). These two are perpendicular, so the resultant magnitude is

\[B = \sqrt{B_a^{\,2} + B_{eq}^{\,2}} = \frac{\mu_o}{4\pi}\,\frac{m}{r^{3}}\,\sqrt{3\cos^{2}\theta + 1} \qquad \text{--- (12.8)}\]

If \(\alpha\) is the angle made by \(\vec{B}\) with \(\vec{r}\), then

\[\tan\alpha = \frac{B_{eq}}{B_a} = \frac{1}{2}\left(\tan\theta\right) \qquad \text{--- (12.9)}\]

and the angle between \(\vec{B}\) and \(\vec{m}\) is \(\left(\theta + \alpha\right)\). Setting \(\theta = 0^\circ\) recovers the axial field, and \(\theta = 90^\circ\) the equatorial field.

Gauss’ Law of Magnetism
  1. The net magnetic flux through any closed Gaussian surface is zero:

\[\Phi_B = \int \vec{B}\cdot\vec{dS} = 0\]

  1. The reason is that magnetic field lines are closed loops: every line entering a closed surface must leave it. Even a closed surface drawn around the N pole of a magnet must enclose an S pole as well, since even a thin slice of a magnet has both poles.
  2. Compare with electrostatics, where \(\Phi_E = \displaystyle\int \vec{E}\cdot\vec{dS} = \dfrac{q}{\varepsilon_0}\) can be non-zero, because a surface can enclose a net charge.
  3. Conclusion: an isolated electric charge exists, but an isolated magnetic pole does not. Only dipoles exist in magnetism.
Earth’s Magnetism
  1. The Earth behaves as a huge magnet — this is terrestrial magnetism. Its magnetic north pole lies below Antarctica and its magnetic south pole below north Canada; the line joining them is the magnetic axis MM′. Unless stated otherwise, directions named (North, South …) are geographic.
  2. The magnetic equator is the great circle perpendicular to the magnetic axis; it passes through India near Thiruvananthapuram.
  3. Geographic meridian: the vertical plane perpendicular to the geographic axis. Magnetic meridian: the vertical plane passing through the magnetic axis; the Earth’s resultant field always lies along or parallel to it.
  4. Magnetic declination (\(\alpha\)): the angle between the geographic and the magnetic meridian. It is small in India — 0° 58′ west at Mumbai, 0° 41′ east at Delhi — so a needle there shows true North accurately.
  5. Magnetic inclination or angle of dip (\(\phi\)): the angle made by the Earth’s resultant field with the horizontal.
  6. Resolving the Earth’s field \(B\) into horizontal and vertical components:

\[B_H = B\cos\phi, \qquad B_V = B\sin\phi\]

\[\frac{B_V}{B_H} = \tan\phi \qquad \text{--- (12.10)}\]

\[B^{2} = B_V^{\,2} + B_H^{\,2} \quad \therefore\ B = \sqrt{B_V^{\,2} + B_H^{\,2}} \qquad \text{--- (12.11)}\]

  1. Special cases:

  2. At the magnetic North pole: \(\vec{B} = \vec{B}_V\), directed upward, \(\vec{B}_H = 0\), \(\phi = 90^{0}\).

  3. At the magnetic south pole: \(\vec{B} = \vec{B}_V\), directed downward, \(\vec{B}_H = 0\), \(\phi = 270^{0}\).

  4. Anywhere on the magnetic great circle (magnetic equator): \(B = B_H\) along South to North, \(B_V = 0\), \(\phi = 0\).

  5. The magnetic elements (\(B_H\), \(\alpha\), \(\phi\)) vary from place to place and with time. Maps of them are magnetic maps; maps joining places of equal value of an element are iso-magnetic charts:

  • Isodynamic lines — equal horizontal component \(B_H\).
  • Isogonic lines — equal declination \(\alpha\).
  • Aclinic lines — equal inclination or dip \(\phi\).
  1. A neutral point is a point where the resultant magnetic field is zero — the magnet’s field there exactly cancels the Earth’s horizontal component.
Worked Results to Remember
  1. A short dipole of \(m = 0.5\) A m² gives \(B_a = 1.25 \times 10^{-5}\) Wb/m² and \(B_{eq} = 0.625 \times 10^{-5}\) Wb/m² at \(r = 0.2\) m — a ratio of 2 : 1.
  2. Taking \(B_{eq} = 4 \times 10^{-5}\) T at the Earth’s equator and \(r = 6.4 \times 10^{6}\) m gives the Earth’s dipole moment as \(m = 1.05 \times 10^{20}\) A m².
  3. Note: in this chapter, \(B\) is called the magnetic field, not “magnetic induction”.

12.7 Check Your Concepts

  1. A student claims to have produced a piece of iron that has only a north pole. Explain, using the properties of magnetic lines of force, why this claim must be false.

  2. Why can two magnetic lines of force never intersect? Would your reasoning also rule out the intersection of two electric lines of force?

  3. The magnetic flux through a closed surface drawn entirely inside the body of a bar magnet is zero, even though the magnetic field there is strong. Explain how both statements can be true at the same time.

  4. State Gauss’ law of magnetism. How would the mathematical form of this law change if a magnetic monopole were discovered? What does the present form of the law tell us about the existence of monopoles?

  5. In the electrostatic analogue, every electrostatic quantity has a magnetic counterpart except Coulomb’s law, which has none. Explain the physical reason for this single missing entry.

  6. Why is the magnetic length of a bar magnet always less than its geometric length? What error is introduced in a calculated magnetic dipole moment if a student uses the geometric length by mistake — will the answer be too large or too small?

  7. At the same distance from the centre of a short bar magnet, the axial field is exactly twice the equatorial field. Explain why, using the way the two poles’ contributions combine in each case.

  8. The formulas for the axial and equatorial fields of a bar magnet carry the condition r ≫ l. What goes wrong if these formulas are applied to a point very close to the magnet?

  9. Why is the equatorial field of a bar magnet written with a negative sign? What physical information would be lost if the sign were dropped?

  10. A point P lies at an angle θ to the axis of a short bar magnet. Explain how P can be treated as an axial point and an equatorial point at the same time, and why this does not lead to a contradiction.

  11. Show that Eq. (12.8) for the field at an arbitrary point reduces to the axial field when θ = 0° and to the equatorial field when θ = 90°. Why is such a check useful?

  12. The Earth’s magnetic north pole lies below Antarctica, not below Canada. Reconcile this with the everyday observation that the north pole of a compass needle points towards geographic north.

  13. An ordinary horizontal compass becomes useless near a magnetic pole of the Earth, while a dip needle stands almost vertical there. Explain both observations in terms of B_H and B_V.

  14. A traveller carries a dip circle from Thiruvananthapuram to northern Canada. Describe how the angle of dip changes along the journey, and explain the change using the relation tan φ = B_V / B_H.

  15. Distinguish clearly between magnetic declination and magnetic inclination. In which plane is each measured?

  16. At Mumbai the declination is 0° 58′ west and at Delhi it is 0° 41′ east. What does this difference in sign mean physically for a compass user at each city?

  17. What is a neutral point? At a neutral point near a bar magnet placed on a table, is the magnetic field of the magnet itself zero? Justify your answer.

  18. Why does the definition of magnetic maps state that the magnetic elements vary “with time” as well as with place? Give one practical consequence of this time variation.

  19. A uniform magnetic field passes normally through a flat surface. If the surface is now tilted so that the field is no longer normal to it, does the flux through it increase, decrease or stay the same? Explain, and state why Eq. (12.1) may no longer be used directly.

  20. Lines of force are described in the chapter as “only a crude way of representing the magnetic field”. State two things the picture of lines of force does convey correctly, and one respect in which it is misleading.

  21. What happens if a bar magnet is cut into two pieces transverse to its length / along its length?

  22. What is a geographic meridian? How does the declination vary with latitude? Where is it minimum?

  23. Define the Angle of Dip. What happens to angle of dip as we move towards magnetic pole from magnetic equator?

12.8 Practice with Numericals

  1. A uniform magnetic field of 0.40 T passes normally through a flat circular surface of radius 10 cm. Calculate the magnetic flux through the surface in weber. Express the same magnetic field in gauss.

  2. A magnetic flux of \(2.4 \times 10^{-3}\) Wb passes normally through a rectangular area of dimensions 30 cm × 20 cm. Calculate the magnitude of the uniform magnetic field over the area.

  3. A bar magnet has a geometric length of 12 cm and a pole strength of 5.0 A m. Calculate its magnetic length and its magnetic dipole moment.

  4. A short bar magnet has a magnetic dipole moment of 2.0 A m². Calculate the magnetic field at a point 25 cm from its centre (i) on its axis and (ii) on its equator. State the direction of the field in each case with respect to \(\vec{m}\). (Take \(\mu_0 = 4\pi \times 10^{-7}\) SI units.)

  5. The magnetic field of a short bar magnet at a point on its axis, 20 cm from the centre, is \(5.0 \times 10^{-6}\) T. Calculate the magnetic dipole moment of the magnet, and the magnitude of the field at an equatorial point at the same distance. (Take \(\mu_0 = 4\pi \times 10^{-7}\) SI units.)

  6. A short bar magnet of magnetic moment 0.6 A m² is placed with its centre at O. Find the magnitude of the magnetic field at a point P, 20 cm from O, where OP makes an angle of 30° with \(\vec{m}\). Also find the angle α that the field at P makes with OP. (Take \(\mu_0 = 4\pi \times 10^{-7}\) SI units.)

  7. For a short bar magnet, at what angle θ from the axis is the magnitude of the magnetic field at a fixed distance r equal to \(\dfrac{\mu_0}{4\pi}\dfrac{m}{r^{3}} \times \sqrt{2.5}\)? Use Eq. (12.8).

  8. At a certain place, the horizontal component of the Earth’s magnetic field is \(2.6 \times 10^{-5}\) T and the vertical component is \(4.5 \times 10^{-5}\) T. Calculate (i) the angle of dip and (ii) the magnitude of the Earth’s resultant magnetic field at that place.

  9. At a place, the Earth’s total magnetic field is \(5.0 \times 10^{-5}\) T and the angle of dip is 30°. Calculate the horizontal and the vertical components of the Earth’s magnetic field there.

  10. At a certain location, the horizontal and vertical components of the Earth’s magnetic field are found to be equal, each of magnitude \(3.0 \times 10^{-5}\) T. Determine the angle of dip and the resultant magnetic field at that place.

  11. A short bar magnet of magnetic moment 1.5 A m² is placed horizontally with its axis along the magnetic East–West direction, at a place where the horizontal component of the Earth’s magnetic field is \(3.6 \times 10^{-5}\) T. Calculate the distance from the centre of the magnet, along its axis, at which a neutral point would be obtained. (Take \(\mu_0 = 4\pi \times 10^{-7}\) SI units.)

  12. Two identical short bar magnets, each of magnetic moment 0.8 A m², are placed with their centres 50 cm apart, with their axes lying along the same straight line and their north poles pointing towards each other. Calculate the magnitude of the resultant magnetic field at the mid-point of the line joining their centres. (Take \(\mu_0 = 4\pi \times 10^{-7}\) SI units.)

  13. The magnetic field on the equator of a planet is \(6.0 \times 10^{-5}\) T and its radius is \(3.4 \times 10^{6}\) m. Assuming the planet behaves as a magnetic dipole, calculate its magnetic dipole moment. (Take \(\mu_0 = 4\pi \times 10^{-7}\) SI units.)

  14. A closed cubical Gaussian surface of side 8.0 cm is placed in a region where a bar magnet produces a strong, non-uniform magnetic field. The cube encloses the north pole end of the magnet. State the value of the net magnetic flux through the cube, and justify your answer using Gauss’ law of magnetism.

  15. A magnetic field is quoted as 0.85 gauss. Express it in tesla. If this field is normal to an area of 0.20 m², find the magnetic flux through that area in weber.

  16. A magnetic pole of a bar magnet with pole strength of 100 A m is 20 cm away from the centre of a bar magnet. The bar magnet has pole strength of 200 A m and has a length 5 cm. If the magnetic pole is on the axis of the bar magnet, find the force on the magnetic pole.

  17. A magnet makes an angle of 45° with the horizontal in a plane making an angle of 30° with the magnetic meridian. Find the true value of the dip angle at the place.

  18. Two small and similar bar magnets have magnetic dipole moment of 1.0 A m² each. They are kept in a plane in such a way that their axes are perpendicular to each other. A line drawn through the axis of one magnet passes through the center of the other magnet. If the distance between their centers is 2 m, find the magnitude of magnetic field at the mid point of the line joining their centers.

  19. A circular magnet is made with its north pole at the centre, separated from the surrounding circular south pole by an air gap. Draw the magnetic field lines in the gap. [The magnet is a hypothetical magnet.] Draw a diagram to illustrate the magnetic lines of force between the south poles of two such magnets.

  20. Two bar magnets are placed on a straight line with their north poles facing each other on a horizontal surface. Draw magnetic lines around them. Mark the position of any neutral points (points where there is no resultant magnetic field) on your diagram.

12.9 Objective Questions

  1. Let r be the distance of a point on the axis of a bar magnet from its center. The magnetic field at r is always proportional to
    1. 1/r²
    2. 1/r³
    3. 1/r
    4. not necessarily 1/r³ at all points
  2. Magnetic meridian is the plane
    1. perpendicular to the magnetic axis of Earth
    2. perpendicular to geographic axis of Earth
    3. passing through the magnetic axis of Earth
    4. passing through the geographic axis of Earth
  3. The horizontal and vertical component of magnetic field of Earth are same at some place on the surface of Earth. The magnetic dip angle at this place will be
    1. 30°
    2. 45°
    3. 0°
    4. 90°
  4. Inside a bar magnet, the magnetic field lines
    1. are not present
    2. are parallel to the cross sectional area of the magnet
    3. are in the direction from N pole to S pole
    4. are in the direction from S pole to N pole
  5. A place where the vertical components of Earth’s magnetic field is zero has the angle of dip equal to
    1. 0°
    2. 45°
    3. 60°
    4. 90°
  6. A place where the horizontal component of Earth’s magnetic field is zero lies at
    1. geographic equator
    2. geomagnetic equator
    3. one of the geographic poles
    4. one of the geomagnetic poles
  7. A magnetic needle kept nonparallel to the magnetic field in a nonuniform magnetic field experiences
    1. a force but not a torque
    2. a torque but not a force
    3. both a force and a torque
    4. neither force nor a torque