Magnetism and Matter

1. Check this before you revise anything

The "Additional Exercises" section has been removed from this chapter, as from all 14 chapters of the current Class 12 Physics book. The questions run contiguously from 5.1 to 5.7 with no gaps.

That leaves 7 questions for the whole chapter, five of which are variations on the torque and potential energy of a dipole.

The Earth's magnetism is gone entirely. Older editions devoted a full section to it — magnetic declination, angle of dip, the horizontal component, and the Earth as a giant dipole. A full-text search of this chapter returns zero occurrences of the word "Earth", and zero for "declination" and "dip".

Permanent magnets, electromagnets and the hysteresis loop are gone too. Searching for "hysteresis", "retentivity" and "coercivity" returns zero hits each. The chapter now ends at section 5.5.

Curie's law and the Curie temperature have been removed. Zero occurrences of "Curie" appear anywhere in the chapter. Section 5.5.2 still describes paramagnetism qualitatively and mentions that alignment improves at low temperatures, but the inverse-temperature law itself is no longer stated.

If you are revising from an older guide, a past paper, or a coaching sheet, expect it to test several topics this book no longer contains.

A numbering error in the exercises. Exercise 5.4 begins "If the solenoid in Exercise 5.5 is free to turn about the vertical direction..." — but Exercise 5.5 is about a bar magnet. The solenoid is defined in Exercise 5.3, and its magnetic moment of $0.60$ J T${}^{-1}$ is what 5.4 needs. Our solution uses the correct cross-reference and says so explicitly, rather than silently patching the book.

2. The Bar Magnet and Its Field Lines (Textbook 5.2 to 5.2.1)

A bar magnet has two poles, and the properties of its field lines are a standard two-mark question.

Properties of magnetic field lines:

• They form continuous closed loops, running from north to south outside the magnet and from south to north inside it. Electric field lines, by contrast, start and end on charges.
• The tangent at any point gives the direction of $\vec{B}$ there.
• Line density represents field strength: crowded lines mean a strong field.
• Two field lines never intersect. If they did, the field would have two directions at one point, which is impossible.

Poles cannot be separated. Cutting a bar magnet in half does not isolate a north pole; it produces two shorter magnets, each with both poles. This is the experimental fact behind section 5.3.

The bar magnet as an equivalent solenoid (5.2.2). The previous chapter showed that a current loop behaves as a magnetic dipole. Stacking many loops into a solenoid gives an external field pattern indistinguishable from a bar magnet's, with the ends acting as poles.

The magnetic moment of a solenoid of $N$ turns, area $A$, carrying current $I$ is:

$m=NIA$

This is the whole content of Exercises 5.3 and 5.6, and it is the conceptual bridge from Chapter 4 into this one.

3. The Dipole in a Uniform Field (Textbook 5.2.3)

Because the two poles feel equal and opposite forces, a dipole in a uniform field feels no net force — only a torque:

$\vec{\tau }=\vec{m}\times \vec{B},\tau =mB\sin \theta$

Potential energy. Work done in rotating the dipole is stored as potential energy:

$U=-\vec{m}\cdot \vec{B}=-mB\cos \theta$

Work done between two orientations is the difference of the potential energies:

$W=U_2-U_1=-mB(\cos {\theta }_2-\cos {\theta }_1)$

Exercise 5.5 uses exactly this, first for a rotation to $90^{\circ }$ and then to $180^{\circ }$.

Reading the two equilibria off the energy. Setting $\theta =0$ and $\theta =180^{\circ }$ in the energy expression:

Both extremes have zero torque, so torque alone cannot tell them apart. The energy does. A minimum is stable and a maximum unstable, which is what Exercise 5.2 asks you to identify and Exercise 5.5(b)(ii) exploits when the torque comes out as zero at $180^{\circ }$.

The electrostatic analogue (5.2.4). Replacing $\frac{1}{4\pi {\epsilon }_0}\to \frac{{\mu }_0}{4\pi }$ and $\vec{p}\to \vec{m}$ converts every electric dipole result into its magnetic counterpart. For a short bar magnet at distance $r$:

$B_{\text{axial}}=\frac{{\mu }_0}{4\pi }\frac{2m}{r^3},B_{\text{equatorial}}=\frac{{\mu }_0}{4\pi }\frac{m}{r^3}$

The axial field is twice the equatorial field at the same distance, and it points along the moment while the equatorial field points opposite to it. Exercise 5.7 tests both the factor of two and the two directions.

4. Gauss's Law for Magnetism (Textbook 5.3)

The net magnetic flux through any closed surface is zero:

$\oint \vec{B}\cdot d\vec{A}=0$

Compare this with the electric case. Gauss's law in electrostatics gives $q_{\text{enclosed}}/{\epsilon }_0$, which is non-zero whenever the surface contains net charge. The magnetic version has zero on the right always.

The physical statement is that isolated magnetic poles do not exist. Every field line that enters a closed surface must leave it, because field lines are closed loops with no beginning or end. There is no magnetic charge for them to start on.

This is why breaking a magnet never yields a monopole, and it is one of the four Maxwell equations assembled in Chapter 8.

5. Magnetisation and Magnetic Intensity (Textbook 5.4)

Inside a material, the applied field and the material's own response must be separated.

Magnetisation is the net magnetic moment per unit volume:

$\vec{M}=\frac{{\vec{m}}_{\text{net}}}{V}$

Magnetic intensity $\vec{H}$ describes the applied field alone, independent of the medium. The total field inside is the sum of both contributions:

$\vec{B}={\mu }_0(\vec{H}+\vec{M})$

Susceptibility measures how strongly a material responds:

$\vec{M}=\chi \vec{H}$

Relative permeability then follows:

${\mu }_r=1+\chi ,\mu ={\mu }_0{\mu }_r$

$\chi$ is a pure number with no units, and its sign and size are what classify the material. Note that $\vec{H}$ and $\vec{M}$ share the unit A m${}^{-1}$, while $\vec{B}$ is in tesla — mixing them up is the standard error here.

6. Diamagnetism, Paramagnetism and Ferromagnetism (Textbook 5.5)

The three classes are distinguished by the sign and magnitude of $\chi$, and by how they behave in a non-uniform field.

Diamagnetism (5.5.1) exists in every material. The atoms have no permanent moment; an applied field induces one that opposes the field, by the same argument as Lenz's law in the next chapter. It is usually masked by the stronger para- or ferromagnetic response when either is present.

The perfect diamagnet. A superconductor expels magnetic flux entirely, giving $\chi =-1$ and ${\mu }_r=0$ — the Meissner effect, and the extreme end of the diamagnetic range.

Paramagnetism (5.5.2). The atoms carry permanent moments, but thermal motion randomises them so no net magnetisation appears. A strong field at low temperature aligns them, and the field inside is enhanced slightly — the book quotes about one part in $10^5$.

Ferromagnetism (5.5.3). Atomic moments align spontaneously into domains, regions in which vast numbers of moments point the same way. An applied field grows the favourably aligned domains at the expense of the others, producing a very large magnetisation that can persist after the field is removed.

Summary

• Magnetic field lines form closed loops, running north to south outside a magnet and south to north inside; they never intersect.
• Cutting a magnet produces two magnets, never an isolated pole.
• A solenoid is equivalent to a bar magnet, with moment $m=NIA$.
• In a uniform field a dipole feels zero net force but a torque $\vec{\tau }=\vec{m}\times \vec{B}$, of magnitude $mB\sin \theta$.
• $U=-\vec{m}\cdot \vec{B}=-mB\cos \theta$, so work done in rotating is $W=-mB(\cos {\theta }_2-\cos {\theta }_1)$.
• $\theta =0$ is stable with $U=-mB$; $\theta =180^{\circ }$ is unstable with $U=+mB$. Both have zero torque, so use the energy to distinguish them.
• Short bar magnet: $B_{\text{axial}}=\frac{{\mu }_0}{4\pi }\frac{2m}{r^3}$ along the moment, and $B_{\text{equatorial}}=\frac{{\mu }_0}{4\pi }\frac{m}{r^3}$ opposite to it — a factor of two apart.
• Gauss's law for magnetism: $\oint \vec{B}\cdot d\vec{A}=0$ always, because isolated magnetic poles do not exist.
• $\vec{M}={\vec{m}}_{\text{net}}/V$; $\vec{B}={\mu }_0(\vec{H}+\vec{M})$; $\vec{M}=\chi \vec{H}$; ${\mu }_r=1+\chi$.
• $H$ and $M$ are both in A m${}^{-1}$ while $B$ is in tesla; $\chi$ is dimensionless.
• Diamagnetic: $\chi$ small and negative, ${\mu }_r<1$, repelled towards weaker field, no permanent atomic moment.
• Paramagnetic: $\chi$ small and positive, ${\mu }_r$ slightly above 1, attracted to stronger field, permanent moments randomised by heat.
• Ferromagnetic: $\chi$ large and positive, ${\mu }_r\gg 1$, domain structure, magnetisation can persist.
• A superconductor is a perfect diamagnet with $\chi =-1$ and ${\mu }_r=0$, the Meissner effect.
• The Earth's magnetism, permanent magnets and electromagnets, the hysteresis loop and Curie's law have all been removed from this chapter.
• Exercise 5.4's reference to "the solenoid in Exercise 5.5" is a misprint; the solenoid is in Exercise 5.3.