Magnetism and Matter

Written and reviewed by Akanksha Tiwari, Physics & Chemistry faculty at UniDrill (M.Sc. Physics, Banaras Hindu University · 8 years teaching Class XI–XII science). Based on NCERT Physics Class XII, Chapter 5.

Break a magnet in half. Do you get a north pole and a south pole?

No. You get two smaller magnets, each with its own north and south. Cut again and the same thing happens. You can slice a bar magnet down to single atoms and every piece is still a dipole, with no isolated north or south anywhere. This is not a failure of our scissors. It is a deep fact about nature, and it is the reason magnetism needs a different Gauss's law from electricity.

This chapter builds on that fact. It treats a bar magnet as a magnetic dipole, shows that it behaves like a coil of current, and then asks why some materials are strongly pulled by magnets, some barely react, and a few are pushed away.

The bar magnet and its field

Magnetism has been known for more than two and a half thousand years from lodestone, a naturally magnetic rock. A freely suspended bar magnet settles along the geographic north–south line. The end that points north is its north pole, and the other is its south pole. Like poles repel and unlike poles attract. Iron filings around a bar magnet reveal curved field lines, leaving the north pole and entering the south pole outside the magnet.

The field lines of a magnet have four properties, and each one is tested directly.

Compare this with electric field lines, which start on positive charges and end on negative charges. Magnetic field lines have no such start and end, because there are no isolated magnetic charges.

Ampere's idea: all magnetism is circulating charge. A bar magnet is equivalent to a solenoid. The field pattern of a current carrying solenoid is the same as that of a bar magnet, and at large distances along the axis the solenoid field becomes exactly $B = (\mu_0/4\pi) \times 2m/r^3$, where m is the magnetic moment. The magnetic moment of a coil is $m = NIA$, the same quantity you used in the last chapter. The atomic currents inside a magnet add up the same way.

Dipole in a uniform field: torque and energy

Put a bar magnet in a uniform field B at an angle θ. The force on the north pole is equal and opposite to the force on the south pole, so the net force is zero, but together they form a couple:

$\tau = m \times B$, with magnitude $\tau = mB\sin\theta$

The torque tries to line m up with B. To turn the dipole away from alignment, you must work against it, which stores potential energy

$$U = -m \cdot B = -mB\cos\theta$$

The zero of energy is set at $\theta = 90^\circ$. The minimum energy is $-mB$ at $\theta = 0^\circ$, where m is parallel to B (stable equilibrium). The maximum is $+mB$ at $\theta = 180^\circ$ (unstable equilibrium). The work needed to turn the magnet from $\theta_1$ to $\theta_2$ is $W = mB(\cos\theta_1 - \cos\theta_2)$.

Worked example: torque, energy and work. A bar magnet of magnetic moment 0.80 A m² is in a uniform field of 0.25 T, with $m$ at $30^\circ$ to the field. Find (a) the torque, (b) the potential energy, and (c) the work done in turning it from the stable position through 90°.

Common wrong step: writing $U = +mB\cos\theta$, or using the potential energy at the final point as the work (here zero). Work is the change in energy, so the starting value matters.

The dipole fields and the analogy with the electric dipole

A short bar magnet produces a field that looks just like that of an electric dipole, with the replacements $p \to m$ and $1/4\pi\varepsilon_0 \to \mu_0/4\pi$.

On the axis, the field is along m and has magnitude $(\mu_0/4\pi)(2m/r^3)$. On the equatorial line, the field is opposite to m and has half that magnitude, $(\mu_0/4\pi)(m/r^3)$. Both fall with the cube of the distance.

Worked example: axial and equatorial fields. A short bar magnet of magnetic moment 2.0 A m² lies along a table. Find the field at a point 20 cm from its centre on the axis, and at a point 20 cm from its centre on the equatorial line. ($\mu_0/4\pi = 10^{-7}\ \text{T m A}^{-1}$)

Common wrong step: giving the same value at both points. The axial field is always twice the equatorial field at the same distance.

Pole strength and the bent magnet. An older way to describe a bar magnet is by two poles of strength $q_m$ (unit A m) a distance $2l$ apart, so that $m = q_m \times 2l$. If you bend the magnet, the pole strength stays the same, but the distance between the poles changes. A magnet of length L bent into a semicircle has its poles at the ends of a diameter. The arc length $L = \pi R$, so the pole separation is $2R = 2L/\pi$, and the new moment is

$m' = q_m \times (2L/\pi) = 2m/\pi$

For a bar with $m = 1.0$ A m², this gives $m' = 0.64$ A m².

Beyond the textbook. For the exam: a magnet has a north and south pole, and the dipole is the simplest magnetic object. In reality: Earth acts like a giant bar magnet, but its magnetic pole near the geographic north is a magnetic south pole of that magnet. That is why the north pole of a compass needle points there. Also, the Earth's field is not exactly aligned with its rotation axis, so compasses show a small offset (declination) that varies from place to place.

Gauss's law for magnetism

Since field lines are closed loops, every line that enters a closed surface also leaves it. So the net magnetic flux through any closed surface is zero:

$$\varphi_B = \oint B \cdot dS = 0$$

Magnetic flux is measured in the weber ($1\ \text{Wb} = 1\ \text{T m}^2$). Compare this with electricity, where $\oint E \cdot dS = q/\varepsilon_0$ is non-zero when charge is enclosed. The magnetic version has no right hand side because there is no isolated magnetic charge to enclose. It is the mathematical form of the opening observation about cutting a magnet.

Pause and predict: a closed surface is drawn around the north-pole half of a bar magnet, cutting through the middle of the magnet. Is the net flux through it positive, negative or zero?

Check your answer: zero. Lines leave the surface near the north pole, but the same lines re-enter it where the surface cuts through the magnet's middle. Every line that goes out also comes in. A surface that seems to enclose a "north pole" never encloses a net source.

Beyond the textbook. For the exam: monopoles do not exist, so $\oint B \cdot dS = 0$. In reality: this is an experimental fact, not a proof. Theorists such as Dirac showed that monopoles could exist without breaking quantum mechanics, and searches for them have been made for decades, but none has been confirmed. If one were found, Gauss's law for magnetism would need a right hand side.

Magnetisation, intensity and permeability

When a material is placed in a magnetic field, its atoms behave as tiny dipoles. The magnetisation M is the net magnetic moment per unit volume, with unit A m⁻¹.

Think of a solenoid with n turns per metre carrying a current I. In vacuum it gives $B_0 = \mu_0 n I$. Fill it with a material. The material's own atomic dipoles add a field $\mu_0 M$, so the total field is $B = \mu_0 (nI + M)$. It is useful to separate what we supply (the current in the wires) from what the material adds (its magnetisation). The field due to our current is described by the magnetic intensity H:

$H = B/\mu_0 - M$, so $B = \mu_0 (H + M)$, with H in A m⁻¹

In a long solenoid, $H = nI$ regardless of the core. For materials that respond in proportion, $M = \chi H$, where χ is the dimensionless magnetic susceptibility. Then

$B = \mu_0 (1 + \chi) H = \mu_0 \mu_r H = \mu H$, with $\mu_r = 1 + \chi$

$\mu_r$ is the relative permeability (dimensionless) and $\mu = \mu_0 \mu_r$ is the permeability of the material.

Worked example: a solenoid with an iron core. A long solenoid has 1000 turns per metre and carries 2.0 A. Its core has relative permeability 400. Find H, B, χ and M inside the core.

Common wrong step: answering $B = \mu_0 n I = 2.5 \times 10^{-3}$ T. That is the field without the core. The core multiplies the field by μr, here 400 times.

Three kinds of magnetic material

Every substance responds a little to a magnetic field. How it responds depends on what its atoms do.

Diamagnetic materials. Their atoms have no net magnetic moment. A field induces currents in the atoms (Lenz's law at work), and these set up moments that oppose the field. The result is a weak repulsion. The material is pushed from strong field to weak, and field lines are partly expelled from it. χ is small and negative, and μr is slightly less than 1. Examples: bismuth, copper, lead, silicon, water, sodium chloride and nitrogen at STP.

Paramagnetic materials. Their atoms have permanent moments, but heat randomises them, so there is no net moment without a field. A field partly aligns them, so they are weakly attracted and drawn from weak field to strong. Field lines crowd into the material. χ is small and positive and μr is slightly greater than 1. Aligning the moments fights thermal motion, so χ falls as temperature rises (Curie's law, $\chi \propto 1/T$). Examples: aluminium, sodium, calcium, oxygen at STP and copper chloride.

Ferromagnetic materials. Strong quantum mechanical interactions make neighbouring atomic moments line up in regions called domains, each about a millimetre across and holding about 10¹¹ atoms. Without a field, domains point in random directions and cancel. A field makes the well aligned domains grow at the expense of others until, at saturation, the sample acts as one large domain. Then χ and μr are huge (μr may exceed 1000). Examples: iron, cobalt, nickel and gadolinium.

Superconductors are perfect diamagnets: $\chi = -1$ and $\mu_r = 0$. They expel the field completely, which is called the Meissner effect. Diamagnetism is actually present in every substance, but in paramagnets and ferromagnets the stronger response hides it.

The order of susceptibility is: diamagnetic (negative) < paramagnetic (small positive) < ferromagnetic (large positive).

Hard and soft ferromagnets. Once magnetised, a hard ferromagnet such as Alnico keeps its magnetisation when the field is removed, so it makes a permanent magnet. A soft one such as soft iron loses it almost entirely, so it is used for the cores of transformers and motors, where the field must be switched back and forth. A magnet that retains its magnetisation can still be demagnetised by a field in the opposite direction, and the strength of that reverse field, per metre of solenoid, is called the coercivity. Above the Curie temperature thermal agitation destroys the domain order and a ferromagnet turns paramagnetic, not diamagnetic.

Pause and predict: a small sphere is hung from a thread near the end of a strong bar magnet. It is gently pushed away. What can you say about its material?

Check your answer: it is diamagnetic. Only a diamagnet is repelled. Paramagnetic and ferromagnetic materials are attracted. The push is weak because diamagnetism is a feeble effect.

Beyond the textbook. For the exam: diamagnetic substances are weakly repelled. In reality: the repulsion is real enough to lift small objects. A magnet of about 16 T has levitated a live frog, because the water in its body is diamagnetic. The effect is tiny at ordinary field strengths, which is why you never see water or a leaf pushed away by a fridge magnet.

Exam traps

Formula sheet

60-second revision

Where this chapter connects

The idea that a coil is a magnet comes from Moving Charges and Magnetism, and the dipole formulas are the magnetic twins of those in Electric Charges and Fields. Soft iron cores and the flux ideas return in Electromagnetic Induction and in Alternating Current, where transformers depend on them.