Matter responds to magnetic fields in different ways, giving rise to the fascinating subject of magnetism in materials. This chapter begins with the study of magnets as magnetic dipoles - a bar magnet, like a current loop, has a magnetic moment and experiences a torque in a magnetic field. We study the field of a bar magnet and the forces and torques it experiences.
We then examine the magnetic field of the Earth itself, described by magnetic elements such as declination, dip, and horizontal component. The chapter explains how the geomagnetic field affects compass needles and navigation.
Finally, we study the microscopic origin of magnetism in materials. Magnetic moments arise from the orbital and spin motions of electrons, and materials are classified as diamagnetic, paramagnetic, or ferromagnetic according to their response to an external field. The concepts of magnetization, magnetic intensity, susceptibility, and hysteresis explain the behaviour of soft and hard magnetic materials used in transformers and permanent magnets.
A bar magnet has two poles, north and south, and behaves as a magnetic dipole with magnetic moment M directed from the south pole to the north pole:
M = m * (2l)
where m is the pole strength and 2l the distance between the poles. The magnetic field at a distance r from the centre on the axial line (end-on position) is:
B_axial = (mu_0 / 4 pi) * 2 M / r^3
and on the equatorial line (broadside-on position):
B_equatorial = (mu_0 / 4 pi) * M / r^3
For large distances, the axial field is twice the equatorial field. Like the electric dipole, the magnetic dipole field falls off as 1/r^3. If a bar magnet is broken into two pieces, each piece becomes an independent magnet with its own north and south poles; isolated magnetic monopoles do not exist.
When a bar magnet of moment M is placed in a uniform magnetic field B, the forces on the two poles form a couple. The torque is:
tau = M B sin(theta)
where theta is the angle between M and B. In vector form, tau = M x B. The potential energy of the dipole in the field is:
U = -M B cos(theta)
The torque is zero and the energy minimum when the magnet is aligned with the field (theta = 0), and the energy is maximum when anti-aligned. This is why a compass needle, which is a small magnet, aligns with the Earth's magnetic field. The work done in rotating the magnet is stored as potential energy.
The Earth behaves as a giant bar magnet with its magnetic axis slightly tilted relative to the geographic axis. The magnetic field of the Earth at any location is described by three magnetic elements:
If B is the total field, B_H the horizontal component, and delta the angle of dip, then:
B_H = B cos(delta), B_V = B sin(delta)
At the magnetic equator, the dip is zero, and at the magnetic poles, the dip is 90 degrees. The horizontal component is used to compare the Earth's field with fields of magnets using the tangent law, which states that B = B_H tan(theta) when a magnet suspended by a compass needle is in equilibrium under the two fields.
The magnetization M of a material is the magnetic moment per unit volume. When a material is placed in a magnetizing field H, the magnetic induction inside is:
B = mu_0 (H + M)
The magnetic intensity H is a measure of the applied field strength. The magnetic susceptibility chi relates the magnetization to the intensity:
M = chi H
The magnetic permeability mu of the material relates B and H:
B = mu H
The relative permeability is mu_r = mu/mu_0 = 1 + chi. These quantities are determined by the microscopic response of the material's electron currents to the applied field.
Materials are classified by their magnetic behaviour:
Diamagnetic materials (like bismuth and copper) have all electrons paired, so their atoms have zero net magnetic moment. When placed in a field, they develop a very small induced moment opposing the field. They have negative susceptibility (chi of order -10^-5), are weakly repelled by magnets, and slightly weaken the field inside.
Paramagnetic materials (like aluminium and sodium) have atoms with unpaired electrons carrying net moments that are randomly oriented. An external field partially aligns them, giving small positive susceptibility (chi of order 10^-5) that decreases with temperature according to Curie's law chi = C/T. They are weakly attracted by magnets.
Ferromagnetic materials (like iron, cobalt, and nickel) have domains - regions of aligned moments. In an external field, domains grow and align, producing very large positive susceptibility (up to 10^3 or more). Ferromagnetism disappears above the Curie temperature, above which the material becomes paramagnetic.
Ferromagnetic materials exhibit hysteresis: the magnetization does not retrace itself when the applied field is increased and then decreased, but lags behind. The hysteresis loop shows that a material retains some magnetization when the field is removed - this is retentivity. The field needed to reduce magnetization to zero is the coercivity.
Soft magnetic materials (like soft iron) have narrow hysteresis loops, small coercivity, and are easily magnetized and demagnetized. They are used in transformers and electric motor cores where rapid reversals of magnetization occur, because a narrow loop means small energy loss.
Hard magnetic materials (like steel and alnico) have wide hysteresis loops, large coercivity and retentivity, and are difficult to demagnetize. They are used to make permanent magnets, such as in loudspeakers and generators. The area of the hysteresis loop is proportional to the energy loss per cycle.
| Quantity | Formula | Remarks |
|---|---|---|
| Magnetic moment of a magnet | M = m(2l) | From south to north pole |
| Axial field | B_axial = (mu_0/4 pi) 2M/r^3 | End-on position |
| Equatorial field | B_eq = (mu_0/4 pi) M/r^3 | Broadside-on position |
| Torque on dipole | tau = M B sin theta | Max at 90 degrees |
| Potential energy | U = -M B cos theta | Min when aligned |
| Horizontal component | B_H = B cos delta | delta is dip angle |
| Magnetic induction | B = mu_0 (H + M) | Inside material |
| Susceptibility | M = chi H | Dimensionless |
| Permeability | mu = mu_0 (1 + chi) | Relative mu_r |
| Material | Susceptibility | Behaviour | Examples |
|---|---|---|---|
| Diamagnetic | Small, negative (-10^-5) | Repelled, weakens field | Bismuth, copper, water |
| Paramagnetic | Small, positive (+10^-5) | Attracted, chi = C/T | Aluminium, sodium |
| Ferromagnetic | Large, positive (10^3) | Strong attraction, domains | Iron, cobalt, nickel |
This chapter described magnetism in matter. A bar magnet is a magnetic dipole with moment M = m(2l), experiencing torque tau = MB sin theta in a field, with field itself falling off as 1/r^3. The Earth's field was described by declination, dip, and horizontal component. Microscopically, electron currents produce atomic moments, and materials respond differently: diamagnetics repel weakly, paramagnetics attract weakly, and ferromagnetics show strong alignment through domains. Magnetization M, intensity H, and susceptibility chi quantify these responses, and the hysteresis loop explains retentivity and coercivity. Soft magnets with narrow loops serve in transformers, while hard magnets with wide loops make permanent magnets - completing our understanding of how ordinary matter interacts with magnetic fields.