Chapter 5: Magnetism and Matter | Summary

A bar magnet can be thought of as a large collection of circulating currents, in analogy with a solenoid. Cutting a bar magnet in half is analogous to cutting a solenoid — both halves become smaller magnets with weaker properties and continuous field lines. The axial magnetic field of a finite solenoid at large distances matches that of a bar magnet: B = (mu0/4pi)(2m/r^3).

The magnetic moment of a bar magnet equals the magnetic moment of an equivalent solenoid that produces the same field. This analogy can be tested by moving a compass needle near both a bar magnet and a current-carrying solenoid and observing similar deflections.

When a magnetic dipole of moment m is placed in a uniform magnetic field B, it experiences a torque given by tau = m x B, with magnitude tau = mB sin(theta), where theta is the angle between m and B. This is the restoring torque that tends to align the dipole with the field.

The magnetic potential energy is given by U = -m.B. Potential energy is minimum (-mB) at theta = 0 degrees (most stable position, dipole parallel to field) and maximum (+mB) at theta = 180 degrees (most unstable position, dipole antiparallel to field). The zero of potential energy is taken at theta = 90 degrees.

The equations governing a magnetic dipole closely parallel those of an electric dipole. The substitutions E to B, p to m, and 1/(4pi*epsilon0) to mu0/4pi transform electric dipole equations into magnetic ones. The equatorial field of a bar magnet is BE = -(mu0/4pi)(m/r^3) and the axial field is BA = (mu0/4pi)(2m/r^3).

This analogy is summarised in the dipole analogy table: dipole moment p maps to m, torque p x E maps to m x B, and potential energy -p.E maps to -m.B. The analogy holds at large distances from the dipole.

Gauss's law for magnetism states that the net magnetic flux through any closed surface is zero: phi_B = sum B.Delta_S = 0. This is fundamentally different from Gauss's law for electrostatics, where the flux depends on the enclosed charge. The difference reflects the non-existence of isolated magnetic poles (monopoles).

Since magnetic monopoles do not exist, there are no sources or sinks of B. The simplest magnetic element is a dipole or a current loop. All magnetic field lines are continuous and form closed loops. If monopoles did exist, the right-hand side of Gauss's law for magnetism would equal mu0 times the magnetic charge enclosed.

Magnetisation M of a sample is defined as its net magnetic moment per unit volume: M = m_net / V. It is a vector measured in units of A/m. When a magnetic material fills a solenoid, the net field inside is B = B0 + Bm, where B0 = mu0*nI is the applied field and Bm = mu0*M is the field due to the material.

The magnetic intensity H is defined as H = B/mu0 - M, so that B = mu0(H + M). The magnetisation is related to H by M = chi*H, where chi is the dimensionless magnetic susceptibility. The relative permeability is mu_r = 1 + chi, and the magnetic permeability is mu = mu0*mu_r. These three quantities are interrelated and only one is independent.

Materials are classified as diamagnetic (chi negative and small, -1 <= chi < 0), paramagnetic (chi positive and small, 0 < chi < epsilon), or ferromagnetic (chi >> 1). Diamagnetic substances are repelled by magnets and move from stronger to weaker fields; their field lines are expelled. Paramagnetic substances are weakly attracted to magnets and move from weaker to stronger fields; their field lines concentrate inside.

Ferromagnetic substances are strongly magnetised in an external field. Their atoms are grouped into domains of about 10^11 atoms, each with a net magnetisation. Without an external field, domains are randomly oriented. On applying a field B0, domains align and grow, producing a strong net magnetisation. Ferromagnetism depends on temperature; above the Curie temperature, ferromagnets become paramagnets.

Diamagnetism arises because electrons whose orbital magnetic moments are in the direction of the applied field slow down, while those in the opposite direction speed up (Lenz's law). The net result is a magnetic moment opposing the applied field. Diamagnetic materials include bismuth, copper, lead, silicon, water, and sodium chloride. Superconductors are perfect diamagnets with chi = -1 and mu_r = 0; this is the Meissner effect.

Paramagnetic atoms possess permanent magnetic dipole moments. Without an external field, random thermal motion prevents net magnetisation. In a strong external field at low temperatures, dipoles align with the field. Examples include aluminium, sodium, calcium, oxygen, and copper chloride. Susceptibility chi and relative permeability mu_r of paramagnetic materials depend on both the material and temperature. Magnetisation increases with field strength and decreasing temperature until saturation.

In ferromagnetic materials, neighbouring atoms interact via quantum mechanical exchange forces, causing spontaneous alignment of dipoles in macroscopic volumes called domains. Typical domain size is 1 mm containing about 10^11 atoms. Initially domains are randomly oriented. Under external field B0, domains oriented along B0 grow at the expense of others until a single giant domain forms.

Hard ferromagnets (e.g., Alnico, lodestone) retain magnetisation after the external field is removed and are used as permanent magnets and compass needles. Soft ferromagnets (e.g., soft iron) lose magnetisation when the field is removed. Ferromagnetic elements include iron, cobalt, nickel, and gadolinium, with relative permeability exceeding 1000.

Chapter 5 establishes magnetism as a rich and self-contained subject. Beginning with the bar magnet as a magnetic dipole, the chapter develops the analogy between electric and magnetic dipoles, states Gauss's law for magnetism (net flux through any closed surface is zero), and introduces the key quantities — magnetisation M, magnetic intensity H, susceptibility chi, and permeability mu — needed to classify and describe magnetic materials. The three classes of magnetic materials — diamagnetic, paramagnetic, and ferromagnetic — are distinguished by the sign and magnitude of chi, with ferromagnetics uniquely explained through domain theory.

For board examinations, this chapter is high-yield. Questions frequently appear on: the torque and potential energy of a dipole in a uniform field (tau = m x B, U = -m.B), axial and equatorial field formulas, Gauss's law for magnetism and its implications for monopoles, comparison of dia/para/ferromagnetic properties, domain theory, the Meissner effect in superconductors, and numerical problems involving H, M, B, chi, and mu_r. A thorough command of Table 5.1 (dipole analogy) and Table 5.2 (classification of magnetic materials) is essential.