Chapter 7

Magnetic Materials

7.1 Magnetization

Materials respond to an applied magnetic field by developing a magnetization $\mathbf{M}$ — the magnetic dipole moment per unit volume. The bound currents created by $\mathbf{M}$ are:

$$\mathbf{J}_b = \nabla \times \mathbf{M} \qquad \text{(bulk bound current)}$$$$\mathbf{K}_b = \mathbf{M} \times \hat{n} \qquad \text{(surface bound current)}$$

7.2 The H Field

The auxiliary field $\mathbf{H}$ separates free from bound currents:

$$\mathbf{H} = \frac{1}{\mu_0}\mathbf{B} - \mathbf{M}, \qquad \nabla \times \mathbf{H} = \mathbf{J}_f$$

For a linear magnetic material, $\mathbf{M} = \chi_m \mathbf{H}$ where $\chi_m$ is the magnetic susceptibility, giving:

$$\mathbf{B} = \mu_0(1 + \chi_m)\mathbf{H} = \mu_0\mu_r\mathbf{H} = \mu\mathbf{H}$$

Diamagnets

$\chi_m < 0$

Bismuth, copper, water

Weakly repelled by external B. Meissner effect in superconductors is extreme diamagnetism.

Paramagnets

$\chi_m > 0$

Aluminium, platinum

Weakly attracted. Curie's law: M = C·B/T at temperature T.

Ferromagnets

$\chi_m \gg 1$

Iron, nickel, cobalt

Strongly attracted. Exhibit hysteresis and permanent magnetization.

7.3 Boundary Conditions

Normal B (from $\nabla\cdot\mathbf{B}=0$)

$$B_1^\perp = B_2^\perp$$

Normal component of B is continuous.

Tangential H (from $\nabla\times\mathbf{H}=\mathbf{J}_f$)

$$H_1^\parallel - H_2^\parallel = K_f$$

Tangential H is discontinuous by free surface current.

← Vector Potential Faraday's Law →