Electrodynamics/Magnetization

Magnetization Field

We can define two new vector fields to help aide us in further studies of magnetics. The first new field is the magnetization field M. This field represents the magnetic moment per unit volume of a magnetized substance. The second, H is known occasionally as the "magnetic field strength" and is used here primarily as a simplifying agent.

Magnetization

We can define the magnetic moment in terms of the magnetization as:

$\mathbf{m} = \mathbf{M}(\mathbf{r}')dV'$
$\mathbf{A}(\mathbf{r}') = -\int \mathbf{M}(\mathbf{r})' \times \nabla \frac{1}{|\mathbf{r} - \mathbf{r}'|}dV'$
$\mathbf{A}(\mathbf{r}) = \int \frac{\nabla \times \mathbf{M}(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}dV'$
$\mathbf{j}_M = \nabla \times \mathbf{M}$

H Field

$\mathbf{H} = \mathbf{B} - 4\pi\mathbf{M}$
$\nabla \times \mathbf{H} = 4 \pi \mathbf{j}_M$
$\oint_C \mathbf{H} \cdot d\mathbf{l} = 4 \pi \int_C\mathbf{j}_F \cdot \mathbf{n}dA$