Andrea's Crash Course in Chapter 6

Torque on a dipole:

$$\vec{N} = \vec{m}\times\vec{B}$$

Force on a dipole:

$$\vec{F} = \nabla(\vec{m}\cdot \vec{B})$$

Compare to it’s electrical ``twin”: $\vec{F} = \nabla(\vec{p}\cdot \vec{E})$

The “Gilbert” model:

You can do a lot of magnetostatic problems by using electrostatic results and replacing:

• $\vec{p}$ with $\vec{m}$
• $1/\epsilon_0$ with $\mu_0$
• $E$ with $B$

Once you get really close-up to the dipole (where it looks like a ring of current instead of a perfect magnetic dipole) then this breaks down. Griffiths’ advice is to use the Gilbert model to get intuition, but don’t rely on it for quantitative results.

Paramagnetism

The presence of the magnetic field can make the electron speed up in its orbit. The magnetic moment changes by:

$$\Delta \vec{m} = -\frac{1}{2} e(\Delta v)R\hat{z} = -\frac{e^2R^2}{4m_e}\vec{B}$$

Notice the change in $\vec{m}$ is opposite the direction of $\vec{B}$.

Magnetization

$\vec{M}$ is the magnetic dipole moment per unit volume (just like $\vec{P}$ is the electric dipole moment per unit volume.)

The vector potential due to some magnetized matter is:

$$\vec{A}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{\vec{M}(\vec{r}^\prime) \times \hat{\mathscr{r}}} {\mathscr{r}^2} d\tau^\prime$$

In a manner very similar to what we did for electric dipoles, we can define a bound volume current and a bound surface current.

$$\vec{J}_b = \nabla \times \vec{M}$$ $$\vec{K}_b = \vec{M} \times \hat{n}$$

where $\hat{n}$ is the normal unit vector (normal to the surface.)

With those definitions:

$$\vec{A}(\vec{r}) = \frac{\mu_0}{4\pi} \int_V \frac{\vec{J}_b(\vec{r}^\prime)}{\mathscr{r}} d\tau^\prime + \frac{\mu_0}{4\pi} \int_S \frac{\vec{K}_b(\vec{r}^\prime)}{\mathscr{r}} da^\prime$$

The Auxiliary Field

$$\vec{H} = \frac{1}{\mu_0} \vec{B} - \vec{M}$$

Ampere’s law for H:

$$\nabla \times \vec{H} = \vec{J}_f$$

or in integral form:

$$\oint \vec{H}\cdot d\vec{l} = I_{free}$$

Boundary Conditions

The boundary conditions on $\vec{H}$ are:

$$H_{above}^\perp - H_{below}^\perp = M_{above}^\perp - M_{below}^\perp$$ $$\vec{H}_{above}^\| - \vec{H}_{below}^\| = \vec{K}_f \times \hat{n}$$