Useful Equations

Here are some of the equations and ideas I found useful in this class.

Magnetic Fields

$$\begin{array}{rl}\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}& \text{dl along B (Ampere’s law)}\end{array}$$ $$\begin{array}{rl}\vec{B}(\vec{r})& =\frac{{\mu }_0}{4\pi }\int \frac{I\times \hat{R}}{|R{|}^2}d{l}^{\prime }\text{Biot-Savart law}\\ & \equiv \frac{{\mu }_0}{4\pi }I\int \frac{d\vec{l}\times \hat{R}}{|R{|}^2}\text{if constant I}\end{array}$$

$\vec{r}$ from origin to test point, ${\vec{r}}^{\prime }$ origin to test charge $d\vec{l}$, $\vec{R}=\vec{r}-{\vec{r}}^{\prime }$ (Griffiths' script-r)

Magnetism

Force $$\begin{array}{rl}\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})& \\ {\vec{F}}_{\text{loop}}=I\oint (d\vec{l}\times \vec{B})=0& \text{uniform B}\\ \vec{N}=\vec{m}\times \vec{B}& \end{array}$$

For an electron $\Delta \vec{m}=-\frac{e^2{R}^2}{4m_e}\vec{B}$.

Auxiliary field $\vec{H}$ $$\begin{array}{r}\vec{H}=\frac{1}{{\mu }_0}\vec{B}-\vec{M}\\ \vec{B}={\mu }_0(1+{\chi }_m)\vec{H}\\ \vec{B}=\mu \vec{H}\\ \vec{M}={\chi }_m\vec{H}\\ \vec{\nabla }\cdot \vec{H}=-\vec{\nabla }\cdot \vec{M}\end{array}$$

${\chi }_m$ is magnetic susceptibility, $\mu ={\mu }_0(1+{\chi }_m)$ is permeability

$\vec{M}$: $$\begin{array}{r}\vec{M}=\frac{1}{{\mu }_0}\vec{B}\left(1-\frac{1}{1+{\chi }_m}\right)\\ \vec{M}=\vec{B}\left(\frac{1}{{\mu }_0}-\frac{1}{\mu }\right)\end{array}$$

Current densities $\vec{J},{\vec{J}}_b$ and ${\vec{J}}_f$ and surface density ${\vec{K}}_b$: $$\begin{array}{r}\vec{J}={\vec{J}}_b+{\vec{J}}_f\\ {\vec{J}}_f=\vec{\nabla }\times \vec{H}\\ {\vec{J}}_b=\vec{\nabla }\times \vec{M}\\ \vec{\nabla }\cdot {\vec{J}}_b=0\\ \frac{1}{{\mu }_0}(\vec{\nabla }\times \vec{B})=\vec{J}\\ \overrightarrow{K_b}=\vec{M}\times \hat{n}\end{array}$$

${\vec{J}}_b=0$ if uniform $\vec{M}$.

Divergence of curl is always zero.

Magnetic vector potential $\vec{A}(\vec{r})$ $$\begin{array}{rl}\vec{A}(\vec{r})=\frac{{\mu }_0}{4\pi }\frac{\vec{m}\times \hat{R}}{R^2}& \text{single dipole}\\ \vec{A}(\vec{r})=\frac{{\mu }_0}{4\pi }\int \frac{\vec{M}({\vec{r}}^{\prime })\times \hat{R}}{R^2}d\tau & \text{magnetized obj}\\ \vec{A}(\vec{r})=\frac{{\mu }_0}{4\pi }\left[{\int }_V\frac{{\vec{J}}_b({\vec{r}}^{\prime })}{R}d{\tau }^{\prime }+{\oint }_S\frac{{\vec{K}}_b({\vec{r}}^{\prime })}{R}d{a}^{\prime }\right]& \text{mag. obj. J \& K}\end{array}$$

Dimensional analysis

Symbols: time (T), length (L), mass (M), electric current (I)

Del in other coordinates