Useful Equations

Here are (most) of the equations I found useful in this class.

Dimensional analysis

Symbols: time (T), length (L), mass (M), electric current (I), charge (Q = TI), energy (T${}^{-2}$L${}^2$M)

Wikipedia's dimensional analysis page has some other good simple cases.

Electrostatics

Coulomb's law (both $\vec{F}$ and $\vec{E}$) $$\begin{array}{rl}\vec{F}& =\frac{1}{4\pi {ϵ}_0}\frac{qQ}{{\mathcal{R}}^2}\hat{\mathcal{R}}\vec{E}=\frac{1}{4\pi {ϵ}_0}\frac{q}{{\mathcal{R}}^2}\hat{\mathcal{R}}\\ & \equiv \frac{1}{4\pi {ϵ}_0}\frac{qQ}{{\mathcal{R}}^3}\vec{\mathcal{R}}\equiv \frac{1}{4\pi {ϵ}_0}\frac{q}{{\mathcal{R}}^3}\vec{\mathcal{R}}\end{array}$$

Notes:

• $\vec{r}$ points from the origin to point we're interested in, $\overrightarrow{r^{\prime }}$ from the origin to some 'unit charge' $dq$, thus $\vec{\mathcal{R}}=\vec{r}-\overrightarrow{r^{\prime }}$.
  • Imagine from one point charge to another.
• $\vec{F}=Q\vec{E}$

Gauss's law $$\oint \vec{E}\cdot d\vec{a}=\frac{Q_{\text{enc}}}{{ϵ}_0}\vec{\nabla }\cdot \vec{E}=\frac{\rho }{{ϵ}_0}$$

Multipole expansion

Monopole moment is total charge $$Q=\int dq{V}_{\text{mon}}(\vec{r})=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}$$ Dipole moment points from south to north, origin-dependent $$\vec{\rho }=\sum _{i=1}^n{q}_i{\vec{r}}_i^{\prime }\equiv \int {\vec{r}}^{\prime }dq{V}_{\text{dip}}=\frac{1}{4\pi {ϵ}_0}\frac{\vec{\rho }}{r^2}$$ Quadrupole moment describes how charge dist. behaves in nonuniform fields $$Q_{ij}=\int \rho (\vec{r})\left(\frac{3}{2}{r}_i{r}_j-\frac{1}{2}{r}^2{\delta }_{ij}\right)d^3\vec{r}$$

$dq$, the charge integration element, is $$dq=\{\begin{array}{ll}\lambda (\vec{r})dl& \text{linear charges}\\ \sigma (\vec{r})da& \text{surface charges}\\ \rho (\vec{r})d\tau & \text{volume charges}\end{array}$$

The electric potential for all multipole expansion terms is $$V(\vec{r})=\frac{1}{4\pi {ϵ}_0}\sum _{n=0}^{\infty }\frac{1}{r^{n+1}}\int (r^{\prime }{)}^n{P}_n(\cos (\alpha ))dq$$

• $P_n$ represents the $n$th Legendre polynomial
• $\alpha$ is the angle between $\vec{r}$ and ${\vec{r}}^{\prime }$

Legendre polynomials

Calculated according to Rodrigues' formula: $$P_n(x)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}(x^2-1{)}^n$$