EM Final Fun Equation Sheet

Annoying thing

$$\nabla \cdot \frac{\hat{r}}{r^2}=4\pi {\delta }^3(\vec{r})$$

Continuity Equation

$$\frac{\partial p}{\partial t}+\nabla \cdot \vec{j}=0$$

Power Integral for Poynting Vector

$$P=\int \vec{s}\cdot da$$

Electromagnetic Waves

$$E(\vec{r},t)=E_0{e}^{i(\vec{k}\vec{r}-\omega t)}\hat{n}$$

Where $\hat{n}$ is the direction of polarization

$$B(\vec{r},t)=\frac{1}{c}\hat{k}\times \vec{E}$$

Wave-number Eq

$$k=\omega c$$

EM Waves in matter, n is index of refraction

$$V=\frac{c}{n}=\frac{1}{\sqrt{\mu ϵ}}$$

Brewster Angle

$$\tan ({\theta }_b)\approx \frac{n_2}{n_1}$$

In Conducting Material

$$j=\sigma \vec{E}$$

Gauge Transformation

$$\nabla \cdot A=\frac{1}{c^2}\frac{\partial V}{\partial t}$$ $$A^{\prime }=A+\nabla \lambda$$ $$V^{\prime }=V-\frac{\partial \lambda }{\partial t}$$ $${\nabla }^2\Lambda =\frac{1}{c^2}\frac{{\partial }^2\lambda }{\partial t^2}$$

Retarded Potentials and Time

$$V=\frac{1}{4\pi {ϵ}_0}{\int }_{v^{\prime }}\frac{\rho (r^{\prime },t_r)}{r}d{V}^{\prime }$$ $$A=\frac{{\mu }_0}{4\pi }{\int }_{v^{\prime }}\frac{j(r^{\prime },t)}{r}d{V}^{\prime }$$ $$t_r=t-\frac{r}{c}$$

Where

$$r=|r-r^{\prime }|$$