$$ \textbf E=-\nabla V $$
Electric field and potential are easier to calculate with the right coordinate systems and choice of origin. The choice of origin may sometimes not be the area of most symmetry. For example, if the polar form is known, it may be easier.
Both electric field and potential obey the law of superposition, as divergence and curl are linear functions. The superposition of two valid fields produce another valid field.
| Name | Differential equations | Integral form |
|---|---|---|
| Gauss’s law | $\nabla \cdot \textbf E= \rho/\epsilon_0$ | too lazy |
| Maxwell-Faraday equation | $\nabla \times \textbf E = -\partial \textbf B/ \partial t$ | |
| Gauss’s law for magnetism | $\nabla \cdot \textbf B= 0$ | |
| Ampere’s circuital law | $\nabla \times \textbf B = \mu_0 (\textbf J + \epsilon_0 \,\partial \textbf E/ \partial t)$ |
Add this to the Lorentz force and that is basically the entirety of EM
Note that usually we will ignore the change in the E field in Ampere’s law, and this is equivalent to ignoring radiation effects, or a quasistatic approximation which means that the currents do not change quickly enough for effects to be on the order of the speed of light.
Transforming to a frame moving at speed $v$, the parallel component of E and B (to the velocity) is conserved.
$$ \mathbf E_\parallel'=\mathbf E_\parallel \qquad \mathbf B_\parallel'=\mathbf B_\parallel \\ \mathbf E_\perp'=\gamma (\mathbf E_\perp + \mathbf v \times \mathbf B) \qquad \mathbf B_\perp'=\gamma\left (\mathbf B_\perp-\frac{\mathbf v}{c^2}\times \mathbf E\right) $$
Using the divergence theorem, Gauss’s law can be rewritten as:
$$ \Phi=\oiint \mathbf E \cdot \mathrm{d}\mathbf S=\frac{Q}{\epsilon_0} $$
By constructing a Gaussian cylinder around an infinite sheet of charge, the electric field can be obtained as $E=\sigma/2\epsilon_0$
For a thin sheet of charge (not conductor),
$$ P=\sigma E_{ext}=\frac{\sigma(E_{in}+E_{out})}{2} $$
This is easily proven by just adding the electric field of the sheet of charge and the external field