Table of Contents

Equations of Electromagnetism

Electrostatics

Coulomb's law

Attractive force $Q_2$ feels thanks to $Q_1$'s electric field: $$\mathbf F_ = \frac{1}{4\pi\varepsilon_0}\frac{Q_1Q_2}{r^2}\mathbf r_{012}$$

Electric field

Electric field created by a single point charge $Q$: $$\mathbf E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\mathbf r_0$$

Electric field created by multiple point charges $Q_1, Q_2, ..., Q_N$: $$\mathbf E = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^{N}\frac{Q_i}{r^2_i}\mathbf r_{0i}$$

Electric field created by charge smeared across a 1D line $L$, where $Q'$ $[C / m]$ is the charge density: $$\mathbf E = \frac{1}{4\pi\varepsilon_0}\int_L\frac{Q'\mathrm d\ell}{r^2}\mathbf r_0$$

Electric field created by charge smeared across a 2D surface $S$ where $\rho_S$ $[C / m^2]$ is the charge density: $$\mathbf E = \frac{1}{4\pi\varepsilon_0}\int_S\frac{\rho_S\mathrm dS}{r^2}\mathbf r_0$$

Electric field created by charge smeared across a 3D volume $V$ where $\rho$ $[C / m^3]$ is the charge density: $$\mathbf E = \frac{1}{4\pi\varepsilon_0}\int_V\frac{\rho\mathrm dV}{r^2}\mathbf r_0$$

Electric potential

Electric field $\mathbf E$ created by the electric potential $V$: $$\mathbf E = -\nabla V$$ Nabla operator $\nabla$ is defined as: $$\nabla = \frac{\partial}{\partial x}\mathbf i_x + \frac{\partial}{\partial y}\mathbf i_y + \frac{\partial}{\partial z}\mathbf i_z$$

Gauss's law

Gauss's law (integral form): $$\oint_S\mathbf E \cdot\mathrm dS = \frac{1}{\varepsilon_0}\int_V\rho\mathrm dV$$

Gauss's law in differential form: $$\nabla\mathbf E = \frac{\rho}{\varepsilon_0}$$

Poisson and Laplace equations

Electric potential in vaccuum satisfies the Poisson equation: $$\nabla^2 V = -\frac{\rho}{\varepsilon_0}$$

Without charge $\rho$, Poisson equation reduces to Laplace equation: $$\nabla^2 V = 0$$

Nabla squared operator is defined as: $$\nabla = \frac{\partial^2}{\partial x^2}\mathbf i_x + \frac{\partial^2}{\partial y^2}\mathbf i_y + \frac{\partial^2}{\partial z^2}\mathbf i_z$$

Electrostatic dipole

$$V = \frac{p\cos\thta}{}$$