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# (WIP) 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 distributed along a 1D curve $L$, where $Q'$ $[C / m]$ is the charge density: 
$$\mathbf E = \frac{1}{4\pi\varepsilon_0}\int_C\frac{Q'\mathrm d\ell}{r^2}\mathbf r_0$$

Electric field created by charge distributed 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 distributed 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: 
$$\nabla = \frac{\partial}{\partial x}\mathbf i_x +  \frac{\partial}{\partial y}\mathbf i_y +  \frac{\partial}{\partial z}\mathbf i_z$$

Electric potential $V$ created by a single point charge $Q$ 
$$V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$$ 

Electric potential $V$ created by point charges $Q_1, Q_2, ..., Q_N$:
$$V = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^{N}\frac{Q_i}{r_i}$$

Electric potential $V$ created by charge distributed along a 1D curve $C$ with $Q'$ as the charge density:
$$V = \frac{1}{4\pi\varepsilon_0}\int_C\frac{Q'\mathrm d\ell}{r}$$

Electric potential $V$ created by charge distributed across a 2D surface $S$ with $\rho_S$ as the charge density:
$$V = \frac{1}{4\pi\varepsilon_0}\int_S\frac{\rho_S\mathrm dS}{r}$$

Electric potential $V$ created by charge distributed across a 3D volume $V$ with $\rho$ as the charge density:
$$V = \frac{1}{4\pi\varepsilon_0}\int_V\frac{\rho\mathrm dV}{r}$$

Electric potential $V$ as the line integral of an electric field $\mathbf E$ from arbitrary point $P$ to a reference point $R$:
$$V = \int_P^R\mathbf E\cdot\mathrm d\mathbf\ell$$

Electric potential $V'$ when moving the refernce point from $R$ to $R'$ in previous relation:
$$V' = V + \int_{R}^{R'}\mathbf E\cdot\mathrm d\mathbf \ell$$

Electric potential around a charge distributed along a line with charge density Q':
$$V = \frac{Q'}{2\pi\varepsilon_0}\ln\frac{r_R}{r}$$

Potential difference bewteen two points $A$ and $B$:
$$U_{AB} = V_A - V_B = \int_A^B\mathbf E\cdot\mathrm d\mathbf\ell$$

### 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:
$$\nabla\mathbf E = \frac{\rho}{\varepsilon_0}$$

### Maxwell's equations in Electrostatics
Faraday's law in differential form in electrostatics:
$$\nabla\cdot\mathbf E = 0$$

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

Faraday's law in integral form in electrostatics
$$\oint_C\mathbf E\cdot\mathrm d\mathbf\ell = 0$$

Gauss's law in integral form in electrostatics
$$\oint_S\mathbf E\cdot\mathrm d\mathbf S = \frac{1}{\varepsilon_0}\iiint_V\rho\cdot\mathrm dV$$

### 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
Electric potential of an electric dipole:
$$V = \frac{1}{4\pi\varepsilon_0}\left(\frac{Q}{r^+} - \frac{Q}{r^-}\right)$$

Electric field of an electric dipole ($\mathbf E = -\nabla\cdot V$):
$$\mathbf E = -\nabla\cdot V$$