# Maxwell Equations 

##### Definition

The Maxwell Equations are 4 fundamental equations that describe the relation between the quantities of electromagnetism.

Name Integral Version Differential Version
Gauss’s law: $\mathop{\circ\!\!\!\iint}_{\partial \Omega} \boldsymbol E \cdot \,\text{d}A = \frac{1}{\epsilon_0} \iiint_{\Omega} \rho \,\text{d}V$ $\text{div}\boldsymbol D = \varrho$
Faraday’s law of induction $\oint_{\partial A} \boldsymbol E \cdot \,\text{d}\boldsymbol s = - \iint_{A} \frac{\partial \boldsymbol B}{\partial t} \cdot \,\text{d}\boldsymbol A$ $\text{rot}\boldsymbol E + \frac{\partial \boldsymbol B}{\partial t} = 0$
Gauss’s law for magnetism $\mathop{\circ\!\!\!\iint}_{\partial \Omega} \boldsymbol B \cdot \,\text{d}A = 0$ $\text{div}\boldsymbol B = 0$
Ampère’s law $\oint_{\partial A} \boldsymbol H \cdot \,\text{d}\boldsymbol s = \iint_{A} \boldsymbol j \cdot \,\text{d}\boldsymbol A + \iint_{A} \frac{\partial \boldsymbol D}{\partial t} \cdot \,\text{d}\boldsymbol A$ $\text{rot}\boldsymbol H = \boldsymbol j + \frac{\partial \boldsymbol D}{\partial t}$

## Gauss’s law

Explanation: The differential form states that if there exists electric charge somewhere, then the divergence of $D$ at that point is nonzero, otherwise it is equal to zero. The integral form states that the amount of charge inside a volume $V$ of enclosed charge is equal to the total amount of Electric Flux $D$ exiting the surface $S$.