# Maxwell Equations#

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

Name Integral Version Differential Version
Gauss's law: $$\(\oiint_{\partial \Omega} \vec E \cdot \diff A = \frac{1}{\epsilon_0} \iiint_{\Omega} \rho \diff V$$\) $$\(\div \vec D = \varrho$$\)
Faraday's law of induction $$\(\oint_{\partial A} \vec E \cdot \diff \vec s = - \iint_{A} \frac{\partial \vec B}{\partial t} \cdot \diff \vec A$$\) $$\(\rot \vec E + \frac{\partial \vec B}{\partial t} = 0$$\)
Gauss's law for magnetism $$\(\oiint_{\partial \Omega} \vec B \cdot \diff A = 0$$\) $$\(\div \vec B = 0$$\)
AmpĂ¨re's law $$\(\oint_{\partial A} \vec H \cdot \diff \vec s = \iint_{A} \vec j \cdot \diff \vec A + \iint_{A} \frac{\partial \vec D}{\partial t} \cdot \diff \vec A$$\) $$\(\rot \vec H = \vec j + \frac{\partial \vec 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 $$S$$.