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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 \(\(\oiint_{\partial \Omega} \vec B \cdot \diff A = 0\)\) \(\(\div \vec B = 0\)\)
Ampère'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#


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 \(S\).