[[Maxwell's equations]]
# Ampère's circuital Law

**Ampère's circuital law**, with a correction due to Maxwell, relates the [[circulation]] of a magnetic field around an area to the [[Current density]] passing through it:
$$
\begin{align*}
\oint_{\partial\Omega}\vab B\cdot d\vab \ell = \mu_{0}\left(\iint_{\Sigma}\vab J \cdot d\vab a + \epsilon_{0} \frac{ d }{ d t }  \iint_{\Sigma} \vab E \cdot d \vab a\right) = \mu_{0}\left( I_{\Sigma} + \epsilon_{0}\frac{ \partial \Phi_{E,\Sigma} }{ \partial t }  \right)
\end{align*}
$$
^Integral

or equivalently by [[Stokes's theorem]]
$$
\begin{align*}
\vab \nabla \times\vab B = \mu_{0}\left( \vab J +\epsilon_{0} \frac{ \partial\vab E }{ \partial t }  \right) 
\end{align*}
$$
^Differential

See also [[Ampère's law for magnets]]

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#state/tidy | #lang/en | #SemBr