User:Coder Dan/sandbox

Name
Integral equations	Differential equations
Meaning

Gauss's law
${\scriptstyle \partial \Omega }$ $\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {1}{\varepsilon _{0}}}\iiint _{\Omega }\rho \,\mathrm {d} V$	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$
The electric flux through a closed surface is proportional to the charge inside an enclosed volume.

Gauss's law for magnetism
${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$	$\nabla \cdot \mathbf {B} =0$
The magnetic flux through a closed surface is zero (i.e. there are no magnetic monopoles)

Maxwell–Faraday equation (Faraday's law of induction)
$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {l}}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$
The work per unit charge required to move a charge around a closed loop equals the rate of decrease of the magnetic flux through an enclosed surface.

Ampère's circuital law (with Maxwell's addition)
$\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\mu _{0}\left(\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\varepsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} \right)$	$\nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)$
The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (proportional to the rate of change of electric flux) through an enclosed surface.