26 JAN. 2017 LEARNING MAXWELL DIFFERENTIAL +Integral forms.

26 JAN. 2017
LEARNING MAXWELL
DIFFERENTIAL +Integral forms.
+

Formulation in SI units convention

Name	Integral equations	Differential equations	Meaning
Gauss's law	${\displaystyle {\scriptstyle \partial \Omega }}$ ${\displaystyle \mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {1}{\varepsilon _{0}}}\iiint _{\Omega }\rho \,\mathrm {d} V}$	${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}$	The electric flux leaving a volume is proportional to the charge inside.
Gauss's law for magnetism	${\displaystyle {\scriptstyle \partial \Omega }}$ ${\displaystyle \mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}$	${\displaystyle \nabla \cdot \mathbf {B} =0}$	There are no magnetic monopoles; the total magnetic flux through a closed surface is zero.
Maxwell–Faraday equation (Faraday's law of induction)	${\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} }$	${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$	The voltage induced in a closed circuit is proportional to the rate of change of the magnetic flux it encloses.
Ampère's circuital law (with Maxwell's addition)	${\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\mu _{0}\varepsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} }$	${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)}$	The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) it encloses.