Maxwell’s equation

by

Maxwell’s equation in vacuum

NameIntegral equationsDifferential equations
Gauss’s law\oiint{\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}}}
Gauss’s law for magnetism\oiint{\scriptstyle \partial \Omega }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0\nabla \cdot \mathbf {B} =0
Maxwell–Faraday equation(Faraday’s law of induction){\displaystyle \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}}
Ampère’s circuital law (with Maxwell’s addition){\displaystyle {\begin{aligned}\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)\\\end{aligned}}}\nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)

Next, We obtain the electromagnetic wave equation in a vacuum. It is also called Helmholtz equation. Take the Curl of the Faraday equation

\(\nabla \times \left(\nabla \times \mathbf {E} \right)=\nabla \times \left(-{\frac {\partial \mathbf {B} }{\partial t}}\right)=-{\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {B} \right)=-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}\)

Use the Bac-Cab rule for the Vector,

\(\nabla \times \left(\nabla \times \mathbf {E} \right)=\nabla \left(\nabla \cdot \mathbf {E} \right)-\nabla ^{2}\mathbf {E} \)

It is sourceless in the vacuum,

\(\nabla\cdot E=0\)

\(\nabla^2 E-\epsilon_0\mu_0 \frac{\partial^2 E}{\partial t^2}=0\)

In time harmonic wave \(\frac{\partial}{\partial t}=j \omega\)

So the wave equation is

\(\nabla^2 E+\omega^2\epsilon_0\mu_0 E=0\)

Maxwell’s equation in material

NameIntegral equations Differential equations
Gauss’s law\oiint{\scriptstyle \partial \Omega } \mathbf {D} \cdot \mathrm {d} \mathbf {S} =\iiint _{\Omega }\rho _{\text{f}}\,\mathrm {d} V\nabla \cdot \mathbf {D} =\rho _{\text{f}}
Gauss’s law for magnetism\oiint{\scriptstyle \partial \Omega }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0\nabla \cdot \mathbf {B} =0
Maxwell–Faraday equation (Faraday’s law of induction)\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {d}{dt}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}
Ampère’s circuital law (with Maxwell’s addition){\displaystyle {\begin{aligned}\oint _{\partial \Sigma }&\mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=\\&\iint _{\Sigma }\mathbf {J} _{\text{f}}\cdot \mathrm {d} \mathbf {S} +{\frac {d}{dt}}\iint _{\Sigma }\mathbf {D} \cdot \mathrm {d} \mathbf {S} \\\end{aligned}}}\nabla \times \mathbf {H} =\mathbf {J} _{\text{f}}+{\frac {\partial \mathbf {D} }{\partial t}}

Leave a Comment