マクスウェルの方程式 (Maxwell's equations)

Integral form

Gauss's law

∮SD⋅dA=Qf,S\oint_{S} \mathbf{D} \cdot \mathrm{d} \mathbf{A} = Q_{f,S}

Gauss's law for magnetism

∮SB⋅dA=0\oint_{S} \mathbf{B} \cdot \mathrm{d} \mathbf{A} = 0

Maxwell-Faraday equation

∮∂SE⋅dl=−∂ΦB,S∂t\oint_{\partial S} \mathbf{E} \cdot \mathrm{d} \mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t}

Ampere's circuital law

∮∂SH⋅dl=If,S+∂ΦD,S∂t\oint_{\partial S} \mathbf{H} \cdot \mathrm{d} \mathbf{l} = I_{f,S} + \frac {\partial \Phi_{D,S}}{\partial t}

Differential form

Gauss's law

divD=∇⋅D=ρf\mathrm{div} \mathbf{D} = \nabla \cdot \mathbf{D} = {\rho}_{f}

Gauss's law for magnetism

divB=∇⋅B=0\mathrm{div} \mathbf{B} = \nabla \cdot \mathbf{B} = 0

Maxwell-Faraday equation

rotE=∇×E=−∂B∂t\mathrm{rot} \mathbf{E} = \nabla \times \mathbf{E} = - \frac{ \partial \mathbf{B} }{ \partial t}

Ampere's circuital law

rotH=∇×H=Jf+∂D∂t\mathrm{rot} \mathbf{H} = \nabla \times \mathbf{H} = \mathbf{J}_{f} + \frac{ \partial \mathbf{D} }{ \partial t}

Definitions and units

D=ε0E+P=ϵ0EB=μ0(H+M)\begin{aligned} \mathbf{D} & = {\varepsilon}_{0} \mathbf{E} + \mathbf{P} = {\epsilon}_{0} \mathbf{E} \\ \mathbf{B} & = {\mu}_{0} \left( \mathbf{H} + \mathbf{M} \right) \\ \end{aligned}

D\mathbf{D} is the electric flux density. E \mathbf{E} is the electric field. P \mathbf{P} is the polarization density. B\mathbf{B} is the magnetic flux density. H \mathbf{H} is the magnetic field. M \mathbf{M} is the magnetization density. ε0≈5.526 35 …×107[A2s4/kg m3] {\varepsilon}_{0} \approx 5.526\ 35\ \ldots \times 10^{7} [ \mathbf{A}^{2} s^{4} / kg \ m^{3} ] is the permittivity of free space. μ0:=4π×10−7[N/A2] {\mu}_{0} := 4\pi\times 10^{-7} [ \mathbf{N} / \mathbf{A}^{2} ] is the permeability of free space.

Electromagnetic wave equation

(∇2−1c2∂∂t)E=□E=0(∇2−1c2∂∂t)B=□B=0\begin{aligned} \left( {\nabla}^2 - \frac{ 1 }{ c^2 } \frac{ \partial }{ \partial t} \right) \mathbf{E} & = \Box \mathbf{E} = 0 \\ \left( {\nabla}^2 - \frac{ 1 }{ c^2 } \frac{ \partial }{ \partial t} \right) \mathbf{B} & = \Box \mathbf{B} = 0 \\ \end{aligned}

c=c0=1μ0ε0=2.99792458×108[m/s] c = c_0 = { \frac{ 1 }{ \sqrt{ \mu_0 \varepsilon_0 } } } = 2.99792458 \times 10^8 [ m / s ] is the speed of light in vacuum.

□=gμν∂ν∂μ=(∇2−1c2∂∂t)\Box = g^{\mu\nu} \partial_\nu \partial_\mu = \left( {\nabla}^2 - \frac{ 1 }{ c^2 } \frac{ \partial }{ \partial t} \right)

□ \Box is the d'Alembertian operator.