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Electromagnetism
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Electricity · Magnetism
Covariant formulation
Electromagnetic tensor · EM Stress-energy tensor · Four-current · Electromagnetic four-potential

In physics, the electromagnetic stress-energy tensor is the portion of the stress-energy tensor due to the electromagnetic field.

Contents

Definition

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SI units

In free space in SI units, the electromagnetic stress-energy tensor is

T^{\mu\nu} = -\frac{1}{\mu_0}[ F^{\mu \alpha}F_{\alpha}{}^{\nu} + \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}] \,.

where Fμν is the electromagnetic tensor. Note: The tensor Tμν is a symmetric tensor.
And in explicit matrix form:

T^{\mu\nu} =\begin{bmatrix} \frac{1}{2}(\epsilon_0 E^2+\frac{1}{\mu_0}B^2) & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\ S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \ S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix},

with

Poynting vector \vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B},
Electromagnetic field tensor F_{\mu\nu}\!,
Minkowski metric tensor \eta_{\mu\nu}\! = \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}, and
Maxwell stress tensor \sigma_{ij} = \epsilon_0 E_i E_j + \frac{1} {{\mu _0 }}B_i B_j - \frac{1} {2}\left( {\epsilon_0 E^2 + \frac{1} {{\mu _0 }}B^2 } \right)\delta _{ij} .

Note that c^2=\frac{1}{\epsilon_0 \mu_0} where c is light speed.

CGS units

In free space in cgs-Gaussian units, we simply substitute \epsilon_0\, with \frac{1}{4\pi} and \mu_0\, with 4\pi\, :

T^{\mu\nu} = -\frac{1}{4\pi} [ F^{\mu\alpha}F_{\alpha}{}^{\nu} + \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}] \,.

And in explicit matrix form:

T^{\mu\nu} =\begin{bmatrix} \frac{1}{8\pi}(E^2+B^2) & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \ S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \ S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix}

where Poynting vector becomes the form:

\vec{S}=\frac{c}{4\pi}\vec{E}\times\vec{B}.

The stress-energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the unresolved Abraham-Minkowski controversy (however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)).

The element, T^{\mu\nu}\!, of the energy momentum tensor represents the flux of the μth-component of the four-momentum of the electromagnetic field, P^{\mu}\!, going through a hyperplane xν = constan t. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space-time) in general relativity.

Conservation laws

The electromagnetic stress-energy tensor allows a compact way of writing the conservation laws of linear momentum and energy by electromagnetism.

\partial_{\nu}T^{\mu \nu} + \eta^{\mu \rho} \, f_{\rho} = 0 \,

where fρ is the density of the (3D) Lorentz force on matter.

This equation is equivalent to the following 3D conservation laws

\frac{\partial u_{em}}{\partial t} + \vec{\nabla} \cdot \vec{S} + \vec{J} \cdot \vec{E} = 0 \,
\frac{\partial \vec{p}_{em}}{\partial t} - \vec{\nabla}\cdot \sigma + \rho \vec{E} + \vec{J} \times \vec{B} = 0 \,

where

Electromagnetic energy density (joules/meter3) is u_{em} = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2 \,
Poynting vector (watts/meter2) is \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \,
Density of electric current (amperes/meter2) is \vec{J} \,
Electromagnetic momentum density (newton·seconds/meter3) is \vec{p}_{em} = {\vec{S} \over c^2} \,
Maxwell stress tensor (newtons/meter2) is \sigma_{ij} = \epsilon_0 E_i E_j + \frac{1}{{\mu _0 }}B_i B_j - \frac{1}{2}\left( {\epsilon_0 E^2 + \frac{1}{{\mu _0 }}B^2 } \right)\delta _{ij} \,
Density of electric charge (coulombs) is \rho \,.

See also


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