# Electromagnetic stress-energy tensor

Electromagnetic stress-energy tensor

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

Definition

In free space in SI units, the electromagnetic stress-energy tensor is:And in explicit matrix form::,

with:Poynting vector $vec\left\{S\right\}=frac\left\{1\right\}\left\{mu_0\right\}vec\left\{E\right\} imesvec\left\{B\right\}$,:electromagnetic field tensor $F_\left\{mu u\right\}!$,:Minkowski metric tensor $eta_\left\{mu u\right\}!$, and:Maxwell stress tensor $sigma_\left\{ij\right\} = epsilon_0 E_i E_j + frac\left\{1\right\}$mu _0 B_i B_j - frac{1}{2}left( {epsilon_0 E^2 + frac{1}mu _0 B^2 } ight)delta _{ij} .Note that $c^2=frac\left\{1\right\}\left\{epsilon_0 mu_0\right\}$ where "c" is light speed.

CGS

In free space in cgs units, we simply substitute $epsilon_0,$ with $frac\left\{1\right\}\left\{4pi\right\}$ and $mu_0,$ with $4pi,$ ::And in explicit matrix form::

where Poynting vector becomes the form: :$vec\left\{S\right\}=frac\left\{c\right\}\left\{4pi\right\}vec\left\{E\right\} imesvec\left\{B\right\}$.

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, $T^\left\{mu u\right\}!$, of the energy momentum tensor represents the flux of the &mu;th-component of the four-momentum of the electromagnetic field, $P^\left\{mu\right\}!$, going through a hyperplane $x^\left\{ u\right\} = constant$. 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.:$partial_\left\{ u\right\}T^\left\{mu u\right\} + eta^\left\{mu ho\right\} , f_\left\{ ho\right\} = 0 ,$

where $f_\left\{ ho\right\}$ is the density of the (3D) Lorentz force on matter.

This equation is equivalent to the following 3D conservation laws:$frac\left\{partial u_\left\{em\left\{partial t\right\} + vec\left\{ abla\right\} cdot vec\left\{S\right\} + vec\left\{J\right\} cdot vec\left\{E\right\} = 0 ,$:$frac\left\{partial vec\left\{p\right\}_\left\{em\left\{partial t\right\} - vec\left\{ abla\right\}cdot sigma + ho vec\left\{E\right\} + vec\left\{J\right\} imes vec\left\{B\right\} = 0 ,$

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

ee also

*Stress-energy tensor
*Covariant formulation of classical electromagnetism

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