Maxwell stress tensor

In classical electromagnetism, the Maxwell stress tensor (named after James Clerk Maxwell) is the stress tensor of an electromagnetic field. In tensor index notation it is given by:

$\sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2}}\left(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2}\right)\delta _{ij}$

where $\epsilon _{0}$ is the electric constant, $\mu _{0}$ the magnetic constant, E the electric field, B the magnetic field, and $\delta _{ij}$ is the Kronecker delta. In the Gaussian system, it is given by:

$\sigma _{ij}={\frac {1}{4\pi }}\left(E_{i}E_{j}+H_{i}H_{j}-{\frac {1}{2}}\left(E^{2}+H^{2}\right)\delta _{ij}\right)$

where H is the magnetizing field. ( $E^{2}$ etc is the field magnitude squared, though this is slightly non-standard in tensor notation.)

The elements $ij$ have units of force per unit of area (negative pressure), denoting the force parallel to the $i$th axis suffered by a surface normal to the $j$th axis per unit of area. As such the diagonal elements give the tension acting on the plane normal to the corresponding axis and the off-diagonal elements represent shear stress in the plane.

Expressed in vector/dyad notation we would have:

${\overset {\leftrightarrow }{\boldsymbol {\sigma }}}={\frac {1}{4\pi }}\left[\mathbf {E} \otimes \mathbf {E} +\mathbf {H} \otimes \mathbf {H} -{\frac {E^{2}+H^{2}}{2}}\mathbb {I} \right]$

where $\otimes$ is the dyadic product, and the last tensor is the unit dyadic:

$\mathbb {I} \equiv {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}=\mathbf {\hat {x}} \otimes \mathbf {\hat {x}} +\mathbf {\hat {y}} \otimes \mathbf {\hat {y}} +\mathbf {\hat {z}} \otimes \mathbf {\hat {z}}$

In the relativistic formulation of electromagnetism, the nine components of the Maxwell stress tensor appear, negated, as components of the electromagnetic stress–energy tensor, which is the electromagnetic component of the total stress–energy tensor. The latter describes the density and flux of energy and momentum in spacetime.

The following derivation defines the Maxwell stress tensor by way of introducing it so as to show how electromagnetic body forces arise from electrical and magnetic fields. For an explanation more directly as to how the tensor describes stress (in the absence of body forces) see the web article by J.S.Reid .