Maxwell stress tensor

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In physics, the Maxwell stress tensor is the stress tensor of an electromagnetic field. In cgs units, it is given by:

$\sigma_{\alpha\beta}=\frac{1}{4\pi}(E_{\alpha}E_{\beta}+H_{\alpha}H_{\beta}- \frac{1}{2}(E^2+H^2)\delta_{\alpha\beta})$ ,

where E is the electric field, H is the magnetic field and $δ$ is Kronecker's delta.

In SI units, it is given by:

$\sigma _{\alpha \beta } = \varepsilon _o E_\alpha E_\beta + \frac{1} {{\mu _0 }}B_\alpha B_\beta - \frac{1} {2}\left( {\varepsilon _o E^2 + \frac{1} {{\mu _0 }}B^2 } \right)\delta _{\alpha \beta }$ ,

where $\varepsilon _o$ and $μ 0$ are the permittivity and permeability of vacuum respectively.

The element ij of the Maxwell stress tensor has units of momentum per unit of area and unit of time and gives the flux of momentum parallel to the ith axis crossing a surface normal to the jth axis per unit of time.

These units can also be seen as units of force per unit of area (pressure), and the ij element of the tensor can also be interpreted as the force parallel to the ith axis suffered by a surface normal to the jth axis per unit of area. Indeed the diagonal elements give the pressure acting on a differential area element normal to the corresponding axis. Unlike forces due to the pressure of an ideal gas, an area element in the electromagnetic field also feels a force in a direction that is not normal to the element. This shear (rather than pressure) is given by the off-diagonal elements of the stress tensor.