Maxwell Stress Tensor - Equation

Equation

In physics, the Maxwell stress tensor is the stress tensor of an electromagnetic field. As derived above in SI units, it is given by:

\sigma_{i j} = \epsilon_0 E_i E_j + \frac{1} {{\mu _0 }}B_i B_j - \frac{1}{2}\bigl( {\epsilon_0 E^2 + \tfrac{1} {{\mu _0 }}B^2 } \bigr)\delta _{ij} ,

where ε0 is the electric constant and μ0 is the magnetic constant, E is the electric field, B is the magnetic field and δij is Kronecker's delta. In Gaussian cgs unit, more commonly used in physics textbooks, it is given by:

\sigma_{i j}=\frac{1}{4\pi}\left(E_{i}E_{j}+H_{i}H_{j}- \frac{1}{2}(E^2+H^2)\delta_{ij}\right),

where H is the magnetizing field.

An alternative way of expressing this tensor is:

\overset{\leftrightarrow }{ \mathbf{\sigma} } = \frac{1}{4\pi} \left ,

where ⊗ is the dyadic product.

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

These units can also be seen as units of force per unit of area (negative 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 tension (pulling) 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 is given by the off-diagonal elements of the stress tensor.

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