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:

where is the electric constant, the magnetic constant, E the electric field, B the magnetic field, and is the Kronecker delta. In the Gaussian system, it is given by:

where H is the magnetizing field. ( etc is the field magnitude squared, though this is slightly non-standard in tensor notation.)

The elements have units of force per unit of area (negative pressure), denoting the force parallel to the th axis suffered by a surface normal to the 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:

in SI units or

in CGS units where is the dyadic product, and the last tensor is the unit dyadic:

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 .

Derivation

As outlined below, the electromagnetic force is written in terms of and . Using vector calculus and Maxwell's equations, symmetry is sought for in the terms containing and , and introducing the Maxwell stress tensor simplifies the result.

The derivation follows Griffiths and applies in the case of a vacuum.

+ Maxwell's equations in SI units in vacuum (for reference) Name Differential form Gauss's law (in vacuum) Gauss's law for magnetism Maxwell–Faraday equation(Faraday's law of induction) Ampère's circuital law (in vacuum)(with Maxwell's correction)

In magnetostatics If the field is only magnetic (which is largely true in motors, for instance), some of the terms drop out, and the equation in SI units becomes:

In electrostatics

In electrostatics the effects of magnetism are not present. In this case the magnetic field vanishes, i.e. , and we obtain the electrostatic Maxwell stress tensor. It is given in component form by

and in symbolic form by

where is the appropriate identity tensor usually .

Eigenvalue The eigenvalues of the Maxwell stress tensor are given by:

These eigenvalues are obtained by iteratively applying the matrix determinant lemma, in conjunction with the Sherman–Morrison formula.

Noting that the characteristic equation matrix, , can be written as

where

we set

Applying the matrix determinant lemma once, this gives us

Applying it again yields,

From the last multiplicand on the RHS, we immediately see that is one of the eigenvalues.

To find the inverse of , we use the Sherman-Morrison formula:

Factoring out a term in the determinant, we are left with finding the zeros of the rational function:

Thus, once we solve

we obtain the other two eigenvalues.

Maxwell stress tensor

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