The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor in three dimensions that is used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum.
In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law.
When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines.
It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.
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.
As outlined below, the electromagnetic force is written in terms of
Using vector calculus and Maxwell's equations, symmetry is sought for in the terms containing
, and introducing the Maxwell stress tensor simplifies the result.
the force per unit volume is in the above relation for conservation of momentum,
is the momentum flux density and plays a role similar to
(both free and bounded charges and currents).
For the case of nonlinear materials (such as magnetic iron with a BH-curve), the nonlinear Maxwell stress tensor must be used.
is the dyadic product, and the last tensor is the unit dyad: The element
th axis (in the negative direction) per unit of time.
element of the tensor can also be interpreted as the force parallel to the
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.
It has recently been shown that the Maxwell stress tensor is the real part of a more general complex electromagnetic stress tensor whose imaginary part accounts for reactive electrodynamical forces.
[2] 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: For cylindrical objects, such as the rotor of a motor, this is further simplified to: where
is the shear in the radial (outward from the cylinder) direction, and
In electrostatics the effects of magnetism are not present.
, and we obtain the electrostatic Maxwell stress tensor.
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.
, 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
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.