The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor 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 impossibly 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.
As outlined below, the electromagnetic force is written in terms of E and B, using vector calculus and Maxwell's equations symmetry in the terms containing E and B are sought for, and introducing the Maxwell stress tensor simplifies the result.
Maxwell's equations in SI units in vacuum (for reference)
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)
This expression contains every aspect of electromagnetism and momentum and is relatively easy to compute. It can be written more compactly by introducing the Maxwell stress tensor,
All but the last term of f can be written as the tensor divergence of the maxwell stress tensor, giving:
As in the Poynting's theorem, the second term on the right side of the above equation can be interpreted as the time derivative of the EM field's momentum density, while the first term is the time derivative of the momentum density for the massive particles. In this way, the above equation will be the law of conservation of momentum in classical electrodynamics.
in the above relation for conservation of momentum, is the momentum flux density and plays a role similar to in Poynting's theorem.
The above derivation assumes complete knowledge of both ρ and J (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.
The element ij of the Maxwell stress tensor has units of momentum per unit of area per unit 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.
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 r is the shear in the radial (outward from the cylinder) direction, and t is the shear in the tangential (around the cylinder) direction. It is the tangential force which spins the motor. Br is the flux density in the radial direction, and Bt is the flux density in the tangential direction.
In electrostatics the effects of magnetism are not present. In this case the magnetic field vanishes, , 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 ).
The eigenvalues of the Maxwell stress tensor are given by: