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Concept
Maxwell stress tensor
Summary
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 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 Maxwell's tensor appears as a part 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 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.
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 and (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.
In physics, the Maxwell stress tensor is the stress tensor of an electromagnetic field. As derived above in SI units, it is given by:
where is the electric constant and is the magnetic constant, is the electric field, is the magnetic field and is Kronecker's delta. In Gaussian cgs unit, it is given by:
where is the magnetizing field.
Official source
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