Maxwell's Stress Tensor and Conservation of Momentum in Electromagnetic Field

Somnath Datta

NCERT (retired)

Maxwell's Stress Tensor owes its origin to the notion prevailing before the advent of relativity that `action at a distance' is actually a mechanical interaction, like push and pull, and is transmitted by an assumed mechanical property of the Aether which pervaded all space, in particular vacuum. Even after withrawal of Aether this tensor has a useful role to play not only in formulating conservation of momentum in a time varying electromagnetic field, but also in simplifying several problems in electrostatics and magnetostatics, by removing the distinction between the field caused by `external sources' and the total field surrounding a distribution of charges and currents. This tensor is to be constructed on the principle that the force acting on unit volume of a distribution of electric charges and currents equals the divergence of the stress tensor.. Our derivations of the stress tensors corresponding to electrostatic field, magnetostatic field, and time varying electromagnetic field respectively, are based on a single vector identity and application of Maxwell's field equations. We have worked out two examples of how the force on an isolated system can be calculated by surrounding it with a sphere of some radius r and integrating the stress vector over the entire surface, namely, an isolated electric charge in the electrostatic field of another charge, and an isolated magnetic dipole in the magnetostatic field of another magnetic dipole. We have taken the stress tensor to its logical end by writing momentum conservation in a time varying electromagnetic field.