From the electromagnetic force equation (13.105), we can write the equation for the force density f as follows:

Assume the medium is free space. We can eliminate the volume charge density ρ and the current density J by using Maxwell’s equations, and after using some vector identities, we can write [1]

where the tensor $\overline{T}$, called the Maxwell stress tensor, in free space medium is given by [1,2,3]

The first term on the right side of (13A.3) can be called the electric stress tensor [T_e]. If written out in matrix form in Cartesian coordinates,

The second term on the right side (13A.3) can be similarly written out. The term S in (13A.2) is the usual Poynting vector in free space given by (1.26).

The force F on a volume V bounded by a closed surface s in free space medium containing charges and currents can be obtained by integrating (13A.2). After using an identity similar to the vector divergence theorem involving a tensor of second order [3, p. 99] $\overline{T}$,

we can write

Equation 13A.6 is the statement of conservation of momentum, parallel to that of Poynting theorem for conservation of energy given by (1.27). Thus, one interprets

as the electromagnetic momentum density contained in the fields. Let us denote the electromagnetic momentum in the volume V by G. Substituting (13A.7) in (13A.6),

Equation 13A.8 in words states that the time rate of increase of the total momentum (the sum of the mechanical momentum and electromagnetic momentum) is equal to the momentum carried inward [3, p. 104] of the volume V through its bounding surface s. Proper interpretation of the abbreviated tensor divergence theorem stated in (13A.5) is given in [3, p. 104] to explain the inward word.

The Maxwell stress tensor can be used to calculate the radiation pressure due to a plane electromagnetic wave obliquely incident on a conductor or dielectric half-space moving with relativistic speed (see Appendix 14B).