Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 13A: Maxwell Stress Tensor and Electromagnetic Momentum Density

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

(13A.1)$$f=\left[\mathrm{\rho }E+J\times B\right].$$

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]

(13A.2)$$f=\left[\mathrm{\rho }E+J\times B\right]=\nabla \cdot \overline{T}-{\mathrm{\epsilon }}_0{\mathrm{\mu }}_0\frac{\partial S}{\partial t},$$

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

(13A.3)$$T_{ij}={\mathrm{\epsilon }}_0\left(E_i{E}_j-\frac{1}{2}{\mathrm{\delta }}_{ij}{E}^2\right)+\frac{1}{{\mathrm{\mu }}_0}\left(B_i{B}_j-\frac{1}{2}{\mathrm{\delta }}_{ij}{B}^2\right).$$

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,

(13A.4)$$\left[T_e\right]={\mathrm{\epsilon }}_0[\begin{array}{cc}\frac{1}{2}\left(E_x^2-E_y^2-E_z^2\right)& E_x{E}_y\end{array}$$