# Interpretation of surface forces from the Maxwell Stress tensor

I am studying classical electromagnetics, and reading about the Maxwell Stress tensor. But I’m unsure about what it means for a field to exert forces on a volume element. As I understand it, $$T^{ij}dS_j$$ (no summation) is the $$i$$ component of the force acting on a surface element oriented in the $$j$$ direction. What I’m trying to understand is what this means, for example, for a volume that contains no charge.

For brevity, I’ll assume no currents, so we can deal only with the electrostatic part of the tensor. Say we have a parallel plate capacitor supporting a uniform electric field $$E_0\mathbf{\hat{z}}$$. Then for a small cube inside the field, if we blindly follow the math, I believe we get $$T = \frac{1}{2}\varepsilon_0\begin{bmatrix}E_0^2&0&0\\0&E_0^2&0\\0&0&-E_0^2\end{bmatrix},$$ (using the inward-is-positive sign convention).

This is, of course, divergenceless, so the net force on the volume element is zero (as it must be, with no charges!), and this chimes with the total surface integral, since any force one one face is balanced by that on the other. But the individual components suggest that the field is exerting a “stretching” stress on the faces normal to the $$z$$-axis, and a “compressing” stress on the faces normal to the $$x$$- and $$y$$-axes.

Is this correct thinking? Is there a genuine pressure on such volume elements, or should I not be taking the tensor components too literally, and only interpreting them as part of a total integral?