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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?

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Yes, you can take the tensors and their interpretation literally. At least if you know the material parameters well. However, it often take a special material and thousands of volts to see the effect.

There is work on electroactive polymers for example where you can apply a flexible carbon paste on either sides of an acrylic foam. If the material is held in a tension in a frame, and you apply a high voltage to the electrodes you can watch the Maxwell's stress, be converted to rather large strains. When you apply the voltage you can see the thickness of the material between the electrodes change some, but the lateral area of the electrode will change a lot.

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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?

It boils down to what you understand to be "genuine pressure". EM field can push on material bodies, and if the body has clear cut surfaces, we can think of this as pressure force. If you define tensions and pressures in vacuum via total stress tensor, which includes the EM field contribution even if not matter is present, then yes.

But if genuine pressure means some matter is present and experiences force, then no. Maxwell tensor in vacuum can be thought about as the set of quantities giving rate of EM momentum passing through any imaginable surface element. The quantities in the tensor need not be thought of as pressures and tensions in the continuum mechanics sense. It's just EM momentum flying around.

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