Maxwell’s Stress Tensor
The total electromagnetic force on the charges in volume is
Then, the force per volume is
Using the Maxwell equations, we can get rid of and using the Maxwell’s equations:
The vector calculus identity for differentiating cross products is:
Using Faraday’s Law
Plugging into the above identity gives
Hence, we can write
Using the last Maxwell equation, we can write this to be more symmetrical. Throw in , since
We can simplify the above equation using another vector calculus identity:
A similar derivation is done for . Then,
Which has some symmetry, but is somewhat tedious to look at. Using the Maxwell stress tensor:, which has two indicies
The divergence of has
and thus the force per volume is written in the elegant form:
where is the Poynting vector.
Integrating both sides, we have
If no energy is leaving or entering, we can say the system is static. Then, we have that
Interpretation of the Stress Tensor
is the force per unit area acting on the surface. The first index represents the th direction the force is coming from, and the second index represents the normal vector for the surface in the th direction.
- The first index tells you the direction of the force.
- The second index tells you the direction of the surface.
In a matrix this would be written as:
Suppose I would like to know what is the stress for a given surface, not all surfaces like the stress tensor. This is a stress vector, and you simply get it by taking a dot product with your desired surface. The reason it is to the right, is because we are summing over the first index which has information about the surfaces, and we want to project surface normal vectors to surface normal vectors.
What is the stress in the direction?
What if we want to know the total force? Recall that stress is force per area. Therefore, we can integrate the individual stresses over a surface , since :
which is the net force in the static case!