Maxwell Stress Tensor: Engineering Question Answered

[mind_inertia]

Hello!

I was talking with a friend today about electrical motors and we started talking about theoretical designs. One question came up which was could the Maxwell Stress Tensor be used to calculate the torque on a rotor of a motor where the airgap is held constant and the magnetic circuit length changes instead of the airgap like motors are designed with.

Could someone who is more physics savy than myself answer my theoretical engineering question?

 

[Cryo]

I had a quick look. Here is what I think. Stress tensor arises as a result of invariance with respect to translations. So it contains linear momentum density flow (momentum and energy).

Torque is the rate of change of angular momentum. Angular momentum conservation corresponds to invariance with respect to rotations. So you need to go back to the Lagrangian density of the electromagnetic field and see what conservation law you get if you consider invariance under rotations. This will give you the current for the flow of angular momentum density, and this will give you the torque.

So in short. My answer is no. Maxwell stress tensor is for linear momentum, not angular one. But I could be wrong as this is the first time I did this rough calculation.

Of course what is possible, is to calculate linear force based on Maxwell Stress Tensor, and then convert it to torque through knowledge of the geometry of the object. I.e. if you find that field exerts more linear force on one of the stick, the stick will start spinning. This will allow you to find the torque

 

[mind_inertia]

Thank you for your analysis Cryo! In electromagnetic simulation packages the Maxwell Stress Tensor is industry standard to calculate torques but from your answer I feel you are saying that Torques shouldn't be able to be calculated from the Maxwell Stress Tensor. Do I understand you correctly?

If so I wish to highlight the focus of my question. The torque of an electric motor is related to the change of airgap, the motor moves to minimize reluctance by shortening the air gap. Another method of minimizing reluctance and thereby deriving motion is by changing the effective magnetic circuit length. I feel that the MST should be able to calculate it but since this is a theoretical problem I thought it is best to approach physicists, particularly because I am unsure of how to use Tensors effectively on paper. I am slowly digesting the works of Kron in regards to applying Tensors to electrical engineering problems but it's a slow journey for me!

Thanks again.