Dear students,

After yesterday's lecture, we have now covered Chapter 1-3 in complete and Chapter 5 except for the parts presenting the details of the finite element method. On the next lecture, I aim at covering Chapter 4 in complete.

An interesting question (question 2 below) arose that I, on the lecture, wasn't able to provide a good enough answer to. The question(s) went along these lines:

The force (due to the magnetic field only) is given by Lorenz force law, i.e., \(\mathbf{F}\!=\!q\left(\mathbf{v}\times\mathbf{B}\right)\Rightarrow d\mathbf{F}\!=\!\mathbf{J}\times\mathbf{B} dV\).

Question 1: Why does \(\Rightarrow\) hold here; i.e., can you illustrate how the force on a travelling charge \(q\) travelling with the speed \(\mathbf{v}\) is related to the current density \(\mathbf{J}\)?

Question 2: \(\mathbf{J}\) is identically zero in the air gap. Then, \(d\mathbf{F}\) is also zero. Hence, computing the torque by integrating along the air gap would yield zero torque. Why is this not the case?

"Answer": In this document (proper reference on first page, relevant parts in yellow) it is shown why \(\Rightarrow\) holds. Further, the force on a conductor carrying a current and subjected to an external magnetic field is calculated both by integrating the total force in the region where \(\mathbf{J}\!\neq\!0\) (i.e., inside the conductor) as well as using Maxwell's stress tensor outside the conductor (where \(\mathbf{J}\!=\!0\)). The result of the two calculations are identical and while this only is a verification by example, it represents a good argument why Maxwell's stress tensor can be applied to calculate torque of electric machinery by integrating in the air gap.

Best,

Oskar