I’m trying to find the force per unit length between two parallel wires carrying the same current in the same direction and a distance of 2a apart. I need to use the Maxwell stress tensor and am working in CGS units.
If I set the wires parallel to the z-axis and have both lying in the y-z plane I see the magnetic field on the left wire flows in the positive x direction.
I see the tensor is only non-zero on the diagonal with values:
$$T_{xx} = \frac{-B^2}{8\pi}$$ $$T_{yy} = \frac{B^2}{8\pi}$$ $$T_{zz} = \frac{B^2}{8\pi}$$
My confusion comes from when I go to integrate it. I don’t know what the area should be. So for
$$ \vec{F} = \int T\cdot dA $$
What should the area be? Since the solution needs to be force per unit length, it would seem one direction is the length of the wire, but I have no reason to pick anything for the other direction except knowing what the answer should be. Since the solution, I think, should be
$$ \frac{F}{l} \sim \frac{I^2}{a} $$
and since
$$ B = \frac{I}{ca} $$
it would seem that the force should be something like
$$ F_y = \frac{1}{8\pi}\left(\frac{I}{ca}\right)^2\int_0^l\int_0^{2a}dydz $$
in order to get the right units. Unfortunately I don’t have a good reason why this would be chosen, if it’s even close to correct.