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I am doing some work where I need to look at a simplified situation regarding a conductor for which the conductivity distribution does not change along the z-direction of an infinite cylinder. The distribution itself is not symmetric in any way. Presume 2 infinitely long electrodes along the z-direction with I_1 = - I_2 at every point along those electrodes. Thus the current density J = J(x,y) is independent of z.

I am looking to simplify the calculation of the z-component of the resulting magnetic field inside the conductor in a slice, z_0, and hoping that I do not need the full volume integration of the Biot-Savart law.

Anyone got some insight as to how I can abuse the knowledge of J not changing in the z-direction. Or rather if it is possible. I have easily concluded that I at least only need to integrate half the volume. I would do, however, hope that it is possible to only do the integration over a slice,z_0. Presuming that the r/|r-r'|^3 goes to zero and thus for any current distribution can be pre-calculated and multiplied on the B_z component of the slice.