EvC Forum isn't the best place to pose this question, but I'm sure there are some electrical engineers here. I'm an EE myself, but I've forgotten more field theory than I ever learned.
I wonder if the geometry of your cable is irrelevant, and that it's instead the cross sectional area that you have to pay attention to. Or perhaps it's only the current that matters, though you don't mention current as part of the problem.
Your idea of integrating across the flats from 0 to b also sounds promising.
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--EvC Forum Administrator