(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I know the current distribution through a rectangular cross section of metal. It varies only with respect to y. I need to find a heating function for the metal. (This is only the relevant part of a larger problem).

2. Relevant equations

Q=I^{2}R

3. The attempt at a solution

I attempted this problem twice and got two answers, depending on how I integrated. First I integrated current over the entire surface and plugged that in to Q=I^{2}R. On my second try, I found Q as a function of y and integrated that over the surface. The answers didn't match. Consider this: If a block with uniform current 4A is evaluated as a whole, the Joule heating equation gives Q=16R. However, if it were divided up into four pieces, each with 1A current, Joule heating would give Q=1R for each piece, for a total of Q=4R. So my question is: if I know the current through the material, how can I integrate in a way that makes sense?

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# How to integrate Joule's First Law (Joule Heating)?

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