# Current density

Hi

I am a research student looking at electrical current flow through various materials although my undergraduate degree was in mechanical engineering so electrical stuff isn't my strong point.

I believe that current flow will flow through the path that offers the least resistance, and is analogous to heat flow, in other words, if you imagine a flat conductive material like the letter 'L' then the highest current density would be around the inside corner.

FEA simulations I have conducted show this, but I need to find an analytical solution to validate the results. Are there standard models to determine electrical current density in this manner? i.e. something that proves non-linear current density/distribution in the scenario described?

Any help appreciated!!

Which forms of Ohm's Law are you using, this one?

$$\vec{J}$$ = $$\sigma$$ (del) E

Since Ohm's Law is a constitutive equation and your geometry is not continuous, I do not believe you will be able to find a simple analytical solution. I could be wrong, but I just can't think of a way you could analytically solve for such geometry.

Yes that variation of ohm's law seems the most appropriate, but I am quit stumped about how to validate my results. Common sense would indeed say that my FEA results are correct as you have these current 'hotspots' around the 90 degree bend in my example, however I appreciate it may not be easy to prove analytically without resorting to numerical finite element methods (which I don't want to do).

I have found one paper, regarding 90 degree bends in thin strip conductors, which states that the maximum current density is inversely proportional to the cube root of the inside corner radius so this may yield some analytical results. I was just wondering if there were any other solutions out there; ideally rather than just calculating a maximum I would like to calculate a distribution of current emanating from the maximum point. This solution I have seems quite abstract to be basing my FEA work on.

Thank you for reading and taking the time to respond!