Friday 12 January 2018

electromagnetism - Calculating the resistance of a 3D shape between two points



If I have an arbitrarily shaped object made of a uniform material of some specified resistivity, how would I go about calculating the resistance between two measurement points with known contact geometry?


Is there a general formula for this? (other than just Maxwell's equations) If so, where would I find a derivation?


Edit: Some simulations re answer below: enter image description here enter image description here



Answer



Well, yes you can, but it is usually very hard. Here are the steps:




  1. Solve the Laplace equation: $$ \nabla^2V = 0 \, .$$ In your case, find the general solution in spherical coordinates. Try to use every simplification you can. You might wonder why you don't solve Poisson's equation:
    $$ \epsilon\nabla^2V = \rho \, .$$ That's because a conductor is an equal number of lattice positive and moving negative, so you have a net null density of charge.





  2. Find the electric field with:
    $$ E = -\nabla V $$ You should still have a term dependent on your $V_0$, the potential difference between your two points.




  3. Find the current density with: $$ J = \sigma E \, .$$




  4. Find the total current $I$ by integrating over any closed surface containing only one of your two contact point.





  5. $$V_0 = R I \, !$$




That's it. I have used it to find the resistance for certain geometries when you could do lots of simplifications to find the solution of $V$, but I don't know how it can be applied for more general problems.


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