I'm looking for the electric equivalence of the (second order linear parabolic) heat equation in order to calculate transient processes.
At the phenomenological level, there is a strict analogy between conductive heat flow, the diffusive motion of particles and the electric current. Fourier’s law, Fick’s 1st law and Ohm’s law are equivalent, etc. However, I can’t find the general equation for the electric potential which is analogous to the so called heat equation or diffusion equation (without sources) One reason can be that it is not valid for a general 3D body, because electrons tend to go to the surface of the body.
I guess you are looking for Poisson's equation.
https://en.wikipedia.org/wiki/Poisson%27s_equation#Electrostatics
https://en.wikipedia.org/wiki/Laplace%27s_equation#Electrostatics
Dear Vikash Pandey, I know, even teach the Poisson eq., but it is for the stationary solution without time. The equation I'm looking for is about how the potential changes in time if, for example, the initial distribution of charge is not uniform on the surface. Thank you anyway!
@ Endre, thank you for clarifying. But why would you think that the potential will change in time when the resultant potential from any distribution of charges will be a function of distance alone, unless the charges themselves decay for some reason with respect to time.
Of course the potential changes because the charge is flowing. Like in the case of heat, if T is larger somewhere, heat (energy) will flow to make T uniform. If you have capacitors, each of them has one plate grounded, the other is initially charged to different voltages and then connected through wires (nonzero resistance), electrons will gradually move. This is a transient process and the voltages will finally equalize. This is the discretized version of the above example.
Today I realized that due to the "curvature effect" (I do not know the proper English name of the effect for sure) the charge density is larger where the curvature of a metal surface is larger, thus I guess the equation is not valid for a general surface, unless the curvature is uniform.
But I'm still full of uncertainty about this question...
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