What is a smart way to calculate current densities in 3D objects with unorthodox boundary conditions?

For example, J(x,y,z) in a cube with constant resistivity with applied voltages Va in a vertex and Vb in a plane?

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Assuming these are constant voltages, you need to solve Laplace’s equation, $$\nabla^2 \phi = 0,$$ for the potential $\phi$, with appropriate boundary conditions. That gives you the electric field, $$\mathbf{E} = -\nabla \phi,$$ from which you get the current density by Ohm’s law in differential form $$\mathbf{J} = \sigma \mathbf{E},$$ where $\sigma$ is the conductivity, assumed constant throughout the volume. Boundary conditions may be Dirichlet $\phi = \phi_0$ (constant voltage) or Neumann $\partial \phi/\partial n = c$, in which the normal derivative of the potential is specified. In particular, when the medium adjacent to the object is non-conductive, no current can flow into it and the corresponding boundary condition is $\partial \phi/ \partial n = 0$.

To tackle the situation you depicted you could start with the vertex voltage defined on a small but finite part of the boundary that includes the vertex. For any finite area, the solution is unique and well-defined. The solution you seek would be obtained in the limit when the size of this area shrinks to zero.

  • $\begingroup$ Nice, thanks for the answer. And how would you deal with the conductivity when it is not uniform, but a known function of the spatial coordinates? $\endgroup$ – Gyromagnetic Oct 15 at 12:38
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    $\begingroup$ When the current flow is stationary, you have from the continuity equation div J = 0. Inserting into this equation Ohm’s law and $E =-\nabla \phi$, you obtain $\nabla \cdot (\sigma(r)\nabla \phi) = 0$, which is a generalization of Laplace’s equation. $\endgroup$ – Amit Hochman Oct 15 at 14:20
  • $\begingroup$ I think that in the limit indicated in the answer the field will be similar to that of a point charge at the vertex. This must be so because a finite current must pass through a single point, so the field must behave like 1/(distance from vertex). $\endgroup$ – Amit Hochman Oct 15 at 14:43

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