Volume charge distribution in 4D

Question

Would the electric field from a volume charge distribution suffer a discontinuity at the boundary of a conductor in four dimensions, just as the field from a surface charge distribution suffers discontinuity at the boundary of conductor in three dimensions?

Answer 1

If by 'volume' in 4-dimensions you mean a three-dimensional surface (just as a surface in 3-dimensions is two-dimensional), then yes. Gauss's law says $\nabla\cdot E = \rho$. In an $N$-dimensional space, if you integrate Gauss's law over a thin box enclosing a section of the $N-1$ dimensional surface, you will get the discontinuity in the electric field across that surface. If there is a finite charge within this thin box, there will be a discontinuity.

Here is another PSE question which touches upon other aspects of Maxwell's equations in higher dimensions.