Charge and current density fields

Question

The charge and current density fields in classical electromagnetism are scalar real number fields on space time manifold. But these fields diverge/become infinite in case of point charges, how is this justified and mathematically consistent ?

Answer 1

But these fields diverge/become infinite in case of point charges, how is this justified and mathematically consistent ?

If the charge $q$ of the particle is finite, both densities have to be singular at the point where the particle is, because a regular (everywhere finite) density would need to be non-zero in a region of non-zero volume to give finite charge.

The diverging fields $\rho, \mathbf j$ for a point charged particle are described by three-dimensional distributions proportional to three-dimensional delta distribution; if the point particle is at point $\mathbf r$ and has velocity $\mathbf v$, the charge density is described by the distribution

$$ \rho(\mathbf x) = q\delta^{(3)}(\mathbf x - \mathbf r) $$ and the current density is described by the distribution

$$ \mathbf j(\mathbf x) = q\mathbf v \delta^{(3)}(\mathbf x - \mathbf r). $$

where the three-dimensional delta distribution $\delta^{(3)}(\mathbf x - \mathbf r)$ has the property

$$ \int f(\mathbf x)\delta^{(3)}(\mathbf x - \mathbf r)\,d^3\mathbf x = f(\mathbf r) $$

for any function $f(\mathbf x)$ regular at $\mathbf r$.

These distributions work well with the Maxwell equations; for example, there is an infinity of solutions of these equations for the delta-like $\rho,\mathbf j$. Most often the retarded solutions are used.