How can someone mathematically prove that a spherical wave equation solution cannot be a electromagnetic wave?

Question

It's well documented that spherical electromagnetic waves doesn't actually exists. However, most textbooks just brings this concept without any formal demonstration. So, for a spherical wave equation solution $$\psi(r,t) = \frac{A}{r}e^{i(kr\mp\omega t)}$$ How would someone be able to make a proof that doesn't satisfy a EMW (or any transversal wave). I know that the hairy ball theorem can be used, but i don´t know how viable it would be to consolidate this qualitative approach.

Answer 1

You don't even need the hairy ball theorem. The proof is even simpler.

Let's start with the electric field. At each point in space is a vector $\vec{E}$. One component of this vector is the radial component $E_r$. Because this is a pure EM wave, with no matter, there is no charge inside of the Gaussian sphere. By spherical symmetry, this implies $E_r = 0$.

Now, what about the other two components pointing tangential to the sphere? Let's take, for instance, the point at the very top of the sphere, where the tangential components are $E_x$ and $E_y$. If you perform a rotation around the $z$ axis by $\theta$, the field components at that point change to $\cos \theta E_x + \sin \theta E_y$ and $-\sin \theta E_x + \cos \theta E_y$. Unless $E_x$ and $E_y$ are $0$, these rotated field components won't be equal to the original ones. If you assume your field is spherically symmetric, i.e. that rotations don't change $\vec{E}$, then we must conclude that the tangential components are $0$, meaning $\vec{E} = 0$.

For the magnetic field its basically the same. By Gauss's law for magnetism $\vec{\nabla} \cdot \vec{B} = 0$, we know that $B_r = 0$. We can use the same argument as $\vec{E}$ to also prove that the tangential components are also $0$.

Answer 2

Because charge within a volume cannot change without charge flowing into or out of the volume, an electromagnetic wave cannot be radial and spherically symmetriic. And yes, that is because there is no spherically symmetric way to wrap field lines around a sphere.