# Are magnetic field and electric field perpendicular is spherical waves?

The only way for a vector field to have strict spherical symmetry is for it to be purely in the radial direction. For, if it had a non-radial component then that component would have to be preserved under rotations, but you cannot construct a vector field which has that property everywhere on the surface of a sphere. I provide a proof below. (This is closely related to, but not exactly the same as, the hairy ball theorem.) So the only type of vector field which has strict spherical symmetry is a purely radial one, such as a Coulomb field. Such a field cannot be an electromagnetic wave. So it is not possible to have exactly a spherical electromagnetic wave (i.e. one with no change at all under rotations).

You can have a wave which in the limit $$r \rightarrow \infty$$ has spherical wavefronts and is transverse, but I suppose the question is not about that limit, since it amounts to adopting a plane wave approximation for each part of the spherical wavefront.

You can have an oscillating field which has spherical wavefronts, where a wavefront is a locus of a fixed value of the phase of the oscillation. Such a field is not exactly transverse everywhere.

A proof of the claim (I just made up this proof; I'm adding it to see if anyone likes it or tells me it is not good enough.)

Take a sphere, and put a vector $$\bf E$$ at some point P on it. Let's define the 'equator' of our sphere to be the great circle running through P and parallel to $$\bf E$$ there. Now rotate the sphere through 90 degrees, carrying P and $$\bf E$$ up to the north pole. The vector is pointing in a direction we shall agree to call $$x$$.

Now return to the initial condition, and this time rotate the ball by 90 degrees about an axis through the poles, thus carrying P around the equator, and $$\bf E$$ with it. Then rotate again, carrying P up to the north pole. Now $$\bf E$$ is sitting at the north pole and pointing in a direction $$y$$, at right angles to the direction we got in the first rotation. But if we had been able to paint a vector field onto our sphere such that it had spherical symmetry, then these two transformations should both give no net effect on the whole sphere, and therefore both should carry $$\bf E$$ to a direction at the pole which would be the same in both cases. But it is not the same, so we have a contradiction, and the false step was the assumption that a vector field could be painted on the sphere in a spherically symmetric way.

Write Ignatowsky's equations (erroneously referred to as Jefimenko's equations in [1]) in the form as follows [2]:

$$\mathbf{B} = \frac{1}{c}\int d^3\mathbf{x}'\frac{([\mathbf{J}]\times\hat {\mathbf{n}})}{R^2} +\frac{1}{c^2}\int d^3\mathbf{x}'\frac{([\dot{\mathbf{J}}]\times\hat {\mathbf{n}})}{R} \tag{4}\label{4}$$ $$\mathbf{E} = \int d^3\mathbf{x}' \frac{[\rho] \hat {\mathbf{n}}}{R^2} +\frac{1}{c}\int d^3\mathbf{x}'\frac{([\mathbf{J}]\cdot\hat {\mathbf{n}})\hat {\mathbf{n}}}{R^2} +\frac{1}{c^2}\int d^3\mathbf{x}'\frac{([\dot{\mathbf{J}}]\times\hat {\mathbf{n}})\times \hat {\mathbf{n}}}{R} \tag{5}\label{5}$$

The brackets $$[]$$ mean retarded time. While assuming that the current density $$\mathbf{J}$$ is localized in space for large $$R=|\mathbf{x}-\mathbf{x}'|$$ only the terms whose integrand is proportional to $$1/R$$ will matter representing the radiation field, while terms having $$1/R^2$$ is the near field; so the radiation field is: $$\mathbf{B} \approx \frac{1}{c^2}\int d^3\mathbf{x}'\frac{([\dot{\mathbf{J}}]\times\hat {\mathbf{n}})}{R} \approx -\hat {\mathbf{n}}\times \frac{1}{c^2}\frac{1}{R}\int d^3\mathbf{x}'[\dot{\mathbf{J}}] \\ \mathbf{E} \approx \frac{1}{c^2}\int d^3\mathbf{x}'\frac{([\dot{\mathbf{J}}]\times\hat {\mathbf{n}})\times \hat {\mathbf{n}}}{R} \approx \frac{1}{c^2}\frac{1}{R}\hat {\mathbf{n}}\times \left(\hat {\mathbf{n}}\times \left(\int d^3\mathbf{x}'[\dot{\mathbf{J}}]\right)\right)$$ from which it is obvious that in the radiation field $$\mathbf{E} \perp \mathbf{B} \perp \hat {\mathbf{n}}$$ where $$\hat {\mathbf{n}}$$ is the unit vector in the direction of propagation.

[1]: https://en.wikipedia.org/wiki/Jefimenko%27s_equations

[2]: Kirk T. McDonald, The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips, American Journal of Physics 65 (11) (1997), 1074-1076