# 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