Are all recursive interactions between electric and magnetic fields always orthogonal to each other?

Question

ecursively, this implies that changes in the electric field generate the magnetic field, and vice versa, akin to the plane wave solution resulting from specific boundary conditions intentionally imposed on Maxwell's equations. Naturally, there exist other solutions to Maxwell's equations where both fields are not necessarily perpendicular to each other. An example is the scenario involving a stationary single charge placed in a uniform magnetic field. However, in instances where changes in one field induce changes in the other (recursively), are all these cases inherently characterized by perpendicularity?

Answer 1

There is not a causal relation between electric and magnetic fields.

According to Jefimenko's Eqs (https://en.wikipedia.org/wiki/Jefimenko%27s_equations), $\vec E$ ($\vec B$) is caused by charge, and the time derivative of charges and currents (currents and their time derivative) on the past light-cone, and that is it.

Ofc, the solutions to those equations satisfy Maxwell's Eqs.

From a special relativity point-of-view, $E, B$ are frame dependent components of the covariant EM field tensor:

$$F_{\mu\nu}=\partial_{\mu}A_{\nu} - \partial_{nu}A_{\mu}$$

so to say a temporal (spatial) change in one causes a spatial (temporal) change in the other is not valid. In any frame:

$$ A_{\mu} = (\phi, \vec A) $$

and of course $\phi$ and $\vec A$ are due to

$$j_{\mu} = (\rho, \vec J)$$

Also: solutions in a vacuum are homogeneous, you can add them together, so any EM wave with $\vec E \cdot \vec B=0$ propagates just fine through a static magnetic field with any orientation (axions notwithstanding), so you can have any dot product you want.

Nevertheless, $\vec E \cdot \vec B$ is a Lorentz invariant (along with $E^2-B^2$), so fields perpendicular in one frame, are perpendicular in all frames.

Now if you are an engineer, none of this matters, and you can continue believing fields induce each other, as it seems a clearer way to design motors, generators and all that.

Answer 2

Not necessarily. Consider a waveguide, the TM and TE modes it presents have $\mathbf{E}$ and $\mathbf{H}$ in general not orthogonal to each other. The functional form of the fields depends on the geometry of the waveguide.