How can electric wave and magnetic waves have any angle between them?

Question

I have been taught that the electromagnetic length of a wave is a geometrical property that can be measured in units of length, while its amplitude (both the electric and magnetic) is not a geometrical property, like if one of them was a level of taxes and the other level of population growth (Note here is the analogy was is intentionally non-geometrical). Now if I want to emphasize that there is a relationship between the high of taxes and the population growth (and there is), I do not have to represent both graphs as perpendicular. I can overlay them, I can put them one above the other, and many more graphical techniques.

Now according to my analogy above, one can say that also in the electromagnetic field, there is no need to represent the sine waves perpendicularly to each other since it is only a graphical instrument, but I have been told by chat GPT (Confess I do it many times) that they are really orthogonal, and not for a graphical reasons, I cannot understand how non-geometric things can have any angle between them, particularly 90 degrees.

Answer 1

In 3-dimensional space a monochromatic electromagnetic wave can be characterized by 3 directions. First is the direction of propagation, in other more technical words this is the direction of the phase advance of the wave. The two other directions are the vector of electric field strength and the vector of magnetic field strength. The value of the amplitude of the wave has indeed no geometrical meaning, it is field strength, but the field strength amplitude has a direction which is orthogonal to the direction of propagation.

The field strengths are vectors: A vector has a direction, but it has also a value which could anything, here it is field strength's absolute value.

An example: an electron can be excited by the electric field amplitude of the EM wave, if this occurs it will move parallel to the direction of the field strength. The electron's excitation is stronger if the field strength is stronger -- that's the meaning of field strength.
The motion of the excitation is perpendicular to the direction of propagation of the wave which shows that the amplitude of the electric field has a direction. The magnetic field amplitude is orthogonal to the direction of propagation too, but also orthogonal to the direction of the electrical field strength amplitude. So the 3 directions of space have a meaning for the electromagnetic wave. Of course the field strengths cannot be measured by geometrical units. Their value can be measured by the excitation of motion of particles (well for the magnetic field it is bit more subtle as it acts via Lorentz force. Actually it can act on the particle spin -- if the particle has one which is the case for electrons for instance).