How can we actually measure the angle between the $E$ and $B$ fields in a light beam? I know this angle is 90 degrees but I have hard time accepting it without experimental proof. Can one prove for example this is not 91 degrees?
One can receive the electromagnetic plane wave with an antenna that is responsive to either the electric field or to the magnetic field but not to both within a reasonable engineering accuracy. For example, the standard electric dipole (thin wire linear antenna) or magnetic dipole (thin wire loop antenna) is essentially unresponsive to the magnetic or the electric field, respectively, when are properly aligned in space. It just turns out experimentally for a linearly polarized EM wave the maximum energy is received when the electric dipole is parallel with the electric field and the magnetic dipole is perpendicular to the electric field of the plane wave. Granted the accuracy of such orthogonality for practical antennas is not within 1 degree for simple dipoles but if you instead consider really high gain antennas whose beamwidth is fraction of a degree you do get to angular precision on the order of millidegrees.