Let be a parameter and define new electric and magnetic field vectors as linear combinations of the usual electric and magnetic field vectors: E'

Let θ be a parameter and define “new” electric and magnetic field vectors as linear combinations of the usual electric and magnetic field vectors:

E' = E' cos θ + cBsin θ
cB' = −E sin θ + cB cos θ.

(a) Show that E' and B' satisfy the Maxwell equations without sources (ρ = j = 0) if E and B satisfy these equations.

(b) Discuss the implications of this result for source-free solutions of the Maxwell equations where E ⊥ B everywhere in space.