Chapter 31, Problem 63GP

The electric and magnetic fields of a certain EM wave in free space are given by $\stackrel{\to }{E}=E_0\sin (kx-\omega t)\hat{j}+E_0\cos (kx-\omega t)\hat{k}$ and $\stackrel{\to }{B}=B_0\cos (kx-\omega t)\hat{j}-B_0\sin (kx-\omega t)\hat{k}.$ (a) Show that $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$ are perpendicular to each other at all times. (b) For this wave, $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$ are in a plane parallel to the yz plane. Show that the wave moves in a direction perpendicular to both $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$. (c) At any arbitrary choice of position x and lime t, show that the magnitudes of $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$ always equal E₀ and B₀, respectively. (d) At x = 0, draw the orientation of $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$ in the yz plane at t = 0. Then qualitatively describe the motion of these vectors in the yz plane as lime increases. [Note: The EM wave in this Problem is “circularly polarized.”]