The diagram shows two electromagnetic beams intersecting at right angles. (E_H, B_H) propagates in the +x-direction. (E_V,B_V) propagates in the +y-direction. For simplicity, each beam is taken as a pure plane wave (with ω = ck = 2πc/λ) cut off transversely so its cross section is a perfect square of area λ²:

The beams overlap in a cube (volume λ³) centered at the origin where the total fields are E = E_H + E_V and B = B_H + B_V.

(a) Calculate the time-averaged energy density uEM(r) for the horizontal (H) beam alone, for the vertical (V) beam alone, and for the total field in the overlap region. Show that the last of these takes its minimum value on the plane x = y shown dashed in the diagram. Compute E and B on this plane.
(b) Calculate the time-averaged Poynting vector S(r) for the H beam, the V beam, and the total field as in part (a). Make a careful sketch of S(x, y) everywhere the fields are defined.
(c) Explain why the behavior of both E and B in the vicinity of the x = y plane is exactly what you would expect if that plane were a perfect conductor. This shows that the interfering beams behave as if they had specularly reflected from each other.

EH-Eo exp[i (kx - wt)]2 BH +Eo exp[i(kx - wt)]y Ey Eo expli(ky - wt)]2 cBy = Eo exp[i (ky - wt)]x = = |y| /2, |z| /2, ly] /2, z X/2, |x 2/2, z 2/2, |x 2/2, z /2.