Question

A plane EM wave travelling along z direction is described by E = E_o sin(kz –ωt )i and  B = B_o sin(kz –ωt )j . Show that

(i) The average energy density of the wave is given by

uav=1/4 ɛ_oE²_o+1/4B²_o/μ_o

(ii) The time averaged intensity of the wave is given by

I_av=1/2 cɛ_oE²_o.

Answer 1

The values of E² and B² vary from point to point and from moment to moment. Hence, the effective values of E² and B² are their time averages.

The graph of sin²θ and cos²θ are identical in shape but shifted by π/2, so the average values of sin²θ and Cos²θ are also equal over any integral multiple of π.

and also sin²θ + cos²θ =1

So by symmetry the average of sin²θ = average of cos²θ =1/2