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

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

uav=1/4 ɛoE2o+1/4B2o/μo

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

Iav=1/2 cɛoE2o.

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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

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