Question

Find the mean Plynting vector `( : S: )` of a plane electromagnetice wave `E=E_(m) cos ( omegat - kr )` if the wave propagates in vacuum.

Answer 1

`overset rarr(E) = overset rarr(E_(m)) cos (omegat - oversetrarr(k.)oversetrarr(r ))` then as before
`oversetrarr(H) = sqrt((epsilon_(0))/(mu_(0))) (oversetrarr(k)xxoverset rarr(E_(m)))/(k)cos(omegat-overset rarr(k).oversetrarr(r ))`
so `overset rarr(S) = overset rarr(E) xx overset rarr(H) = sqrt((epsilon_(0))/(mu_(0))) oversetrarr(E_(m)) xx (oversetrarr(k)xxoversetrarr(E_(m)))(1)/(k)cos^(2) (omegat- oversetrarr(k).oversetrarr(r))`
`= sqrt((epsilon_(0))/(mu_(0)) oversetrarr(E_(m)^(2)) (oversetrarr(k))/(k)cos^(2) (omega t - overset rarr(k).oversetrarr(r ))`
`lt oversetrarr(S) gt = (1)/(2) sqrt((epsilon_(0))/(mu_(0))) oversetrarr(E_(m)^(2)) (oversetrarr(k))/(k)`