Show that average value of radiant flux density `S` over a single period 'T' is given by `S=1/(2cmu_0)E_0^2`.

Explanation

Radiant flux density `S=(1)/(mu_(0)) (Exx B) =c^(2) epsilon_(0) (E xx B)" "[:.c =(1)/(sqrt(mu_(0) epsilon_(0)))]`
Suppose electromagnetic waves be propagaitng along x-axis. The electric field vector of electromagnetic wave be along y-axis and magnetic field vector be along z-axis . therefore,
`E_(0) =E_(0) " cos "(kx - omegat)`
`B=B_(0) " cos " (kx- omegat)`
`Exx B =(E_(0) xx B_(0)) "cos"^(2)(kx-omegat)`
`S=c^(2) epsilon_(0)(ExxB)`
`=c^(2) epsilon_(0) (E xx B_(0)) " cos"^(2) (kx -omegat)`
Average value of the magnitude of radiant flux density over complete cycle is
`S_(av) =c^(2)epsilon_(0) |E_(0) xx B_(0)|1/T underset(0)overset(T)(int) "cos"^(2) (kx - omegat) dt`
`=c^(2) epsilon_(0) E_(0) B_(0) xx 1/T xx T/2" "[:. underset(0)overset(T)(int) " cos"^(2)(kx - omegat)dt = T/2]`
`rArr " "S_(av) =(c^(2))/(2) epsilon_(0) E_(0) ((E_(0))/(C))" "[AS ,c =(E_(0))/(B_(0))]`
`=(c)/(2) epsilon_(0) E_(0)^(2) =(
c)/(2) xx (1)/(c^(2) mu_(0)) E_(0)^(2)" "[c=(1)/(sqrtmu_(0)epsilon_(0)) " or " epsilon_(0) =(1)/(c^(2)mu_(0))]`
`rArr " "S_(av) (E_(0)^(2))/(2mu_(0)c)`