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A plane EM wave travelling along z direc...

A plane EM wave travelling along z direction is described by
`E=E_0sin (kz-omegat) hati and B=B_0 sin (kz-omegat)hatj`.
show that
(i) The average energy density of the wave is given by `u_(av)=1/4 epsilon_0 E_0^2+1/4 (B_0^2)/(mu_0)`.
(ii) The time averaged intensity of the wave is given by `I_(av)=1/2 cepsilon_0E_0^2`.

A

`c epsi _(0) E_(0) ^(2)`

B

`1/2 c epsi _(0) E _(0) ^(2)`

C

`1/2 c epsi _(0) E _(0)`

D

`1/4 c epsi _(0) E_(0) ^(2)`

Text Solution

Verified by Experts

The correct Answer is:
B
`because E _(0) = c B _(0) and c = (1)/( sqrt ( mu _0) epsi_(0))`
`therefore 1/4 (B _(0) ^(2))/( mu _(0)) = 1/4 ( E _(0) ^(2) // c ^(2))/( mu _(0)) = (E _(0) ^(2))/( 4 mu _(0)) xx mu _(0) epsi _(0) = 1/4 E _(0) ^(2)`
`u _(av) =1/4 epsi _(0) E _(0) ^(2) + 1/4 (B _(0) ^(2))/( mu _(0)) =1/4 epsi _(0) E _(0) ^(2) + 1/4 epsi _(0) E _(0) ^(2) `
`=1/2 epsi _(0) E _(0) ^(2) =1/2 (B _(0) ^(2))/( mu _(0))`
Time averaged intensity
`I _(av) = u _(av) c = (1)/(2) c epsi _(0) E _(0) ^(2)`
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