Home
Class 12
PHYSICS
Show that average value of radiant flux...

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

Text Solution

Verified by Experts

We know that radiant flux density S (called Poynting vector) is given by
`vecS=1/(mu_0) (vecExxvecB)=c^2in_0(vecExxvecB) [ :' c=1/(sqrt(mu_0in_0))]`
Let the e.m. wave be propagating along x-axis. The electric field vector of e.m. wave be along y-axis and
magnetic field vecotr be along z-axis. Therefore,
`vecE=vecE_0cos (kx-omegat) and vecB=vecB_0 cos (kx-omegat) :. vecExxvecB=(vecE_0xxvecB_0)cos^2(kx-omegat)`
`vecS=c^2 epsilon_0(vecExxvecB)=c^2 epsilon_0 (vecE_0xxvecB_0)cos^2 (kx-omegat)`
Average value of the magnitude of radiant flux density over complete cycle is
`s_(av)=c^2in_0|vecE_0xxvecB_0| 1/T int_0^T cos^2(kx-omegat)dt`
`=c^2in_0E_0B_0xx1/TxxT/2 [ :' int_0^T cos^2 (kx-omegat)dt=T/2]`
`s_(av) =(c^2)/2in_0E_0((E_0)/c) [As c=E_0//B_0]`
`=c/2 in_0E_0^2=c/2xx1/(c^2mu_0) E_0^2 [c=1/(sqrt(mu_0 in_r)) or in_0=1/(c^2mu_0)]`
`=(E_0^2)/(2mu_0c)`
Promotional Banner

Topper's Solved these Questions

  • ELECTROMAGNETIC WAVES

    PRADEEP|Exercise Solved Example|4 Videos

  • ELECTROMAGNETIC WAVES

    PRADEEP|Exercise (I) Conceptual problems|1 Videos

  • ELECTROMAGNETIC INDUCTION & ALTERNATING CURRENT

    PRADEEP|Exercise Multiple Choice Questions|1 Videos

  • ELECTRONIC DEVICES

    PRADEEP|Exercise Fill in the Blanks|1 Videos

Similar Questions

Explore conceptually related problems

Poynting vectors vec(S) is defined as vec(S)=(1)/(mu_(0))vec(E)xx vec(B) . The average value of 'vec(S)' over a single period 'T' is given by

The average value of current for the current shown for time period 0 to T/2 s

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 .

Assertion (A) : The average value of lt sin^(2) omega t gt is zero. Reason (R ) : The average value of function F (t) over a period T is lt F (t) gt = (1)/(T) int_(0)^(T) F (t) dt

Show that average energy density of electromagnetic radiation can be written as epsi_0 E_( rms ) ^(2) .

The average value of a saw-tooth voltage V=V_(0) ((2t)/(T)-1) Over 0 to (T)/(2) , (T- time period) is given by

A closed coil having 50 turns, area 300cm^(2) , is rotated from a position where it plane makes an angle of 45^(@) with a magnetic field of flux density 2.0T to a position perpendlcular to the field in a time of 0.1s. What is the average EMF induced in the coil ?

Determine the average value of the electromotive force E_(m) over one period, ie, over the time from t= 0 to t=T, if electromotive force is computed by the formula E = E_(0) " sin "(2pi t)/(T) where T is the duration of the period in seconds, E_(0) the amplitude (the maximum value) of the electromotive force corresponding to the value t= 0.25T. The fraction (2pi t)/(T) is called the phase.