Question

Poynting vectors $\vec{S}$ is defined as $\vec{S}=\frac{1}{{\mu }_0}\vec{E}\times \vec{B}$. The average value of ${}^{\prime }{\vec{S}}^{\prime }$ over a single period ${}^{\prime }{T}^{\prime }$ is given by

• A. $E_0^2/2c{\mu }_0$
• B. $E_0^2/c{\mu }_0$
• C. $2E_0^2/c{\mu }_0$
• D. $E_0^2/c{\mu }_0^2$

Answer 1

Answer: (A)

Poynting vector is given by, $\vec{S}=\frac{\vec{E}\times \vec{B}}{{\mu }_0}$
Average value of magnitude of radiant flux density over one cycle is
$S_{av}=\frac{1}{{\mu }_0}\left|{\vec{E}}_o\times {\vec{B}}_o\right|\left(\frac{1}{T}\right)\underset{0}{\overset{T}{\int }}co{s}^2\left(kx-\omega t\right)dt$
$\left[\because E=E_{0}cos\left(kx-\omega t\right),B=B_{0}cos\left(kx-\omega t\right)\right]$
$=\frac{E_0{B}_0}{{\mu }_0}\left(\frac{1}{T}\right)\left(T/2\right)=\frac{E_0{B}_0}{{\mu }_0}$
$\left[\because \underset{0}{\overset{T}{\int }}co{s}^2\left(kx-\omega t\right)dt=T/2\right]$
$S_{av}=\frac{E_0\left(E_0/c\right)}{2{\mu }_0}=\left(\frac{1}{2c{\mu }_0}\right)E_0^2$
$\left(\because B_0=\frac{E_0}{c}\right)$