Question

Poynting vectors $ \vec{S}$ is defined as $\vec{S} = \frac{1}{\mu_{0}} \vec{E} \times \vec{B}$. The average value of $'\vec{S} '$ over a single period $'T'$ is given by

• A. $E^{2}_{0}/2c\mu_{0}$
• B. $E^{2}_{0}/c\mu_{0}$
• C. $2E^{2}_{0}/c\mu_{0}$
• D. $E^{2}_{0}/c\mu_{0}^2$

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
$S_{av} = \frac{1}{\mu_{0}}\left|\vec{E} _o\times \vec{B}_o \right|\left(\frac{1}{T}\right)\int\limits^{T}_{0} cos^{2}\left(kx-\omega t\right)dt$
$\left[\because E = E_{0}\,cos\left(kx-\omega t\right), B = B_{0}\,cos\left(kx-\omega t\right)\right]$
$= \frac{E_{0}B_{0}}{\mu_{0}}\left(\frac{1}{T}\right)\left(T /2 \right) = \frac{E_{0}B_{0}}{\mu _{0}}$
$\left[\because \int\limits^{T}_{0} \,cos^{2}\left(kx-\omega t\right)dt = T /2 \right]$
$S_{av} = \frac{E_{0}\left(E_{0} / c \right)}{2\mu_{0}} = \left(\frac{1}{2c\mu_{0}}\right)E^{2}_{0}$
$\left(\because B_{0} = \frac{E_{0}}{c}\right)$