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Pointing vectors $\vec S$ is defined as a vector whose magnitude is equal to the wave intensity and whose direction is along the direction of wave propagation. Mathematically, it is given by $\vec S = \frac{1}{{{\mu _0}}}(\vec E \times \vec B)$. Show the nature of $\vec S$ vs $t$ graph.
Solution
In electromagnetic wave let $\overrightarrow{\mathrm{E}}$ is in $y$-direction, $\overrightarrow{\mathrm{B}}$ is in $z$-direction and electromagnetic wave propagate in $x$-direction energy propagating will be in direction of $\overrightarrow{\mathrm{E}} \times \overrightarrow{\mathrm{B}}$ (in $x$-direction).
$\overrightarrow{\mathrm{E}}=\mathrm{E}_{0} \sin (\omega t-k x) \hat{j}$
$\overrightarrow{\mathrm{B}}=\mathrm{B}_{0} \sin (\omega t-k x) \hat{k}$
$\therefore \overrightarrow{\mathrm{S}}=\frac{1}{\mu_{0}}(\overrightarrow{\mathrm{E}} \times \overrightarrow{\mathrm{B}})=\frac{1}{\mu_{0}} \mathrm{E}_{0} \mathrm{~B}_{0} \sin ^{2}(\omega t-k x)(\hat{j} \times \hat{k})$
$\therefore \overrightarrow{\mathrm{S}}=\frac{\mathrm{E}_{0} \mathrm{~B}_{0}}{\mu_{0}} \sin ^{2}(\omega t-k x) \hat{i}[\because \hat{j} \times \hat{k}=\hat{i}]$
Variation in magnitude of $|\overrightarrow{\mathrm{S}}|$ with time is shown in figure shown below.
