#### Poynting vectors 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 $S = \frac{1}{\mu_0} E \times B$. Show the nature of the S versus t graph.

In an electromagnetic waves, Let $\vec{E}$ be varying along y-axis, $\vec{B}$ is along z- axis and propagation of wave be along x-axis. Then $\vec{E} \times \vec{B}$ will tell the direction of propagation of energy flow in electromagnetic wave, along x-axis.

Let

$\\ \vec{E} =E_0 \sin\left ( \omega t -kx \right )\hat{j}\\ \vec{B} =B_0 \sin\left ( \omega t -kx \right )\hat{k}\\ S=\frac{1}{\mu_0}\left ( \vec{E} \times \vec{B} \right )=\frac{1}{\mu_0}\;E_0B_0\sin^{2}\left ( \omega t- kx\right )\left ( \hat{j}\times \hat{k} \right )\\ \Rightarrow S=\frac{E_0B_0}{\mu_0}\sin^{2}\left ( \omega t-kx \right )\hat{i}\; \; \; \; \; \;$

Since $\sin^{2}\left ( \omega t-kx \right )$ is never negative, $\vec{S}\left ( x,t \right )$ always point in the positive X-direction, i.e, in the direction of wave propagation.
The variation of $\left | S \right |$ with time Twill be as given in the figure below: