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Electromagnetic Waves — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsElectromagnetic Waves5 Marks
Question
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 propogation. Mathematically, it is given by $\text{S}=\frac{1}{\mu_0}\text{E}\times\text{B}$. Show the nature of S vs t graph.
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Answer

In an electromagnetic waves, let $\vec{\text{E}}$ be varying along y-axis, $\vec{\text{B}}$ is along z-axis and propagation of wave be along x-axis. Then $\vec{\text{E}}\times\vec{\text{B}}$ will tell the direction of propagation of energy flow in eletromagnetic wave, along x-axis.
Let $\vec{\text{E}}=\text{E}_0\sin(\omega\text{t}-\text{kx})\hat{\text{j}}$
$\vec{\text{B}}=\text{B}_0\sin(\omega\text{t}-\text{kx})\hat{\text{k}}$
$\text{S}=\frac{1}{\mu_0}(\vec{\text{E}}\times\vec{\text{B}})=\frac{1}{\mu_0}\text{E}_0\text{B}_0\sin(\omega\text{t}-\text{kx})(\hat{\text{j}}\times\hat{\text{k}})$
$\Rightarrow\ \text{S}=\frac{\text{E}_0\text{B}_0}{\mu_0}\sin^2(\omega\text{t}-\text{kx})\hat{\text{i}}\ \ (\text{As}\ \hat{\text{j}}\times\hat{\text{k}}=\hat{\text{i}})$
Since $\sin^2(\omega\text{t}-\text{kx})$ is never negative, $\vec{\text{S}}(\text{x},\text{t})$ always points in the positive X-direction, i.e, in the direction of wave propagation.
The variation of |S| with time T will be as given in the figure below:

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