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The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by $\vec{E}=30(2 \hat{x}+\hat{y}) \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{V} \mathrm{m}^{-1}$. Which of the following option($s$) is(are) correct?
[Given: The speed of light in vacuum, $c=3 \times 10^8 \mathrm{~ms}^{-1}$ ]
($A$) $B_x=-2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{Wbm}^{-2}$.
($B$) $B_y=2 \times 10^{-7} \sin \left[2 \pi\left(5 \times 10^{14} t-\frac{10^7}{3} z\right)\right] \mathrm{Wbm}^{-2}$
($C$) The wave is polarized in the $x y$-plane with polarization angle $30^{\circ}$ with respect to the $x$-axis.
($D$) The refractive index of the medium is $2$ .
$A,C,D$
$A,B$
$A,C$
$A,D$
Solution

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$\mathrm{C}_{\text {medium }}=\frac{5 \times 10^{14}}{10^7 / 3}=1.5 \times 10^8 \mathrm{~m} / \mathrm{s} \therefore \mu=2$
$\mathrm{C}_{\text {medium }}=\frac{\mathrm{E}}{\mathrm{B}} \Rightarrow \mathrm{B}=\frac{\mathrm{E}}{\mathrm{C}_{\mathrm{m}}}=\frac{30 \sqrt{5}}{1.5 \times 10^8}=2 \sqrt{5} \times 10^{-7}$
$\overrightarrow{\mathrm{B}}_{\text {dinection }} \equiv \hat{\mathrm{k}} \times(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}) \equiv \frac{2 \hat{\mathrm{j}}-\hat{\mathrm{i}}}{\sqrt{5}}$
$\therefore \overrightarrow{\mathrm{B}}=2 \times 10^{-7}(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}) \sin \left[27\left(5 \times 10^{17} \mathrm{t}-\frac{10^7}{3} \mathrm{z}\right)\right]$
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