A plane electromagnetic wave is propagating along the direction

Question:

A plane electromagnetic wave is propagating along the direction $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$, with its polarization along the direction

$\hat{k}$. The correct form of the magnetic field of the wave would be (here $\mathrm{B}_{0}$ is an appropriate constant):

  1. (1) $\mathrm{B}_{0} \frac{\hat{i}-\hat{j}}{\sqrt{2}} \cos \left(\omega t-k \frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$

  2. (2) $\mathrm{B}_{0} \frac{\hat{j}-\hat{i}}{\sqrt{2}} \cos \left(\omega t+k \frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$

  3. (3) $\mathrm{B}_{0} \hat{k} \cos \left(\omega t-k \frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$

  4. (4) $\mathrm{B}_{0} \frac{\hat{i}+\hat{j}}{\sqrt{2}} \cos \left(\omega t-k \frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$

Correct Option:

Solution:

(1) Direction of polarisation $=\hat{E}=\hat{k}$

Direction of propagation $=\widehat{E} \times \widehat{B}=\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

But $\vec{E} \cdot \vec{B}=0 \quad \therefore \hat{B}=\frac{\hat{i}-j}{\sqrt{2}}$

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