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 $B_0$ is an appropriate constant):

• A. $B_0\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cos (\omega t-k\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right))$
• B. $B_0\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cos (\omega t+k\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right))$
• C. $B_0\hat{k}\cos (\omega t-k\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right))$
• D. $B_0\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)\cos (\omega t-k\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right))$

Answer 1

The correct option is A $B_0\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cos (\omega t-k\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right))$

Direction of polarization $=\hat{E}=\hat{k}$

And, direction of propagation

$=\hat{E}\times \hat{B}=\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

But, $\hat{E}.\hat{B}=0\Rightarrow \hat{B}=\frac{\hat{i}-\hat{j}}{\sqrt{2}}$

$\Rightarrow \overrightarrow{B}=B_0\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$

So, the equation of the EM wave is,

$B_0\left(\frac{\hat{i}-\hat{j}}{\sqrt{2}}\right)\cos (\omega t-k\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right))$

Hence, $\left(A\right)$ is the correct answer.