Question:
The electric field of a plane electromagnetic wave is given by $\vec{E}=E_{0}(\hat{x}+\hat{y}) \sin (k z-\omega t)$ Its magnetic field will be given by:
Correct Option: 1
Solution:
(1) $\vec{E}=E_{0}(\hat{x}+\hat{y}) \sin (k z-\omega t)$
Direction of propagation of em wave $=+\hat{k}$
Unit vector in the direction of electric field, $\hat{E}=\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
The direction of electromagnetic wave is perpendicular to both electric and magnetic field.
$\therefore \hat{k}=\hat{E} \times \hat{B}$
$\Rightarrow \hat{k}=\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right) \times \hat{B} \Rightarrow \hat{B}=\frac{-\hat{i}+\hat{j}}{\sqrt{2}}$
$\therefore \vec{B}=\frac{E_{0}}{c}(-\hat{x}+\hat{y}) \sin (k z-\omega t)$