# In a plane electromagnetic wave,

Question:

In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by $\hat{k}$ and $2 \hat{i}-2 \hat{j}$, respectively. What is the unit vector along direction of propagation of the wave.

1. (1) $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$

2. (2) $\frac{1}{\sqrt{2}}(\hat{j}+\hat{k})$

3. (3) $\frac{1}{\sqrt{5}}(\hat{i}+2 \hat{j})$

4. (4) $\frac{1}{\sqrt{5}}(2 \hat{i}+\hat{j})$

Correct Option: 1

Solution:

(1) Electromagnetic wave will propagate perpendicular to the direction of Electric and Magnetic fields

$\hat{C}=\hat{E} \times \hat{B}$

Here unit vector $\hat{C}$ is perpendicular to both $\hat{E}$ and $\hat{B}$

Given, $\vec{E}=\hat{k}, \vec{B}=2 \hat{i}-2 \hat{j}$

$\therefore \hat{C}=\hat{E} \times \hat{B}=\frac{1}{\sqrt{2}}\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1 \\ 1 & -1 & 0\end{array}\right|=\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

$\Rightarrow \hat{C}=\frac{\hat{i}+\hat{j}}{\sqrt{2}}$