# In a plane electromagnetic wave, the directions of electric field and magnetic field are represented

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

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

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

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

Correct Option: 1

Solution:

$\hat{\mathrm{E}}=\hat{\mathrm{k}}$

$\overrightarrow{\mathrm{B}}=2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}} \Rightarrow \hat{\mathrm{B}}=\frac{\overrightarrow{\mathrm{B}}}{|\mathrm{B}|}=\frac{2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}}{2 \sqrt{2}}$

$\Rightarrow \hat{\mathrm{B}}=\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}-\hat{\mathrm{j}})$

Direction of wave propagation $=\hat{\mathrm{C}}=\hat{\mathrm{E}} \times \hat{\mathrm{B}}$

$\hat{\mathrm{C}}=\hat{\mathrm{k}} \times\left[\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}-\hat{\mathrm{j}})\right]$

$\hat{\mathrm{C}}=\frac{1}{\sqrt{2}}(\hat{\mathrm{k}} \times \hat{\mathrm{i}}-\hat{\mathrm{k}} \times \hat{\mathrm{j}})$

$\hat{\mathrm{C}}=\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}})$