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 the direction of propagation of the wave?

• A. $\frac{1}{\sqrt{2}}\left(\hat{i}+\hat{j}\right)$
• B. $\frac{1}{\sqrt{5}}\left(2\hat{i}+\hat{j}\right)$
• C. $\frac{1}{\sqrt{5}}\left(\hat{i}+2\hat{j}\right)$
• D. $\frac{1}{\sqrt{2}}\left(\hat{j}+\hat{k}\right)$

Answer 1

The correct option is A

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

Step 1: Given data

The direction of the electric field, $\overrightarrow{E}$= $\hat{k}$

The direction of the magnetic field, $\overrightarrow{B}$ = $2\hat{i}-2\hat{j}$

Step 2: Formula used

The formula to find the unit vector is:

$\hat{\mathrm{u}}=\frac{\mathrm{u}}{\left|\mathrm{u}\right|}$
$\hat{\mathrm{u}}=$ unit vector
$\mathrm{u}=$ non-zero vector
$\left|\mathrm{u}\right|=$magnitude (or length) of $\mathrm{u}$

The direction of the vector along the direction of propagation of the wave $=\overrightarrow{E}\times \overrightarrow{B}$

Step 3 : Calculate the Propagation Vector

$$\begin{gathered} \overrightarrow{E}\times \overrightarrow{B}=\hat{k}\times (2\hat{i}-2\hat{j}) \\ =2\hat{k}\times \hat{i}-2\hat{k}\times \hat{j} \\ =2\hat{j}+2\hat{i} \end{gathered}$$

The magnitude of $u$ can be given as,

$\left|u\right|=\frac{2\hat{i}+2\hat{j}}{\sqrt{2^2+2^2}}=\frac{2\hat{i}+2\hat{j}}{2\sqrt{2}}$

Step 4: Calculate the Unit Vector

$$\begin{gathered} Unitvectoralong\overrightarrow{E}\times \overrightarrow{B}=\frac{1}{2\sqrt{2}}\left(2\hat{i}+2\hat{j}\right) \\ =\frac{1}{\sqrt{2}}\left(\hat{i}+\hat{j}\right) \end{gathered}$$

Hence, the unit vector along the direction of propagation of the wave is $\frac{1}{\sqrt{2}}\left(\hat{i}+\hat{j}\right)$.

Hence, option A is correct.