Question

A plane electromagnetic wave is propagating along the direction$\begin{array}{l}\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\end{array}$with the polarization along the direction $\stackrel{\wedge }{k}$. The correct form of the magnetic field of the wave would be ( here $B_0$ is an appropriate constant)

• A. $\begin{array}{l}{B}_0\left(\frac{\stackrel{\wedge }{i}-\stackrel{\wedge }{j}}{\sqrt{2}}\right)cos\left[\omega t-k\begin{array}{l}\left(\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\right)\end{array}\right]\end{array}$
• B. $\begin{array}{l}{B}_0\left(\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\right)cos\left[\omega t+k\left(\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\right)\right]\end{array}$
• C. $\begin{array}{l}{B}_0\left(\frac{\stackrel{\wedge }{i}-\stackrel{\wedge }{j}}{\sqrt{2}}\right)cos\left[\omega t+k\left(\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\right)\right]\end{array}$
• D. $\begin{array}{l}{B}_0\stackrel{\wedge }{k}cos\left[\omega t-k\left(\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\right)\right]\end{array}$

Answer 1

The correct option is A

$\begin{array}{l}{B}_0\left(\frac{\stackrel{\wedge }{i}-\stackrel{\wedge }{j}}{\sqrt{2}}\right)cos\left[\omega t-k\begin{array}{l}\left(\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\right)\end{array}\right]\end{array}$

Explanation for the correct answer is,

Step 1. Given Data:

Polarization of electromagnetic waves denotes the direction of the electric field.

So, Electric field vector is along $\stackrel{\wedge }{k}$

So, unit electric filed vector, $\stackrel{\wedge }{E}=\stackrel{\wedge }{k}$

EM wave is in direction, $\stackrel{\wedge }{C}=\left(\begin{array}{l}\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\end{array}\right)$

Direction of propagation of EM wave is given by, $\begin{array}{l}\overrightarrow{E}\times \overrightarrow{B}\end{array}$

Step 2. Let $\stackrel{\wedge }{B}$ denotes unit magnetic field vector:

$\stackrel{\wedge }{B}=\stackrel{\wedge }{C}\times \stackrel{\wedge }{E}$ Relation between $\stackrel{\wedge }{B},\stackrel{\wedge }{C}$ and $\stackrel{\wedge }{E}$

$$\begin{gathered} \Rightarrow \stackrel{\wedge }{B}=\left(\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\right)\times \stackrel{\wedge }{k} \\ \Rightarrow \stackrel{\wedge }{B}=\frac{\left(\stackrel{\wedge }{i}\times \stackrel{\wedge }{k}\right)+\left(\stackrel{\wedge }{j}\times \stackrel{\wedge }{k}\right)}{\sqrt{2}} \\ \Rightarrow \stackrel{\wedge }{B}={\frac{\left(-{\stackrel{\wedge }{j}}^{}+\stackrel{\wedge }{i}\right)}{\sqrt{2}}}^{} \\ \Rightarrow \stackrel{\wedge }{B}=\frac{(\stackrel{\wedge }{i}-\stackrel{\wedge }{j})}{\sqrt{2}} \end{gathered}$$

$\Rightarrow B=(B_0\stackrel{\wedge }{B})cos(\omega t-\stackrel{\wedge }{C}k)$ General form

$B_0$ magnitude of magnetic field

$\therefore B=\begin{array}{l}{B}_0\left(\frac{\stackrel{\wedge }{i}-\stackrel{\wedge }{j}}{\sqrt{2}}\right)cos\left[\omega t-k\left(\frac{\stackrel{\wedge }{i}+\stackrel{\wedge }{j}}{\sqrt{2}}\right)\right]\end{array}$

Hence, the correct option is (A).