Question

The electric field of a plane electromagnetic wave propagating along the x-direction in a vacuum is $\overrightarrow{E}=E_{\circ }\hat{j}\cos \left(\omega t-kx\right)$.The magnetic field $\overrightarrow{B}$ at the moment $t=0$ is:

• A. $\overrightarrow{B}=\frac{E_{\circ }}{\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}}\cos \left(kx\right)\hat{j}$
• B. $\overrightarrow{B}=\frac{E_{\circ }}{\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}}\cos \left(kx\right)\hat{k}$
• C. $\overrightarrow{B}=E_{\circ }\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}\cos \left(kx\right)\hat{j}$
• D. $\overrightarrow{B}=E_{\circ }\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}\cos \left(kx\right)\hat{k}$

Answer 1

The correct option is D

$\overrightarrow{B}=E_{\circ }\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}\cos \left(kx\right)\hat{k}$

Step 1: Given data

The electric field of a plane electromagnetic wave propagating along the x direction in vacuum is $\overrightarrow{E}=E_{\circ }\hat{j}\cos \left(\omega t-kx\right)$

Step 2: Formula used

The relation between electric and magnetic field is given by the following formula:

$\begin{array}{rcl}Electricfield\left(E\right)& =& Magneticfield\left(B\right)\times speedoflight\left(c\right)\\ B& =& \frac{E}{c}\dots \dots \dots \dots \dots \dots \left(1\right)\end{array}$

It is understood that $c=\frac{1}{\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}}\dots \dots \dots \dots \dots \dots \left(2\right)$

Where ${\mu }_{\circ }=$permeability in a vacuum, ${\epsilon }_{\circ }=$permittivity in a vacuum.

Step 3: Compute magnetic filed

Consider the following figure,

Substitute equation $\left(2\right)$ in equation $\left(1\right)$.

$$\begin{gathered} \overrightarrow{B}=\frac{E_{\circ }}{{\displaystyle \frac{1}{\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}}}} \\ \overrightarrow{B}=E_{\circ }\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}\cos \left(\omega t-kx\right)\hat{k} \\ Whent=0, \\ \overrightarrow{B}=E_{\circ }\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}\cos \left(\omega \times 0-kx\right)\hat{k} \\ \overrightarrow{B}=E_{\circ }\sqrt{{\mu }_{\circ }{\epsilon }_{\circ }}\cos \left(kx\right)\hat{k} \end{gathered}$$

Hence, option D is the correct answer.