Question

Magnetic field in a plane of electromagnetic wave is given by
$\overrightarrow{B}=B_{\circ }\sin (kx+\omega t)\hat{\text{j}}\text{ T}$
Expression for corresponding electric field will be: where c is the speed of light
a)$\overrightarrow{E}=B_{\circ }c\sin (kx+\omega t)\hat{k}\text{ V/m}$
b)$\overrightarrow{E}={\displaystyle \frac{B_{\circ }}{c}}\sin (kx+\omega t)\hat{k}\text{ V/m}$
c)$\overrightarrow{E}=-B_{\circ }c\sin (kx+\omega t)\hat{k}\text{ V/m}$
d)$\overrightarrow{E}=B_{\circ }c\sin (kx-\omega t)\hat{k}\text{ V/m}$

Answer 1

Hint: An electromagnetic wave consists of varying electric and magnetic fields vibrating perpendicular to each other. It is a transverse wave as it oscillates perpendicular to the direction of motion. Because of varying electric and magnetic field, it does not require medium for its propagation
Formula used:
${\displaystyle \frac{{\text{E}}_{\circ }}{{\text{B}}_{\circ }}}=c$

Complete answer:
An electromagnetic wave is a wave radiated by an accelerated charge and which propagates through space as coupled electric and magnetic fields, oscillating perpendicular to each other and to the direction of propagation of the wave.
Maxwell showed that any variation in electric field would generate magnetic field and vice versa both serving as a source for each other. More precisely hence they do not require any medium to travel. Let us say there is an electromagnetic wave propagating along the x-axis and whose electric field vibrates along z-axis and magnetic field along y-axis. Therefore the electric field vector ($\overrightarrow{\text{E}}$)and the magnetic field vector ($\overrightarrow{\text{B}}$) is given by,
$\begin{array}{rl}& \overrightarrow{\text{E}}=E_{\circ }\sin \left[2\pi ({\displaystyle \frac{x}{\lambda }}-{\displaystyle \frac{t}{T}})\right]\hat{\text{k}}\mathrm{\&}\\ & \overrightarrow{\text{B}}=B_{\circ }\sin \left[2\pi ({\displaystyle \frac{x}{\lambda }}-{\displaystyle \frac{t}{T}})\right]\hat{\text{j}}\end{array}$
Where $E_{\circ }$and $B_{\circ }$ are the amplitudes of the electric and magnetic field,
$\lambda$ is the wavelength of the wave, ‘x’ is a point on the wave at any time ‘t’ and T is the time period. The relation between the amplitudes of the above waves is given by
${\displaystyle \frac{{\text{E}}_{\circ }}{{\text{B}}_{\circ }}}=c$
Where ‘c’ is the speed of light. Hence the corresponding value of electric field is, $\overrightarrow{E}=B_{\circ }c\sin (kx+\omega t)\hat{k}\text{ V/m}$
Therefore the correct answer of the above question is option a.

Note:
The electromagnetic waves carry energy while propagating. This energy is equally shared between the electric and magnetic fields. Since the amplitude of electric field is very much greater than magnetic field, electric field is responsible for optical phenomena.