Question

For a plane electromagnetic wave, the magnetic field at a point $x$and time t is$B(x,t)=[1.2\times {10}^{-7}\sin (0.5\times {10}^{-3}x+1.5\times {10}^{11}t)k]T$. The instantaneous electric field $E$ corresponding to $B$ is:

(speed of light $c=3\times {10}^{8}m{s}^{-1}$)

• A. $E(x,t)=[36\sin (1\times {10}^{3}x+1.5\times {10}^{11}t)i]V/m$
• B. $E(x,t)=[36\sin (0.5\times {10}^{3}x+1.5\times {10}^{11}t)k]V/m$
• C. $E(x,t)=[36\sin (1\times {10}^{3}x+0.5\times {10}^{11}t)j]V/m$
• D. $E(x,t)=[-36\sin (0.5\times {10}^{3}x+1.5\times {10}^{11}t)j]V/m$

Answer 1

The correct option is D

$E(x,t)=[-36\sin (0.5\times {10}^{3}x+1.5\times {10}^{11}t)j]V/m$

Step 1: Given data

The magnetic field at a point $x$ and time $t$, $B\left(x,t\right)=\left[1.2\times {10}^{-7}\sin \left(0.5\times {10}^{3}x+1.5\times {10}^{11}t\right)k\right]T$

Speed of light $c=3\times {10}^{8}m{s}^{-1}$

Electric field $=E$

Step 2: To find the instantaneous electric field $E$corresponding to $B$

Wave is traveling along $-x$ axis and $B$ is along $+z$ axis

$E$and $B$ are not in the same direction, in Electromagnetic waves $E$and $B$are always perpendicular to each other.

$E=cB$

$$\begin{gathered} E=3\times {10}^8\left[1.2\times {10}^{-7}\sin \left(0.5\times {10}^{3}x+1.5\times {10}^{11}t\right)k\right] \\ =36\sin \left(0.5\times {10}^{3}x+1.5\times {10}^{11}t\right) \end{gathered}$$

The wavelength is not changed so answers 1 and 3 are incorrect

$E$ and $B$ are not in the same direction so 2nd is incorrect.

Hence $\overrightarrow{E}$ must be along $-y$ axis then $E(x,t)=[-36\sin (0.5\times {10}^{3}x+1.5\times {10}^{11}t)j]V/m$

Therefore the correct answer is Option D