For a plane electromagnetic wave, the magnetic field at a point $x$ and time $t$ is $\vec{B}(x, t)=\left[1.2 \times 10^{-7} \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \hat{k}\right] T$ The instantaneous electric field $\vec{E}$ corresponding to $\vec{B}$ is: (speed of light $c=3 \times 10^{8} \mathrm{~ms}^{-1}$)

(A) $\vec{E}(x, t)=\left[-36 \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \hat{j}\right] \frac{V}{m}$

(B) $\vec{E}(x, t)=\left[36 \sin \left(1 \times 10^{3} x+0.5 \times 10^{11} t\right) \hat{j}\right] \frac{V}{m}$

(C) $\vec{E}(x, t)=\left[36 \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right) \hat{k}\right] \frac{V}{m}$

(D) $\vec{E}(x, t)=\left[36 \sin \left(1 \times 10^{3} x+1.5 \times 10^{11} t\right) \hat{i}\right] \frac{V}{m}$

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · EM Waves

For a plane electromagnetic wave, the magnetic field at a point

x

and time

t

B (x, t) = [1.2 \times 1 0^{- 7} sin (0.5 \times 1 0^{3} x + 1.5 \times 1 0^{11} t) \hat{k}] T

The instantaneous electric field

E

corresponding to

B

is: (speed of light

c = 3 \times 1 0^{8} ms^{- 1}

)

(A) $E (x, t) = [- 36 sin (0.5 \times 1 0^{3} x + 1.5 \times 1 0^{11} t) \hat{j}] \frac{V}{m}$ ✓ Correct
(B) $E (x, t) = [36 sin (1 \times 1 0^{3} x + 0.5 \times 1 0^{11} t) \hat{j}] \frac{V}{m}$
(C) $E (x, t) = [36 sin (0.5 \times 1 0^{3} x + 1.5 \times 1 0^{11} t) \hat{k}] \frac{V}{m}$
(D) $E (x, t) = [36 sin (1 \times 1 0^{3} x + 1.5 \times 1 0^{11} t) \hat{i}] \frac{V}{m}$

Answer: (A)

For a plane electromagnetic wave the field amplitudes are tied by

E_{0} = c B_{0} = (3 \times 1 0^{8}) (1.2 \times 1 0^{- 7}) = 36 V/m

. In the phase

0.5 \times 1 0^{3} x + 1.5 \times 1 0^{11} t

the

x

and

t

terms share the same sign, so the wave advances along the negative

x

direction. The Poynting vector

E \times B

must point that way, and with

B

along

+ \hat{k}

the electric field has to lie along

- \hat{j}

for

E \times B

to come out along

- \hat{i}

. Keeping the same wave number and angular frequency,

E (x, t) = - 36 sin (0.5 \times 1 0^{3} x + 1.5 \times 1 0^{11} t) \hat{j} V/m

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Single correct option