Question

The equation that represents magnetic field of a plane electromagnetic wave which is propagating along x-direction with wavelength 10 mm and maximum electric field 60 V m$^{-1}$ in y-direction is (c – speed of light)

• A. \( (6 \times 10^{-7}) \sin [0.2\pi(ct - x)] \, \hat{k} \) tesla
• B. \( (2 \times 10^{-7}) \sin [200\pi(ct - x)] \, \hat{k} \) tesla
• C. \( (2 \times 10^{-7}) \sin [200\pi(ct - x)] \, \hat{i} \) tesla
• D. \( (6 \times 10^{-7}) \sin [0.2\pi(ct - x)] \, \hat{i} \) tesla

Hint:

Use \(\vec{B} = \frac{E_0}{c}\) for EM waves.
The directions follow right-hand rule: \(\vec{E} \times \vec{B} = \vec{k}\).

Answer 1

The Correct Option is B

The electric and magnetic fields in an electromagnetic wave are related by: \[ B_0 = \frac{E_0}{c} \] Given: \[ E_0 = 60 \text{ V/m}, \quad \lambda = 10 \text{ mm} = 0.01 \text{ m}, \quad c = 3 \times 10^8 \text{ m/s} \] Frequency: \[ f = \frac{c}{\lambda} = \frac{3 \times 10^8}{0.01} = 3 \times 10^{10} \text{ Hz} \] Angular frequency: \[ \omega = 2\pi f = 6\pi \times 10^{10} \] Wave number: \[ k = \frac{2\pi}{\lambda} = 200\pi \] Magnetic field amplitude: \[ B_0 = \frac{60}{3 \times 10^8} = 2 \times 10^{-7} \text{ T} \] Since wave propagates in x-direction and electric field in y-direction, magnetic field is along z-direction: \[ \vec{B} = (2 \times 10^{-7}) \sin(200\pi(ct - x)) \hat{k} \]