The electric field in a plane electromagnetic wave is given by E_y =…

The correct answer is Option (4) → (A), (C), and (D) only

Compare the given equation with the standard form of a plane progressive wave:

    \( E_{y} = E_{0}\sin(kx + \omega t) \)

    \( E_{0} = 50\ \mathrm{V/m} \)
    \( k = 2\pi\times10^{5}\ \mathrm{m^{-1}} \)
    \( \omega = 4\pi\times10^{11}\ \mathrm{rad/s} \)

(A) The frequency of the wave is \( 200\ \mathrm{GHz} \). — Correct.

    \( \displaystyle f = \frac{\omega}{2\pi} = \frac{4\pi\times10^{11}}{2\pi}\ \mathrm{Hz} = 2\times10^{11}\ \mathrm{Hz} = 200\ \mathrm{GHz} \)

(B) The amplitude of the magnetic field of the wave is \( 150\ \mathrm{nT} \) — Incorrect. In an electromagnetic wave \( E_{0} = vB_{0} \), where v is the speed of the wave (found in (C)):

    \( \displaystyle B_{0} = \frac{E_{0}}{v} = \frac{50\ \mathrm{V/m}}{2\times10^{6}\ \mathrm{m/s}} = 2.5\times10^{-5}\ \mathrm{T} = 25\ \mu\mathrm{T} \)

Even with the speed of light, \( B_{0} = 50/(3\times10^{8}) \approx 1.67\times10^{-7}\ \mathrm{T} \approx 167\ \mathrm{nT} \), so 150 nT does not follow either way.

(C) The speed of the wave is \( 2\times10^{6}\ \mathrm{m/s} \). — Correct.

    \( \displaystyle v = \frac{\omega}{k} = \frac{4\pi\times10^{11}\ \mathrm{rad/s}}{2\pi\times10^{5}\ \mathrm{m^{-1}}} = 2\times10^{6}\ \mathrm{m/s} \)

(This is less than c, so the equation describes a wave travelling in a material medium.)

(D) The wavelength of the wave is \( 10\ \mu\mathrm{m} \). — Correct.

    \( \displaystyle \lambda = \frac{2\pi}{k} = \frac{2\pi}{2\pi\times10^{5}}\ \mathrm{m} = 10^{-5}\ \mathrm{m} = 10\ \mu\mathrm{m} \)