An EM wave propagating in $$x$$-direction has a wavelength of 8 mm. The electric field vibrating $$y$$-direction has maximum magnitude of 60 Vm$$^{-1}$$. Choose the correct equations for electric and magnetic fields if the EM wave is propagating in vacuum:

An EM wave propagates in the $$x$$-direction with wavelength $$\lambda = 8$$ mm $$= 8 \times 10^{-3}$$ m. The electric field vibrates in the $$y$$-direction with maximum magnitude $$E_0 = 60$$ V m$$^{-1}$$.

Calculate the wave number $$k$$.

$$k = \frac{2\pi}{\lambda} = \frac{2\pi}{8 \times 10^{-3}} = \frac{\pi}{4} \times 10^3 \text{ m}^{-1}$$

Write the angular frequency $$\omega$$.

$$\omega = kc = \frac{\pi}{4} \times 10^3 \times 3 \times 10^8 = \frac{3\pi}{4} \times 10^{11} \text{ rad s}^{-1}$$

The argument of the sine function can be written as $$k(x - ct) = \frac{\pi}{4} \times 10^3 (x - 3 \times 10^8 t)$$.

Calculate the magnetic field amplitude $$B_0$$.

$$B_0 = \frac{E_0}{c} = \frac{60}{3 \times 10^8} = 2 \times 10^{-7} \text{ T}$$

Write the complete equations.

Since the wave propagates in the $$x$$-direction and $$\vec{E}$$ is along $$\hat{j}$$, the magnetic field $$\vec{B}$$ must be along $$\hat{k}$$ (as $$\vec{E} \times \vec{B}$$ should point in the direction of propagation).

$$E_y = 60 \sin\left[\frac{\pi}{4} \times 10^3(x - 3 \times 10^8 t)\right] \hat{j} \text{ V m}^{-1}$$

$$B_z = 2 \times 10^{-7} \sin\left[\frac{\pi}{4} \times 10^3(x - 3 \times 10^8 t)\right] \hat{k} \text{ T}$$

The correct answer is Option B.