The magnetic field of an electromagnetic wave travelling in a...

The speed of the wave in the medium is $v = \dfrac{c}{\sqrt{\epsilon_r}} = \dfrac{3 \times 10^8}{\sqrt{4}} = 1.5 \times 10^8$ m/s. (Matches A-III)

From the given equation, the wave vector $k = \pi$ rad/m. The wavelength is $\lambda = \dfrac{2\pi}{k} = \dfrac{2\pi}{\pi} = 2$ m. (Matches B-IV)

The amplitude of the electric field is $E_0 = vB_0 = (1.5 \times 10^8) \times (2 \times 10^{-8}) = 3$ V/m. (Matches C-I)

The phase of the wave is $(1.5\pi \times 10^8 t + \pi x)$. The positive sign between the time and space terms indicates that the wave is propagating in the $-x$ direction. (Matches D-V)

Thus, the correct matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(V).

Answer: (A)-(III), (B)-(IV), (C)-(I), (D)-(V)