Question

A plane electromagnetic wave is represented by the equation \[ E = 100 \cos(6 \times 10^8 t + 4x) \, \text{volt/metre}. \] Calculate the following:
(i) The refractive index of the medium
(ii) The velocity of electromagnetic wave in the medium
(iii) The expression for magnetic field

Hint:

For plane electromagnetic waves, the velocity can be calculated using \( v = \frac{\omega}{k} \) and the magnetic field using \( B = \frac{E}{c} \).

Answer 1

i. The refractive index of the medium Step 1: The given equation of the wave is: \[ E = 100 \cos(6 \times 10^8 t + 4x) \] Step 2: Comparing with the general wave equation \( E = E_0 \cos(\omega t + kx) \), we get: \[ \omega = 6 \times 10^8, \quad k = 4 \] Step 3: Using the relation \( v = \frac{\omega}{k} \): \[ v = \frac{6 \times 10^8}{4} = 1.5 \times 10^8 \, \text{m/s} \] Step 4: The refractive index is: \[ n = \frac{c}{v} = \frac{3 \times 10^8}{1.5 \times 10^8} = 2 \] \[ \boxed{n = 2} \] ii. The velocity of electromagnetic wave in the medium Step 1: From the previous step, we obtained the velocity as: \[ v = \frac{\omega}{k} = 1.5 \times 10^8 \, \text{m/s} \] \[ \boxed{v = 1.5 \times 10^8 \, \text{m/s}} \] iii. The expression for magnetic field Step 1: In an electromagnetic wave, the magnetic field \( B \) is related to the electric field \( E \) by: \[ B = \frac{E}{c} \] Step 2: Substituting the given electric field expression: \[ B = \frac{100}{3 \times 10^8} \cos(6 \times 10^8 t + 4x) \] \[ B = \frac{E_0}{c} \cos(6 \times 10^8 t + 4x) \] \[ \boxed{B = \frac{E_0}{c} \cos(6 \times 10^8 t + 4x)} \]