Question

The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by, 
\[ E_y = 20 \sin (3 \times 10^6 x - 4.5 \times 10^{14} t) \, \text{V/m} \] (where \(x\), \(t\) and other values have S.I. units). The dielectric constant of the medium is ____________.

Hint:

For electromagnetic waves in non-magnetic media, dielectric constant is equal to the square of the refractive index.

Answer 1

Correct Answer: 4

The general equation of a plane electromagnetic wave is:
\[ E = E_0 \sin (kx - \omega t) \] Comparing with the given equation:
\[ k = 3 \times 10^6 \, \text{m}^{-1} \] \[ \omega = 4.5 \times 10^{14} \, \text{rad/s} \] Step 1: Calculate the speed of the wave in the medium.
\[ v = \frac{\omega}{k} = \frac{4.5 \times 10^{14}}{3 \times 10^6} \] \[ v = 1.5 \times 10^8 \, \text{m/s} \] Step 2: Find refractive index of the medium.
Speed of light in vacuum: \[ c = 3 \times 10^8 \, \text{m/s} \] \[ n = \frac{c}{v} = \frac{3 \times 10^8}{1.5 \times 10^8} = 2 \] Step 3: Calculate dielectric constant.
Since the medium is non-magnetic: \[ \varepsilon_r = n^2 = 2^2 = 4 \] Final Answer: \[ \boxed{4} \]