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 given electric field of a plane electromagnetic wave is:
\[ E_y = 20 \sin (3 \times 10^6 x - 4.5 \times 10^{14} t) \, \text{V/m} \]

The wave equation of an electromagnetic wave in a medium is:
\[ E = E_0 \sin(kx - \omega t) \]

Here, \(k\) is the wave number, \(\omega\) is the angular frequency. From the given equation, we can identify:
\(k = 3 \times 10^6 \, \text{m}^{-1}\)
\(\omega = 4.5 \times 10^{14} \, \text{rad/s}\)

The speed of the wave in the medium, \(v\), can be calculated as:
\[ v = \frac{\omega}{k} = \frac{4.5 \times 10^{14}}{3 \times 10^6} = 1.5 \times 10^8 \, \text{m/s} \]

The speed of light in a medium is related to the speed of light in vacuum (\(c = 3 \times 10^8 \, \text{m/s}\)) and the refractive index (\(n\)) of the medium by:
\[ v = \frac{c}{n} \]

Rearranging for \(n\), we have:
\[ n = \frac{c}{v} = \frac{3 \times 10^8}{1.5 \times 10^8} = 2 \]

The refractive index (\(n\)) is related to the dielectric constant (\(\varepsilon_r\)) by:
\[ n = \sqrt{\varepsilon_r} \]

Substituting the value of \(n\):
\[ 2 = \sqrt{\varepsilon_r} \]

Squaring both sides, we find:
\[ \varepsilon_r = 4 \]

The dielectric constant of the medium is confirmed as 4, which is within the expected range of 4 to 4.