Answer to Question #89920 in Electricity and Magnetism for Shivam Nishad

Question

A uniform plane wave has a wavelength of 6cm in vacuum and 2 cm in a non-magnetic dielectric having permeability of free space. Calculate the dielectric constant of the dielectric and the phase velocity of propagation of the wave through it.

Answer 1

Wavelength in vacuum:

$$\lambda_0 = \frac{c}{\nu} = 6\,\text{cm}$$

Frequency $\nu$ of both waves are equal.

In dielectric medium, speed of light equals to

$$v = \frac{c}{n} = \frac{c}{\sqrt{\varepsilon \mu}}$$

As dielectric is non-magnetic, wavelength equals to

$$\lambda_m=\frac{v}{\nu} = \frac{c}{\nu\sqrt{\varepsilon}} = 2\,\text{cm}$$

Therefore, dielectric constant

$$\sqrt{\varepsilon} = \frac{\lambda_0}{\lambda_m} \Rightarrow \varepsilon = \frac{\lambda_0^2}{\lambda_m^2} = 9$$

phase velocity of propagation of the wave:

$$v = \frac{c}{\sqrt{\varepsilon }} = 10^8 \,\text{m/s}$$