Question

A light wave has a 670 nm wavelength in air. Its wavelength in a transparent solid is 420 nm. a. What is the speed of light in this solid? b. What is the light’s frequency in the solid?

Short answer

Part a

The speed of light in the transparent solid is $1.88\times {10}^{8}m/s$.

Part b

The frequency of the light in the transparent solid is$4.48\times {10}^{14}Hz$.

Step-by-step solution

Given information

The wavelength of light in air ${\lambda }_{air}=670nm$.

The wavelength of light in transparent solid is${\lambda }_{solid}=420nm$.

Part a

The speed of light in a material is given by $v_{material}=\frac{c}{n}\dots \left(1\right)$.

Where $c$is the speed of light in a vacuum and $n$the refractive index of the material.

The refractive index is also given as role="math" localid="1649963673118" $n=\frac{{\lambda }_{vacuum}}{{\lambda }_{material}}\dots \left(2\right)$

where, ${\lambda }_{vacuum}$is the wavelength of light in a vacuum and ${\lambda }_{material}$is the wavelength of light in the material.

Here the material is transparent solid.

From (1) and (2)

$$\begin{gathered} v_{solid}=c\left(\frac{{\lambda }_{solid}}{{\lambda }_{vaccum}}\right) \\ =3\times {10}^8\times \left(\frac{420}{670}\right) \\ =1.88\times {10}^{8}m/s \end{gathered}$$

Therefore, the speed of light in this solid is$1.88\times {10}^{8}m/s$.

Part b

The relation between speed, wavelength, and frequency of the wave is given by $v=f\lambda$.

The light’s frequency in the transparent solid is

$$\begin{gathered} f=\frac{v_{solid}}{{\lambda }_{solid}} \\ =\frac{1.88\times {10}^8}{420\times {10}^{-9}} \\ =4.48\times {10}^{14}Hz \end{gathered}$$

Therefore, the frequency of the light in the transparent solid is$4.48\times {10}^{14}Hz$.