Chapter 31, Problem 29P

(a)

To determine

The speed of the helium neon laser light in air, water and glass.

Answer to Problem 29P

The speed of the helium neon laser light in air is $3\times {10}^8\text{m}/\text{s}$ , in water is $2.25\times {10}^8\text{m}/\text{s}$ and in glass is $2\times {10}^8\text{m}/\text{s}$ .

Explanation of Solution

Given:

The wavelength of the helium-neon laser is $\lambda =632.8\text{nm}$ .

Formula used:

The expression for the speed is given as,

  $v=\frac{c}{n}$

Here, $c$ is the speed of light and $n$ is the refractive index of that particular material.

Calculation:

The value of speed of light is $c=3\times {10}^8\text{m}/\text{s}$ .

The refractive index of air is $n_a=1$ .

The value of speed of helium neon laser light in air can be calculated as,

  $\begin{array}{c}{v}_a=\frac{c}{n_a}\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1}\\ =3\times {10}^8\text{m}/\text{s}\end{array}$

The refractive index in water is $n_w=1.33$ .

The value of speed of helium neon laser light in water can be calculated as,

  $\begin{array}{c}{v}_w=\frac{c}{n_w}\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1.33}\\ =2.25\times {10}^8\text{m}/\text{s}\end{array}$

The refractive index in glass is $n_g=1.50$ .

The value of speed of helium neon laser light in glass can be calculated as,

  $\begin{array}{c}{v}_g=\frac{c}{n_g}\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1.5}\\ =2\times {10}^8\text{m}/\text{s}\end{array}$

Conclusion:

Therefore, the speed of the helium neon laser light in air is $3\times {10}^8\text{m}/\text{s}$ , in water is $2.25\times {10}^8\text{m}/\text{s}$ and in glass is $2\times {10}^8\text{m}/\text{s}$ .

(b)

To determine

The wavelength of the helium neon laser light in air, water and glass.

Answer to Problem 29P

The wavelength of the helium neon laser light in air is $6.329\times {10}^{-7}\text{m}$ , in water is $4.57\times {10}^{-7}\text{m}$ and in glass is $4.21\times {10}^{-7}\text{m}$ .

Explanation of Solution

Formula used:

The expression for the frequency is given as,

  $f=\frac{c}{\lambda }$

The expression for the wavelength is given as,

  $\lambda =\frac{c}{nf}$

Calculation:

The value of frequency can be calculated as,

  $\begin{array}{c}f=\frac{c}{\lambda }\\ =\frac{3\times {10}^8\text{m}/\text{s}}{632.8\text{nm}\left(\frac{1\text{m}}{{10}^9\text{nm}}\right)}\\ =4.74\times {10}^{14}\text{Hz}\end{array}$

The value of wavelength of the helium neon laser light in air can be calculated as,

  $\begin{array}{c}{\lambda }_a=\frac{c}{n_{a}f}\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1\times 4.74\times {10}^{14}\text{Hz}}\\ =6.329\times {10}^{-7}\text{m}\end{array}$

The value of wavelength of the helium neon laser light in water can be calculated as,

  $\begin{array}{c}{\lambda }_w=\frac{c}{n_{w}f}\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1.33\times 4.74\times {10}^{14}\text{Hz}}\\ =4.57\times {10}^{-7}\text{m}\end{array}$

The value of wavelength of the helium neon laser light in glass can be calculated as,

  $\begin{array}{c}{\lambda }_g=\frac{c}{n_{g}f}\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1.5\times 4.74\times {10}^{14}\text{Hz}}\\ =4.21\times {10}^{-7}\text{m}\end{array}$

Conclusion:

Therefore, the wavelength of the helium neon laser light in air is $6.329\times {10}^{-7}\text{m}$ , in water is $4.57\times {10}^{-7}\text{m}$ and in glass is $4.21\times {10}^{-7}\text{m}$ .

(c)

To determine

The frequency of the helium neon laser light in air, water and glass.

Answer to Problem 29P

The frequency of the helium neon laser light in air is $4.74\times {10}^{14}\text{Hz}$ , in water is $3.56\times {10}^{14}\text{Hz}$ and in glass is $3.16\times {10}^{14}\text{Hz}$ .

Explanation of Solution

Formula used:

The expression for the frequency can be given as,

  $f=\frac{c}{n\lambda }$

Calculation:

The value of frequency of the helium neon laser light in air can be calculated as,

  $\begin{array}{c}{f}_a=\frac{c}{n_a\lambda }\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1\times 632.8\text{nm}\left(\frac{1\text{m}}{{10}^9\text{nm}}\right)}\\ =4.74\times {10}^{14}\text{Hz}\end{array}$

The value of frequency of the helium neon laser light in water can be calculated as,

  $\begin{array}{c}{f}_w=\frac{c}{n_w\lambda }\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1.33\times 632.8\text{nm}\left(\frac{1\text{m}}{{10}^9\text{nm}}\right)}\\ =3.56\times {10}^{14}\text{Hz}\end{array}$

The value of frequency of the helium neon laser light in glass can be calculated as

  $\begin{array}{c}{f}_g=\frac{c}{n_g\lambda }\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1.50\times 632.8\text{nm}\left(\frac{1\text{m}}{{10}^9\text{nm}}\right)}\\ =3.16\times {10}^{14}\text{Hz}\end{array}$

Conclusion:

Therefore, the frequency of the helium neon laser light in air is $4.74\times {10}^{14}\text{Hz}$ , in water is $3.56\times {10}^{14}\text{Hz}$ and in glass is $3.16\times {10}^{14}\text{Hz}$ .