Chapter 38, Problem 10PQ

Figure P38.10 on the next page shows a monochromatic beam of light of wavelength 575 nm incident on a slab of crown glass surrounded by air. Use a protractor to measure the angles of incidence and refraction. a. What is the speed of the beam of light within the glass slab? b. What is the frequency of the beam of light within the glass slab? c. What is the wavelength of the beam of light within the glass slab?

FIGURE P38.10

To determine

The speed of the beam of light within the glass slab.

Answer to Problem 10PQ

The speed of the beam of light within the glass slab is $1.99\times {10}^8\text{m}/\text{sec}$.

Explanation of Solution

Write the expression given by Snell’s for the light ray travelling in two different medium.

    $n_{\text{air}}\sin {\theta }_{\text{i}}=n_{\text{glass}}\sin {\theta }_{\text{t}}$                                                                                    (I)

Here, $n_{\text{water}}$ is the refractive index of liquid, $n_{\text{air}}$ is the refractive index of air medium, ${\theta }_{\text{t}}$ is the angle of refraction in glass and ${\theta }_{\text{i}}$ is the angle of incidence in air.

Write the expression to calculate the speed of light.

    $n_{\text{glass}}=\frac{c}{v}$                                                                                                   (II)

Here, $n_{\text{glass}}$ is the refractive index of the glass slab, $v$ is the speed of light in the glass slab and $c$ is the speed of light in vacuum.

Conclusion:

By the use of protractor the angle of incidence is measured as $65°$.

Substitute $65°$ for ${\theta }_{\text{i}}$, $1.0002926$ for $n_{\text{air}}$ and $37°$ for ${\theta }_{\text{t}}$ in equation (I) to calculate the value of $n_{\text{glass}}$.

    $\begin{array}{c}1.0002926\sin 65°=n_{\text{glass}}\sin 37°\\ n_{\text{glass}}=\frac{1.0002926\sin 65°}{\sin 37°}\\ =1.5064\end{array}$

Substitute $1.5064$ for $n_{\text{glass}}$ and $3\times {10}^8\text{m}/\text{sec}$ for $c$ in equation (II) to calculate the value of $v$.

    $\begin{array}{c}1.5064=\frac{3\times {10}^8\text{m}/\text{sec}}{v}\\ v=\frac{3\times {10}^8\text{m}/\text{sec}}{1.5064}\\ =1.99\times {10}^8\text{m}/\text{sec}\end{array}$

Therefore, the speed of the beam of light within the glass slab is $1.99\times {10}^8\text{m}/\text{sec}$.

To determine

The frequency of the beam of light within the glass slab.

Answer to Problem 10PQ

The frequency of the beam of light within the glass slab is $5.21\times {10}^{14}\text{Hz}$.

Explanation of Solution

Write the expression to calculate the frequency.

    $f=\frac{v}{{\lambda }_{\text{glass}}}$                         (III)

Here, $v$ is the velocity of light in the glass, $f$ is the frequency in the glass slab and ${\lambda }_{\text{glass}}$ is the wavelength of the light travelling in the glass slab.

Write the expression to calculate the wavelength of the light travelling in the glass medium.

    ${\lambda }_{\text{glass}}=\frac{{\lambda }_{\text{air}}}{n_{\text{glass}}}$                            (IV)

Here, ${\lambda }_{\text{air}}$ is the wavelength of the light beam travelling in the air.

Conclusion:

Substitute $575\text{nm}$ for ${\lambda }_{\text{air}}$ and $1.5064$ for $n_{\text{glass}}$ in equation (IV) to calculate the value of ${\lambda }_{\text{glass}}$.

    $\begin{array}{c}{\lambda }_{\text{glass}}=\frac{\left(575\text{nm}\times \frac{{10}^{-9}\text{m}}{1\text{nm}}\right)}{1.5064}\\ =3.817\times {10}^{-7}\text{m}\\ \approx 3.82\times {10}^{-7}\text{m}\end{array}$

Substitute $3.817\times {10}^{-7}\text{m}$ for ${\lambda }_{\text{glass}}$ and $1.99\times {10}^8\text{m}/\text{sec}$ in equation (III) to calculate the value of $f$.

    $\begin{array}{c}f=\frac{1.99\times {10}^8\text{m}/\text{sec}}{3.817\times {10}^{-7}\text{m}}\\ =5.21\times {10}^{14}\text{Hz}\end{array}$

Therefore, the frequency of the beam of light within the glass slab is $5.21\times {10}^{14}\text{Hz}$.

To determine

The wavelength of the beam of light within the glass slab.

Answer to Problem 10PQ

The wavelength of the beam of light within the glass slab is $3.82\times {10}^{-7}\text{m}$.

Explanation of Solution

Conclusion:

Substitute $575\text{nm}$ for ${\lambda }_{\text{air}}$ and $1.5064$ for $n_{\text{glass}}$ in equation (IV) to calculate the value of ${\lambda }_{\text{glass}}$.

    $\begin{array}{c}{\lambda }_{\text{glass}}=\frac{\left(575\text{nm}\times \frac{{10}^{-9}\text{m}}{1\text{nm}}\right)}{1.5064}\\ =3.82\times {10}^{-7}\text{m}\end{array}$

Therefore, The wavelength of the beam of light within the glass slab is $3.82\times {10}^{-7}\text{m}$.