Chapter 35, Problem 35.39P

To determine

The angular spread of visible light passing through a prism.

Answer to Problem 35.39P

The angular spread of visible light passing through a prism is $4.61°$.

Explanation of Solution

Given info: The index of refraction for violet light in silica flint glass is $1.66$ and that for red light is $1.62$. The apex angle of prism is $60.0°$ and the angle of incidence is $50.0°$.

The figure of light ray passing through the prism is shown as below,

The expression of the Snell’s law is,

$n_1\sin {\theta }_1=n_{\text{2}}\sin {\theta }_2$ (1)

Here,

$n_{\text{1}}$ is the index of refraction of first medium.

${\theta }_1$ is the angle of incidence of the light ray.

$n_2$ is the index of refraction of second medium.

${\theta }_2$ is the angle of refraction of the transmitted ray.

For the red color:

Consider the angle of refraction on the first face of the prism is $r_{\text{r}}$.

Substitute $50°$ for ${\theta }_1$, $1.62$ for $n_2$ and $1$ for $n_1$ in equation (1).

$\begin{array}{c}1\times \sin 50°=1.62\times \sin r_{\text{r}}\\ \sin r_{\text{r}}=\frac{0.766}{1.62}\\ r_{\text{r}}={\sin }^{-1}\left(\frac{0.766}{1.62}\right)\\ =28.22°\end{array}$

Thus, the angle of refraction on the first face of the prism is $28.22°$.

The angle of incidence on the second face of the prism is,

$i_2=\varphi -r_{\text{r}}$

Here,

$\varphi$ is the apex angle of prism.

Substitute $60.0°$ for $\varphi$ and $28.22°$ for $r_{\text{r}}$ in above equation.

$\begin{array}{c}{i}_2=60.0°-28.22°\\ =31.78°\end{array}$

Consider the angle of refraction on the second face of the prism is $e_{\text{r}}$.

Substitute $31.78°$ for ${\theta }_1$, $1.62$ for $n_1$ and $1$ for $n_2$ in equation (1).

$\begin{array}{c}1.62\times \sin 31.78°=1\times \sin e_{\text{r}}\\ \sin e_{\text{r}}=1.62\times 0.527\\ e_{\text{r}}={\sin }^{-1}\left(0.853\right)\\ =58.56°\end{array}$

Thus, the angle of refraction on the second face of the prism is $58.56°$.

The angle of deviation is,

${\delta }_{\text{r}}=i-r_{\text{r}}+e_{\text{r}}-i_2$

Here,

$i$ is the angle of incidence at first face of prism.

Substitute $50.0°$ for $i$, $28.22°$ for $r_{\text{r}}$, $58.56°$ for $e_{\text{r}}$ and $31.78°$ for $i_2$ in above equation.

$\begin{array}{c}{\delta }_r=50.0°-28.22°+58.56°-31.78°\\ =48.56°\end{array}$

Thus, the angle of deviation for red color is $48.56°$.

For the violet color:

Consider the angle of refraction on the first face of the prism is $r_{\text{v}}$.

Substitute $50°$ for ${\theta }_1$, $1.66$ for $n_2$ and $1$ for $n_1$ in equation (1).

$\begin{array}{c}1\times \sin 50°=1.66\times \sin r_{\text{v}}\\ \sin r_{\text{v}}=\frac{0.766}{1.66}\\ r_{\text{v}}={\sin }^{-1}\left(\frac{0.766}{1.66}\right)\\ =27.48°\end{array}$

Thus, the angle of refraction on the first face of the prism is $27.48°$.

The angle of incidence on the second face of the prism is,

$i_2=\varphi -r_{\text{v}}$

Substitute $60.0°$ for $\varphi$ and $27.48°$ for $r_{\text{v}}$ in above equation.

$\begin{array}{c}{i}_2=60.0°-27.48°\\ =32.52°\end{array}$

Consider the angle of refraction on the second face of the prism is $e_{\text{v}}$.

Substitute $32.52°$ for ${\theta }_1$, $1.66$ for $n_1$ and $1$ for $n_2$ in equation (1).

$\begin{array}{c}1.66\times \sin 32.52°=1\times \sin e_{\text{v}}\\ \sin e_{\text{v}}=1.66\times 0.538\\ e_{\text{v}}={\sin }^{-1}\left(0.8923\right)\\ =63.17°\end{array}$

Thus, the angle of refraction on the second face of the prism is $63.17°$.

The angle of deviation is,

${\delta }_{\text{v}}=i-r_{\text{v}}+e_{\text{v}}-i_2$

Substitute $50.0°$ for $i$, $27.48°$ for $r_{\text{v}}$, $63.17°$ for $e_{\text{v}}$ and $32.52°$ for $i_2$ in above equation.

$\begin{array}{c}{\delta }_v=50.0°-27.48°+63.17°-32.52°\\ =53.17°\end{array}$

Thus, the angle of deviation for violet color is $53.17°$.

The angular spread of visible light passing through a prism is,

$\delta ={\delta }_{\text{v}}-{\delta }_{\text{r}}$

Substitute $53.17°$ for ${\delta }_{\text{v}}$ and $48.56°$ for ${\delta }_{\text{r}}$ in above equation.

$\begin{array}{c}\delta =53.17°-48.56°\\ =4.61°\end{array}$

Conclusion:

Therefore, the angular spread of visible light passing through a prism is $4.61°$.