Chapter 35, Problem 39P

To determine

The angular spread of the visible light passing through a prism of apex angle $60.0°$ for the angle of incidence of $50.0°$.

Answer to Problem 39P

The angle of dispersion is $4.61°$.

Explanation of Solution

Write the expression for the index of refraction on the first face of the prism.

    $\frac{\sin i_1}{\sin r_1}=n$

Here, $i_1$ is the angle of incidence on the first face of the prism, $n$ is the index of refraction and $r_1$ is the angle of refraction on the first face of the prism.

Rewrite the above equation for $r_1$.

    $\begin{array}{c}\frac{\sin i_1}{\sin r_1}=n\\ \sin r_1=\frac{\sin i_1}{n}\\ r_1={\sin }^{-1}\left(\frac{\sin i_1}{n}\right)\end{array}$                                                                                                    (I)

Write the expression for apex angle.

    $\varphi =i_2+r_1$                                                                                                                     (II)

Here, $\varphi$ is the apex angle and $i_2$ is the incident angle on the second face of the prism.

The emergent angle on the second face is,

    $\begin{array}{c}\sin e_2=n\sin i_2\\ e_2={\sin }^{-1}\left(n\sin i_2\right)\end{array}$                                                                                               (III)

Here, $e_2$ is the emergent angle on the second face.

Write the expression for the angle of deviation.

    ${\delta }_{\text{i}}=i_1-r_1+e_2-i_2$                                                                                                     (IV)

Here, ${\delta }_{\text{i}}$ is the angle of deviation.

Write the expression for angular dispersion.

    $\delta ={\delta }_1-{\delta }_2$                                                                                                                 (V)

Here, $\delta$ is the angular dispersion and ${\delta }_1$ and ${\delta }_2$ are the respective angle of deviation.

Conclusion:

Angle of refraction for red on first face of the prism is.

Substitute $1.62$ for $n$ and $50.0°$ for $i_1$ in Equation (I) to calculate $r_{\text{r1}}$.

    $\begin{array}{c}{r}_{\text{r1}}={\sin }^{-1}\left(\frac{\sin 50.0°}{1.62}\right)\\ ={\sin }^{-1}\left(\frac{0.766}{1.62}\right)\\ ={\sin }^{-1}\left(0.472\right)\\ =28.22°\end{array}$

Calculate the angle of incidence for red on second face of the prism.

Substitute $60.0°$ for $\varphi$ and $28.22°$ for $r_{\text{1}}$ in Equation (II) to calculate $i_{\text{r2}}$.

    $\begin{array}{c}60.0°=i_{\text{r2}}+28.22°\\ i_{\text{r2}}=60.0°-28.22°\\ i_{\text{r2}}=31.78°\end{array}$

Substitute $1.62$ for $n$ and $31.78°$ for $i_{\text{2}}$ in Equation (III) t calculate $e_{\text{r}2}$.

    $\begin{array}{c}{e}_{\text{r}2}={\sin }^{-1}\left(\left(1.62\right)\sin 31.78°\right)\\ ={\sin }^{-1}\left(0.853\right)\\ =58.56°\end{array}$

Substitute $58.56°$ for $e_2$, $31.78°$ for $i_2$, $50.0°$ for $i_1$ and $28.22°$ for $r_{\text{1}}$ in Equation (IV) to calculate ${\delta }_{\text{r}}$.

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

Calculate the angle of refraction for violet on the first face of the prism.

Substitute $1.66$ for $n$ and $50.0°$ for $i_1$ in Equation (I) to calculate $r_{\text{v1}}$.

    $\begin{array}{c}{r}_{\text{v1}}={\sin }^{-1}\left(\frac{\sin 50.0°}{1.66}\right)\\ ={\sin }^{-1}\left(\frac{0.766}{1.66}\right)\\ ={\sin }^{-1}\left(0.461\right)\\ =27.48°\end{array}$

Calculate the angle of incidence for violet on second face of the prism.

Substitute $60.0°$ for $\varphi$ and $27.48°$ for $r_{\text{1}}$ in Equation (II) to calculate $i_{\text{v2}}$.

    $\begin{array}{c}60.0°=i_{\text{v2}}+27.48°\\ i_{\text{v2}}=60.0°-27.48°\\ i_{\text{v2}}=32.52°\end{array}$

Substitute $1.66$ for $n$ and $32.52°$ for $i_{\text{2}}$ in Equation (III) t calculate $e_{\text{v}2}$.

    $\begin{array}{c}{e}_{\text{v}2}={\sin }^{-1}\left(\left(1.66\right)\sin 32.52°\right)\\ ={\sin }^{-1}\left(0.892\right)\\ =63.17°\end{array}$

Substitute $63.17°$ for $e_2$, $32.52°$ for $i_2$, $50.0°$ for $i_1$ and $27.48°$ for $r_{\text{1}}$ in Equation (IV) to calculate ${\delta }_{\text{r}}$.

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

Substitute $53.17°$ for ${\delta }_1$ and $48.56°$ for ${\delta }_2$ in Equation (V) to calculate $\delta$.

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

Therefore, angle of dispersion is $4.61°$.