Chapter 36, Problem 55P

To determine

The difference in positions of two rays crossing the principal axis.

Answer to Problem 55P

The difference in positions of two rays crossing the principal axis is $\underset{\_}{21.3\text{cm}}$.

Explanation of Solution

The following figure gives the ray diagram of the two rays.

Write the expression to find the angle of incidence of the first ray.

    ${\theta }_1={\sin }^{-1}\left(\frac{h_1}{R}\right)$                                                                                                            (I)

Here, $h_1$ is the distance of the first ray from the principle axis before passing through the lens.

Write the relation between the angle of incidence of the first ray and the second ray.

    $1.00\sin {\theta }_2=1.60\sin {\theta }_1$                                                                                              (II)

Write the expression for the focal length of the first ray

    $f_1=\frac{h_1}{\tan \left({\theta }_2-{\theta }_1\right)}$                                                                                                    (III)

Write the expression for the distance crossed by the principal axis from the vertex by the first ray.

    $x_1=f_1-R\left(1-\cos {\theta }_1\right)$                                                                                             (IV)

Write the expression to find the angle of incidence of the second ray.

    ${\theta }_1={\sin }^{-1}\left(\frac{h_2}{R}\right)$                                                                                                         (V)

Here, $h_2$ is the distance of the second ray from the principle axis before passing through the lens.

Write the relation between the angle of incidence of the first ray and the second ray.

    $1.00\sin {\theta }_2=1.60\sin {\theta }_1$                                                                                             (VI)

Write the expression for the focal length of the second ray

    $f_2=\frac{h_2}{\tan \left({\theta }_2-{\theta }_1\right)}$                                                                                                   (VII)

Write the expression for the distance crossed by the principal axis from the vertex by the second ray.

    $x_2=f_2-R\left(1-\cos {\theta }_2\right)$                                                                                         (VIII)

Write the expression for the difference in positions of two rays crossing the principal axis .

    $\Delta x=x_1-x_2$                                                                                                            (IX)

Conclusion:

Substitute $0.500\text{cm}$ for $h_1$ and $20.0\text{cm}$ for $R$ in equation (I).

    $\begin{array}{c}{\theta }_1={\sin }^{-1}\left(\frac{0.500\text{cm}}{20.0\text{cm}}\right)\\ =1.43°\end{array}$

Substitute ${\sin }^{-1}\left(\frac{h_1}{R}\right)$ for ${\theta }_1$, $0.500\text{cm}$ for $h_1$ and $20.0\text{cm}$ for $R$ in equation (II).

    $\begin{array}{c}1.00\sin {\theta }_2=1.60\sin \left({\sin }^{-1}\left(\frac{0.500\text{cm}}{20.0\text{cm}}\right)\right)\\ {\theta }_2=2.29°\end{array}$

Substitute $0.500\text{cm}$ for $h_1$, $1.43°$ for ${\theta }_1$ and $2.29°$ for ${\theta }_2$ in equation (III).

    $\begin{array}{c}{f}_1=\frac{0.500\text{cm}}{\tan \left(2.29°-1.43°\right)}\\ =33.3\text{cm}\end{array}$

Substitute $33.3\text{cm}$ for $f_1$, $1.43°$ for ${\theta }_1$ and $20.0\text{cm}$ for $R$ in equation (IV).

    $\begin{array}{c}{x}_1=33.3\text{cm}-\left(20.0\text{cm}\right)\left(1-\cos 1.43°\right)\\ =33.3\text{cm}-0.00625\text{cm}\\ =33.3\text{cm}\end{array}$

Substitute $12.0\text{cm}$ for $h_2$ and $20.0\text{cm}$ for $R$ in equation (V).

    $\begin{array}{c}{\theta }_1={\sin }^{-1}\left(\frac{12.0\text{cm}}{20.0\text{cm}}\right)\\ =36.9°\end{array}$

Substitute ${\sin }^{-1}\left(\frac{h_2}{R}\right)$ for ${\theta }_1$, $12.0\text{cm}$ for $h_2$ and $20.0\text{cm}$ for $R$ in equation (VI).

    $\begin{array}{c}1.00\sin {\theta }_2=1.60\sin \left({\sin }^{-1}\left(\frac{12.0\text{cm}}{20.0\text{cm}}\right)\right)\\ {\theta }_2=73.7°\end{array}$

Substitute $12.0\text{cm}$ for $h_2$, $36.9°$ for ${\theta }_1$ and $73.7°$ for ${\theta }_2$ in equation (VII).

    $\begin{array}{c}{f}_2=\frac{12.0\text{cm}}{\tan \left(73.7°-36.9°\right)}\\ =16.0\text{cm}\end{array}$

Substitute $16.0\text{cm}$ for $f_2$, $36.9°$ for ${\theta }_1$ and $20.0\text{cm}$ for $R$ in equation (VIII).

    $\begin{array}{c}{x}_2=\left(16.0\text{cm}\right)-\left(20.0\text{cm}\right)\left(1-\cos 36.9°\right)\\ =12.0\text{cm}\end{array}$

Substitute $33.3\text{cm}$ for $x_1$ and $12.0\text{cm}$ for $x_2$ in equation (IX).

    $\begin{array}{c}\Delta x=33.3\text{cm}-12.0\text{cm}\\ =21.3\text{cm}\end{array}$

Therefore, the difference in positions of two rays crossing the principal axis is $\underset{\_}{21.3\text{cm}}$.