Chapter 25, Problem 35P

To determine

The focal length and the radius of curvature of the main objective mirror in the reflecting astronomical telescope.

Answer to Problem 35P

Solution:

The focal length of the objective mirror is found to be 4.7 m and the radius of curvature of the mirror is found to be 9.3 m.

Explanation of Solution

The magnification M of a telescope is the ratio of the focal lengths of the objective and eyepiece.

$M=-\frac{f_o}{f_e}$

Here, the focal length of the objective is $f_o$ and that of the eyepiece is $f_e$.

In reflecting telescopes, concave mirrors are used as objective. If the mirror fas a radius of curvature R, then the focal length is related to the radius of curvature, as,

$R=2f_o$

Given:

The magnification of the telescope $M=150\times$

The focal length of the eyepiece $f_e=3.1\text{ cm}$

Formula:

$f_o=-M\times f$

$R=2f_o$

Calculation:

Use the given value of magnification and the focal length of the eyepiece, calculate the focal length of the objective. Affix a negative sign to the magnification, since the image formed by a reflecting telescope is inverted.

$\begin{array}{c}{f}_o=-M\times f_e\\ =-\left(-150\right)\left(3.1\text{ cm}\right)\\ =4.65\text{ m}\\ =4.7\text{ m}\end{array}$

Calculate the radius of curvature of the mirror, using the calculated value of the focal length of the eye piece.

$\begin{array}{c}R=2f_o\\ =2\left(4.65\text{ m}\right)\\ =9.3\text{ m}\end{array}$