Question

You have two lenses of focal length \(25.0 \mathrm{cm}\) (lens 1 ) and $5.0 \mathrm{cm}$ (lens 2 ). (a) To build an astronomical telescope that gives an angular magnification of \(5.0,\) how should you use the lenses (which for objective and which for eyepiece)? Explain. (b) How far apart should they be?

Short answer

Short Answer: To achieve the desired angular magnification of 5, lens 1 (focal length: 25 cm) should be used as the objective lens and lens 2 (focal length: 5 cm) should be used as the eyepiece lens. The lenses should be placed 30 cm apart.

Step-by-step solution

Identify the formula for angular magnification

For a telescope, the angular magnification (M) is equal to the ratio of the focal length of the objective lens (F_obj) to the focal length of the eyepiece lens (F_eye): M = F_obj / F_eye We are given the angular magnification (M) and the focal lengths of the two lenses (lens 1 and lens 2). Our task is to arrange these lenses to act as the objective and eyepiece to achieve the desired magnification.

Calculate the possible combinations

First, let's consider the two possible ways to arrange the lenses and see which combination gives the desired magnification. 1. Lens 1 as objective and Lens 2 as eyepiece: M = F_obj / F_eye = 25 cm / 5 cm = 5 2. Lens 2 as the objective and Lens 1 as the eyepiece: M = F_obj / F_eye = 5 cm / 25 cm = 1/5 Since the desired magnification is 5, the first combination (Lens 1 as objective and Lens 2 as eyepiece) is the correct one.

Answer part (a)

To build an astronomical telescope with an angular magnification of 5, lens 1 (focal length: 25 cm) should be the objective lens, and lens 2 (focal length: 5 cm) should be the eyepiece lens.

Calculate lens separation

To find the distance between the lenses, we will use the lensmaker's equation for a system of two lenses. The equivalent focal length (F) of the two-lens system is given by the formula 1/F = 1/F_obj + 1/F_eye. The distance (d) between the lenses is the sum of their respective focal lengths. First, let's find the equivalent focal length (F): 1/F = 1/25 cm + 1/5 cm = 6/25 cm F = 25 cm / 6 = 4.17 cm Now that we have the equivalent focal length, we can calculate the distance between the lenses (d) by summing their respective focal lengths: d = F_obj + F_eye = 25 cm + 5 cm = 30 cm

Answer part (b)

To build the astronomical telescope with the given lenses and desired angular magnification of 5, the lenses should be placed 30 cm apart.