Question

A hobby telescope has an objective lens with a focal length of 75 cm and an eyepiece with a focal length of 8.2 mm. We view the planet Jupiter with this telescope. We do this at the time of year when we are closest to Jupiter, a distance of 6.28 × 1011 m. The diameter of Jupiter is 1.40 × 108 m.

(a) Find the diameter (mm) of the image of Jupiter produced by the objective lens.
(b) How far (cm) should I place a marble (1-cm diameter) from my eye so that it appears to be the same size as Jupiter as viewed through the eyepiece of the telescope? Give the answers to 2 significant figures.

Answer 1

To make a marble appear the same size as Jupiter through the telescope, it should be placed approximately 91.46 cm away from the eye.

Step-by-step explanation:

To find the distance at which a marble should be placed from the eye so that it appears the same size as Jupiter through the telescope, we can use the formula for angular magnification:

Angular Magnification (M) = -fobj/feye

Where fobj is the focal length of the objective lens and feye is the focal length of the eyepiece. In this case, fobj = 75 cm and feye = 8.2 mm = 0.82 cm.

Plugging in the values, we get: M = -75/0.82 = -91.46

Since the angular magnification is equal to the ratio of the image distance to the object distance, we can set up the following equation:

M = -di/do

Where di is the image distance (unknown) and do is the object distance (1 cm).

Since the marble is 1 cm in diameter, we want the image of the marble to be the same size as Jupiter, which has a diameter of 1.40 x 10^8 m. Therefore, the image distance should be proportional to the size of Jupiter:

di = M * do

Plugging in the values, we get: di = -91.46 * 1 cm = -91.46 cm

Since the distance cannot be negative, the marble should be placed approximately 91.46 cm away from the eye to appear the same size as Jupiter through the telescope.