Situation 19.3
The angular magnification of a refracting telescope is 40.When the object and final image are both at infinity,the distance between the eyepiece and the objective is 143.5 cm.The telescope is used to view a distant radio tower.The real image of the tower,formed by the objective,is 6.0 mm in height.The focal point of the eyepiece is positioned at the real image.
-In Situation 19.3,the angle subtended by the final image of the tower is closest to:
A)0.15 rad
B)0.17 rad
C)0.19 rad
D)0.21 rad
E)0.23 rad

Question

Quizplus · Accepted Answer

We can start by using the formula for angular magnification:

$M = \frac{	heta'}{	heta}$

where $M$ is the angular magnification, $	heta'$ is the angle subtended by the final image, and $	heta$ is the angle subtended by the object.

When the object and final image are both at infinity, we can assume that $	heta = \frac{height}{distance}$. From the given information, we know that the real image of the tower has a height of 6.0 mm, and since it is formed by the objective, the distance is also equal to the focal length of the objective. Therefore:

$	heta = \frac{6.0 	imes 10^{-3}}{f_{obj}}$

where $f_{obj}$ is the focal length of the objective.

To find $	heta'$, we can use the formula for image height in a refracting telescope:

$h' = h_{obj} 	imes (-M)$

where $h_{obj}$ is the height of the object, which we can assume is negligible compared to the height of the real image. Therefore:

$h' = -M 	imes \frac{6.0 	imes 10^{-3}}{f_{obj}}$

We also know that the distance between the eyepiece and the objective is 143.5 cm. If we assume that the final image is formed at the focal point of the eyepiece, then the distance between the final image and the eyepiece is also equal to the focal length of the eyepiece. Therefore:

$	heta' = \frac{h'}{f_{ep}} = \frac{-M 	imes \frac{6.0 	imes 10^{-3}}{f_{obj}}}{f_{ep}}$

Substituting in the given values, we get:

$40 = \frac{	heta'}{	heta} \approx \frac{	heta'}{0.06} \Rightarrow 	heta' \approx 2.4	ext{ mrad}$

To convert to radians, we divide by 1000:

$	heta' \approx 0.0024	ext{ rad}$

This is not one of the answer choices, but we can see that it is closest to B) 0.17 rad. Therefore, the best choice is B).

Situation 19.3 the Angular Magnification of a Refracting Telescope Is 40.When the 40.When