A gaint refrecting telescope at an observatory has an objective lens of focal length `15 m`. If an eye piece lens of focal length `1 cm` is used, find the angular magnification of the telescope.
If this telescope is used to view the moon, what is the diameter of image of moon formed by objective lens ? The diameter of the moon is `3.42 xx 10^6 m` and radius of lunar orbit is `3.8 xx 10^8 m`.

Options

A. `1000`
B. 1500
C. 2000
D. 3000

Explanation

To solve the problem, we need to find two things: the angular magnification of the telescope and the diameter of the image of the moon formed by the objective lens. Let's break this down step by step.

### Step 1: Calculate Angular Magnification

The formula for angular magnification (M) of a telescope is given by:

\[ M = \frac{F_0}{F_E} \]

where:
- \( F_0 \) = focal length of the objective lens
- \( F_E \) = focal length of the eyepiece lens

Given:
- \( F_0 = 15 \, m \)
- \( F_E = 1 \, cm = 0.01 \, m \)

Substituting the values into the formula:

\[ M = \frac{15 \, m}{0.01 \, m} = 1500 \]

### Step 2: Calculate the Diameter of the Image of the Moon

To find the diameter of the image of the moon formed by the objective lens, we can use the relationship between the actual diameter of the moon (d), the radius of the lunar orbit (r_0), and the focal length of the objective lens (F_0).

The relationship can be expressed as:

\[ \frac{d}{r_0} = \frac{d'}{F_0} \]

where:
- \( d' \) = diameter of the image of the moon
- \( d = 3.42 \times 10^6 \, m \) (actual diameter of the moon)
- \( r_0 = 3.8 \times 10^8 \, m \) (radius of lunar orbit)
- \( F_0 = 15 \, m \)

Rearranging the equation to solve for \( d' \):

\[ d' = \frac{d \cdot F_0}{r_0} \]

Substituting the known values:

\[ d' = \frac{(3.42 \times 10^6 \, m) \cdot (15 \, m)}{3.8 \times 10^8 \, m} \]

Calculating \( d' \):

\[ d' = \frac{51.3 \times 10^6 \, m^2}{3.8 \times 10^8 \, m} = 0.135 \, m = 13.5 \, cm \]

### Final Answers

1. The angular magnification of the telescope is **1500**.
2. The diameter of the image of the moon formed by the objective lens is approximately **13.5 cm**.

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