An astronomical telescope of ten-fold angular magnification has a length of `44 cm`. The focal length of the objective is

Options

A. 44 cm
B. 440 cm
C. 4 cm
D. 40 cm

Answer

To find the focal length of the objective lens (F₀) of an astronomical telescope with a given angular magnification and length, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Angular magnification (M) = 10 (ten-fold)
- Length of the telescope (L) = 44 cm

2. **Write the Equations:**
- The length of the telescope is given by the sum of the focal lengths of the objective and the eyepiece:
\[ L = F₀ + Fₑ \quad \text{(Equation 1)} \]
- The angular magnification of the telescope is given by:
\[ M = \frac{F₀}{Fₑ} \quad \text{(Equation 2)} \]

3. **Substitute the Magnification Value:**
- From Equation 2, since M = 10, we can write:
\[ 10 = \frac{F₀}{Fₑ} \]
- Rearranging gives:
\[ F₀ = 10 Fₑ \quad \text{(Equation 3)} \]

4. **Substitute Equation 3 into Equation 1:**
- Substitute \( F₀ \) from Equation 3 into Equation 1:
\[ L = 10 Fₑ + Fₑ \]
- This simplifies to:
\[ L = 11 Fₑ \]

5. **Solve for Fₑ:**
- Now substitute the value of L:
\[ 44 = 11 Fₑ \]
- Dividing both sides by 11 gives:
\[ Fₑ = \frac{44}{11} = 4 \text{ cm} \]

6. **Find F₀ using Fₑ:**
- Now substitute \( Fₑ \) back into Equation 3 to find \( F₀ \):
\[ F₀ = 10 Fₑ = 10 \times 4 = 40 \text{ cm} \]

### Final Answer:
The focal length of the objective lens (F₀) is **40 cm**.

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