The focal lengths of the objective and the eyepiece of an astronomical telescope are 20 cm and 5 cm respectively. If the final image is formed at a distance of 30 cm from the eye piece, find the separation between the lenses for distinct vision:

Options

A. 32.4cm
B. 42.3cm
C. 24.3cm
D. 30.24cm

Answer

To solve the problem, we need to find the separation between the objective lens and the eyepiece of an astronomical telescope for distinct vision. We will use the formula that relates the focal lengths of the lenses and the distance for distinct vision.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Focal length of the objective lens, \( f_O = 20 \, \text{cm} \)
- Focal length of the eyepiece lens, \( f_E = 5 \, \text{cm} \)
- Distance of the final image from the eyepiece, \( V_E = 30 \, \text{cm} \)

2. **Understand the Concept of Distinct Vision:**
- The least distance of distinct vision (D) is typically taken as \( D = 25 \, \text{cm} \) for a normal human eye, but in this case, we are given that the final image is formed at \( V_E = 30 \, \text{cm} \).

3. **Use the Formula for Separation of Lenses:**
- The formula for the distance \( L \) between the objective and the eyepiece for distinct vision is given by:
\[ L = f_O + D \cdot \frac{f_E}{D + f_E} \]
- Here, \( D \) is the distance of the final image from the eyepiece, which is \( 30 \, \text{cm} \).

4. **Substitute the Values into the Formula:**
\[ L = 20 \, \text{cm} + 30 \, \text{cm} \cdot \frac{5 \, \text{cm}}{30 \, \text{cm} + 5 \, \text{cm}} \]

5. **Calculate the Denominator:**
\[ D + f_E = 30 \, \text{cm} + 5 \, \text{cm} = 35 \, \text{cm} \]

6. **Calculate the Fraction:**
\[ \frac{f_E}{D + f_E} = \frac{5 \, \text{cm}}{35 \, \text{cm}} = \frac{1}{7} \]

7. **Calculate the Product:**
\[ 30 \, \text{cm} \cdot \frac{1}{7} = \frac{30}{7} \approx 4.29 \, \text{cm} \]

8. **Add to the Focal Length of the Objective:**
\[ L = 20 \, \text{cm} + 4.29 \, \text{cm} \approx 24.29 \, \text{cm} \]

9. **Final Answer:**
The separation between the lenses for distinct vision is approximately \( 24.3 \, \text{cm} \).