A astronomical telescope has objective and eyepiece of focal lenghts 40 cm 4 cm respectively. To view an object 200 cm away from the objectiv, the lenses must be speparted by a distance

Options

A. 46.0 cm
B. 50.0 cm
C. 54.0 cm
D. 37.3 cm

Answer

To solve the problem of finding the separation between the objective and eyepiece lenses of an astronomical telescope, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Focal length of the objective lens, \( F_o = 40 \, \text{cm} \)
- Focal length of the eyepiece lens, \( F_e = 4 \, \text{cm} \)
- Object distance from the objective lens, \( U = -200 \, \text{cm} \) (the negative sign indicates that the object is on the same side as the incoming light).

2. **Use the Lens Formula**:
The lens formula is given by:
\[ \frac{1}{V} - \frac{1}{U} = \frac{1}{F} \]
where \( V \) is the image distance from the lens, \( U \) is the object distance, and \( F \) is the focal length of the lens.

3. **Calculate the Image Distance for the Objective Lens**:
Plugging in the values for the objective lens:
\[ \frac{1}{V_o} - \frac{1}{(-200)} = \frac{1}{40} \]
Rearranging gives:
\[ \frac{1}{V_o} = \frac{1}{40} - \frac{1}{200} \]
Finding a common denominator (200):
\[ \frac{1}{V_o} = \frac{5}{200} - \frac{1}{200} = \frac{4}{200} \]
Therefore,
\[ V_o = \frac{200}{4} = 50 \, \text{cm} \]

4. **Determine the Position of the Image**:
The image formed by the objective lens is 50 cm from the objective lens.

5. **Position of the Eyepiece**:
For normal adjustment of the telescope, the image formed by the objective should be at the focal point of the eyepiece. Therefore, the distance from the image to the eyepiece lens should be equal to the focal length of the eyepiece:
\[ d_{ie} = F_e = 4 \, \text{cm} \]

6. **Calculate the Separation Between the Lenses**:
The total separation \( S \) between the two lenses is the distance from the objective lens to the image plus the distance from the image to the eyepiece:
\[ S = V_o + d_{ie} = 50 \, \text{cm} + 4 \, \text{cm} = 54 \, \text{cm} \]

### Final Answer:
The separation between the objective and eyepiece lenses should be **54 cm**.

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