Four lenses of focal length `+15 cm , +20 cm, +150 cm` and `+250cm` are available for making an astronomical telescope. To produce the largest magnification, the focal length of the eye-piece should be

Options

A. 250 cm
B. 150 cm
C. 20 cm
D. 15 cm, 5 cm

Explanation

To find the focal length of the eyepiece that produces the largest magnification for an astronomical telescope, we can follow these steps:

### Step 1: Understand the Magnification Formula
The magnification \( M \) of an astronomical telescope is given by the formula:
\[ M = \frac{f_o}{f_e} \]
where \( f_o \) is the focal length of the objective lens and \( f_e \) is the focal length of the eyepiece.

### Step 2: Determine the Relationship for Maximum Magnification
From the formula, we can see that:
- The magnification \( M \) is directly proportional to the focal length of the objective lens \( f_o \).
- The magnification \( M \) is inversely proportional to the focal length of the eyepiece \( f_e \).

To achieve the largest magnification:
- We need the focal length of the objective lens \( f_o \) to be as large as possible.
- We need the focal length of the eyepiece \( f_e \) to be as small as possible.

### Step 3: Identify the Available Focal Lengths
The available focal lengths for the lenses are:
- \( +15 \, \text{cm} \)
- \( +20 \, \text{cm} \)
- \( +150 \, \text{cm} \)
- \( +250 \, \text{cm} \)

### Step 4: Choose the Focal Length of the Objective Lens
To maximize the magnification, we should select the largest focal length for the objective lens. Thus, we choose:
\[ f_o = 250 \, \text{cm} \]

### Step 5: Choose the Focal Length of the Eyepiece
Next, we need to select the smallest focal length for the eyepiece to maximize the magnification. Therefore, we choose:
\[ f_e = 15 \, \text{cm} \]

### Step 6: Conclusion
The focal length of the eyepiece that produces the largest magnification is:
\[ \text{Focal length of eyepiece} = 15 \, \text{cm} \]

### Final Answer
Thus, the focal length of the eyepiece should be \( 15 \, \text{cm} \).

---