Two convex lenses of focal length `10 cm` and `20 cm` respectively placed coaxially and are separated by some distance d. The whole system behaves like a concave lens. One of the possible value of d is

Options

A. `15 cm`
B. `20 cm`
C. `25 cm`
D. `40 cm`

Explanation

To solve the problem of two convex lenses behaving like a concave lens when placed coaxially and separated by a distance \( d \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Focal Lengths**:
- Let the focal length of the first lens \( f_1 = +10 \, \text{cm} \) (convex lens).
- Let the focal length of the second lens \( f_2 = +20 \, \text{cm} \) (convex lens).

2. **Understand the Equivalent Focal Length**:
- When two lenses are placed in combination, the equivalent focal length \( F \) can be calculated using the formula:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} \]
- Here, \( d \) is the distance between the two lenses.

3. **Condition for Concave Lens**:
- For the system to behave like a concave lens, the equivalent focal length \( F \) must be negative:
\[ F < 0 \]

4. **Set Up the Inequality**:
- From the formula, we can rearrange it to find the condition for \( d \):
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} < 0 \]
- This implies:
\[ \frac{d}{f_1 f_2} > \frac{1}{f_1} + \frac{1}{f_2} \]

5. **Calculate the Right Side**:
- Find \( \frac{1}{f_1} + \frac{1}{f_2} \):
\[ \frac{1}{f_1} = \frac{1}{10} = 0.1 \, \text{cm}^{-1} \]
\[ \frac{1}{f_2} = \frac{1}{20} = 0.05 \, \text{cm}^{-1} \]
\[ \frac{1}{f_1} + \frac{1}{f_2} = 0.1 + 0.05 = 0.15 \, \text{cm}^{-1} \]

6. **Calculate \( f_1 f_2 \)**:
- Multiply the focal lengths:
\[ f_1 f_2 = 10 \times 20 = 200 \, \text{cm}^2 \]

7. **Set Up the Final Inequality**:
- Substitute into the inequality:
\[ d > f_1 f_2 \cdot (0.15) \]
\[ d > 200 \cdot 0.15 = 30 \, \text{cm} \]

8. **Conclusion**:
- Therefore, one possible value of \( d \) must be greater than \( 30 \, \text{cm} \). If we check the options, \( d = 40 \, \text{cm} \) is a valid choice.

### Final Answer:
One of the possible values of \( d \) is \( 40 \, \text{cm} \).

---