A convex lens of focal length `10cm` and concave lens of focal length `20 cm` are kept `5 cm` apart. The focal length of the equivalent lens is

Options

A. `(120)/(3)cm`
B. `18cm`
C. `30 cm`
D. `(40)/(3)`

Explanation

To find the focal length of the equivalent lens formed by a convex lens and a concave lens kept apart, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Focal Lengths**:
- The focal length of the convex lens (f1) is +10 cm (positive because it is a convex lens).
- The focal length of the concave lens (f2) is -20 cm (negative because it is a concave lens).

2. **Identify the Distance Between the Lenses**:
- The distance (d) between the two lenses is given as 5 cm.

3. **Use the Lens Combination Formula**:
- Since there is a distance between the lenses, we will use the formula:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 \cdot f_2} \]

4. **Substitute the Values into the Formula**:
- Substitute \( f_1 = 10 \, \text{cm} \), \( f_2 = -20 \, \text{cm} \), and \( d = 5 \, \text{cm} \) into the formula:
\[ \frac{1}{f} = \frac{1}{10} + \frac{1}{-20} - \frac{5}{10 \cdot (-20)} \]

5. **Calculate Each Term**:
- Calculate \( \frac{1}{10} = 0.1 \)
- Calculate \( \frac{1}{-20} = -0.05 \)
- Calculate \( 10 \cdot (-20) = -200 \) so \( \frac{5}{-200} = -0.025 \)

6. **Combine the Terms**:
- Now substitute these values back into the equation:
\[ \frac{1}{f} = 0.1 - 0.05 - (-0.025) \]
- This simplifies to:
\[ \frac{1}{f} = 0.1 - 0.05 + 0.025 = 0.075 \]

7. **Find the Equivalent Focal Length**:
- To find \( f \), take the reciprocal of \( 0.075 \):
\[ f = \frac{1}{0.075} = \frac{100}{7} \approx 14.29 \, \text{cm} \]

### Final Answer:
The focal length of the equivalent lens is approximately \( 14.29 \, \text{cm} \).

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