A convex lens of focal length 30 cm and a concave lens of 10 cm focal length are placed so as to have the same axis. If a parallel beam of light falling on convex lens leaves concave lens as a parallel beam, then the distance between two lenses will be

Options

A. 40 cm
B. 30 cm
C. 20 cm
D. 10 cm

Explanation

To solve the problem, we need to find the distance \( x \) between the convex lens and the concave lens such that a parallel beam of light entering the convex lens exits the concave lens as a parallel beam.

### Step-by-Step Solution:

1. **Identify the Focal Lengths**:
- Focal length of the convex lens, \( f_1 = +30 \) cm (positive for convex lens).
- Focal length of the concave lens, \( f_2 = -10 \) cm (negative for concave lens).

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

3. **Consider the Convex Lens**:
For the convex lens, since the incoming light is parallel, the object distance \( u \) is at infinity. Therefore, the image distance \( v_1 \) can be calculated as follows:
\[ \frac{1}{f_1} = \frac{1}{v_1} - \frac{1}{\infty} \]
This simplifies to:
\[ \frac{1}{30} = \frac{1}{v_1} \]
Thus, \( v_1 = 30 \) cm. This means the image formed by the convex lens is 30 cm away from it.

4. **Distance Between the Lenses**:
Let \( x \) be the distance between the two lenses. The image formed by the convex lens acts as a virtual object for the concave lens. The object distance for the concave lens will be:
\[ u_2 = -(x - 30) \]
(The negative sign indicates that the object is on the same side as the incoming light).

5. **Apply the Lens Formula for the Concave Lens**:
For the concave lens, we have:
\[ \frac{1}{f_2} = \frac{1}{v_2} - \frac{1}{u_2} \]
Since we want the light to exit the concave lens as a parallel beam, the image distance \( v_2 \) is at infinity:
\[ \frac{1}{-10} = 0 - \frac{1}{-(x - 30)} \]
Simplifying gives:
\[ -\frac{1}{10} = \frac{1}{x - 30} \]
Cross-multiplying yields:
\[ -(x - 30) = 10 \]
Thus, we have:
\[ x - 30 = -10 \implies x = 20 \text{ cm} \]

6. **Conclusion**:
The distance between the two lenses is \( x = 20 \) cm.

### Final Answer:
The distance between the two lenses is **20 cm**.