What is the focal length of a convex lens of focal length `30 cm` in contact with a concave lens of focal length `20 cm`. Is the system a converging or a diverging lens ? Ignore thickness of the lenses.

Options

A. 30 cm
B. 60 cm
C. -30 cm
D. -60 cm

Explanation

To find the focal length of a system consisting of a convex lens and a concave lens in contact, we can use the formula for the combined focal length of lenses in contact.

### Step-by-Step Solution:

1. **Identify the Focal Lengths**:
- The focal length of the convex lens (F1) is given as +30 cm (positive because it is a converging lens).
- The focal length of the concave lens (F2) is given as -20 cm (negative because it is a diverging lens).

2. **Use the Formula for Lenses in Contact**:
The formula for the focal length (F) of two lenses in contact is:
\[ \frac{1}{F} = \frac{1}{F_1} + \frac{1}{F_2} \]
Substituting the values:
\[ \frac{1}{F} = \frac{1}{30} + \frac{1}{-20} \]

3. **Calculate the Right-Hand Side**:
- Find a common denominator for the fractions. The least common multiple (LCM) of 30 and 20 is 60.
- Rewrite the fractions:
\[ \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{-20} = \frac{-3}{60} \]
- Now, substitute these values into the equation:
\[ \frac{1}{F} = \frac{2}{60} - \frac{3}{60} = \frac{-1}{60} \]

4. **Find the Resultant Focal Length**:
- Taking the reciprocal gives:
\[ F = -60 \text{ cm} \]

5. **Determine the Nature of the Lens System**:
- Since the focal length is negative, this indicates that the system behaves as a diverging lens.

### Final Answer:
The focal length of the lens system is -60 cm, and the system is a diverging lens.