An object placed at 20 cm from a lens, forms an image on a screen placed 60 cm on the other side of the lens. What is the focal length and type of the lens ?

Options

A. 15 cm and convex lens
B. `-15` cm and concave lens
C. 18 cm and convex lens
D. `-18` cm and concave lens

Explanation

To solve the problem, we will use the lens formula, which is given by:

\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

where:
- \( f \) is the focal length of the lens,
- \( v \) is the image distance,
- \( u \) is the object distance.

### Step-by-Step Solution:

1. **Identify the Object and Image Distances**:
- The object distance \( u \) is given as 20 cm. According to the sign convention, since the object is placed on the same side as the incoming light, we take \( u = -20 \) cm.
- The image distance \( v \) is given as 60 cm. Since the image is formed on the opposite side of the lens, we take \( v = +60 \) cm.

2. **Substitute Values into the Lens Formula**:
- Using the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substitute \( v = 60 \) cm and \( u = -20 \) cm:
\[ \frac{1}{f} = \frac{1}{60} - \frac{1}{-20} \]

3. **Calculate Each Term**:
- Calculate \( \frac{1}{60} \):
\[ \frac{1}{60} = 0.01667 \]
- Calculate \( \frac{1}{-20} \):
\[ \frac{1}{-20} = -0.05 \]

4. **Combine the Terms**:
- Now, combine the two fractions:
\[ \frac{1}{f} = 0.01667 + 0.05 = 0.06667 \]

5. **Calculate the Focal Length**:
- To find \( f \), take the reciprocal:
\[ f = \frac{1}{0.06667} \approx 15 \text{ cm} \]

6. **Determine the Type of Lens**:
- Since the focal length \( f \) is positive, this indicates that the lens is a **convex lens**.

### Final Answer:
- The focal length of the lens is **15 cm** and the type of lens is **convex**.