Question

A convex lens $(f = 30 \, \text{cm})$ is in contact with a concave lens $(f = 20 \, \text{cm}).$ The object is placed on the left side at a distance of $20 \, \text{cm}.$ Find the image distance.
 

• A. 10 cm
• B. 20 cm
• C. 15 cm
• D. 25 cm

Hint:

When two lenses are in contact, the effective focal length is calculated by adding the reciprocals of the individual focal lengths. Use the lens formula to find the image distance.

Answer 1

The Correct Option is C

We are given a convex lens and a concave lens in contact, and we need to find the image distance. The object is placed 20 cm away from the lens system.
The focal length of the system is the effective focal length (\( f_{\text{eff}} \)) of the combination of the convex and concave lenses. The formula for the effective focal length of two thin lenses in contact is: \[ \frac{1}{f_{\text{eff}}} = \frac{1}{f_1} + \frac{1}{f_2} \] where \( f_1 = 30 \, \text{cm} \) (for the convex lens) and \( f_2 = -20 \, \text{cm} \) (for the concave lens).
Substituting the values: \[ \frac{1}{f_{\text{eff}}} = \frac{1}{30} + \frac{1}{-20} = \frac{1}{30} - \frac{1}{20} = \frac{2 - 3}{60} = -\frac{1}{60}. \] Thus, the effective focal length is: \[ f_{\text{eff}} = -60 \, \text{cm}. \] Now, using the lens formula \( \frac{1}{f_{\text{eff}}} = \frac{1}{v} - \frac{1}{u} \), where \( u = -20 \, \text{cm} \) (object distance), we need to find the image distance \( v \).
Substituting the values into the lens formula: \[ \frac{1}{-60} = \frac{1}{v} - \frac{1}{-20}, \] \[ \frac{1}{-60} = \frac{1}{v} + \frac{1}{20}. \] Simplifying: \[ \frac{1}{v} = \frac{1}{-60} - \frac{1}{20} = \frac{-1 - 3}{60} = \frac{-4}{60} = \frac{-1}{15}. \] Thus, the image distance is: \[ v = -15 \, \text{cm}. \] Therefore, the image is formed at a distance of 15 cm on the opposite side of the object. Thus, the correct answer is (3) 15 cm.