Question

A convex lens (f = 30 cm) is in contact with a concave lens (f = -20 cm). An object is placed on the left side at a distance of 20 cm. Find the image distance.

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

Hint:

For systems of lenses in contact, calculate the effective focal length using the formula \( \frac{1}{f_{\text{eff}}} = \frac{1}{f_1} + \frac{1}{f_2} \), and then 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, so we can consider the effective focal length of the system using the formula for lenses in contact: \[ \frac{1}{f_{\text{eff}}} = \frac{1}{f_1} + \frac{1}{f_2} \] where \( f_1 = 30 \, \text{cm} \) (for the convex lens) and \( f_2 = -20 \, \text{cm} \) (for the concave lens).
Substituting the values: \[ \frac{1}{f_{\text{eff}}} = \frac{1}{30} + \frac{1}{-20} = \frac{1}{30} - \frac{1}{20} \] Finding a common denominator: \[ \frac{1}{f_{\text{eff}}} = \frac{2}{60} - \frac{3}{60} = -\frac{1}{60} \] Thus, the effective focal length is: \[ f_{\text{eff}} = -60 \, \text{cm} \] Now, to find the image distance, we use the lens formula: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] where \( f = -60 \, \text{cm} \) (effective focal length) and \( u = -20 \, \text{cm} \) (object distance). Substituting the known values: \[ \frac{1}{-60} = \frac{1}{v} - \frac{1}{-20} \] Simplifying: \[ \frac{1}{-60} = \frac{1}{v} + \frac{1}{20} \] \[ \frac{1}{v} = \frac{1}{-60} - \frac{1}{20} = -\frac{1}{60} - \frac{3}{60} = -\frac{4}{60} \] Thus, the image distance is: \[ v = -\frac{60}{4} = -15 \, \text{cm} \] So, the image is formed at a distance of 15 cm on the same side as the object.