Question

A convex lens of focal length 30 cm is placed in contact with a concave lens of focal length 20 cm. An object is placed at 20 cm to the left of this lens system. The distance of the image from the lens in cm is ____ .

• A. \( \frac{60}{7} \) cm
• B. 30 cm
• C. 15 cm
• D. 45 cm

Hint:

When dealing with a system of lenses in contact, first find the combined focal length using the formula \( \frac{1}{F_{\text{total}}} = \frac{1}{F_1} + \frac{1}{F_2} \), and then apply the lens formula to find the image distance.

Answer 1

The Correct Option is C

When two lenses are in contact, their combined focal length \( F_{\text{total}} \) is given by: \[ \frac{1}{F_{\text{total}}} = \frac{1}{F_1} + \frac{1}{F_2} \] Where: - \( F_1 \) is the focal length of the convex lens, which is \( +30 \, \text{cm} \), - \( F_2 \) is the focal length of the concave lens, which is \( -20 \, \text{cm} \) (since the concave lens has a negative focal length). Substituting the values: \[ \frac{1}{F_{\text{total}}} = \frac{1}{30} + \frac{1}{-20} \] \[ \frac{1}{F_{\text{total}}} = \frac{1}{30} - \frac{1}{20} = \frac{2}{60} - \frac{3}{60} = \frac{-1}{60} \] Thus: \[ F_{\text{total}} = -60 \, \text{cm} \] Now, using the lens formula for the combined lens system: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] Where: - \( f = F_{\text{total}} = -60 \, \text{cm} \) (combined focal length), - \( u = -20 \, \text{cm} \) (object distance, taken as negative for an object placed to the left of the lens), - \( v \) is the image distance, which we need to calculate. Substitute the known values: \[ \frac{1}{-60} = \frac{1}{v} - \frac{1}{-20} \] \[ \frac{1}{-60} = \frac{1}{v} + \frac{1}{20} \] Simplifying: \[ \frac{1}{v} = \frac{1}{-60} - \frac{1}{20} = \frac{-1}{60} - \frac{3}{60} = \frac{-4}{60} \] \[ v = \frac{60}{4} = 15 \, \text{cm} \] Thus, the distance of the image from the lens is 15 cm. Therefore, the correct answer is Option (3).