Question

A point object is placed at a distance of 15 cm from a convex lens. The image is formed on the other side of the lens at a distance of 30 cm from the lens. When a concave lens is placed in contact with the convex lens, the image shifts away further by 30 cm. Calculate the focal lengths of the two lenses.

Hint:

When two lenses are placed in contact, the total focal length of the system is the reciprocal sum of the focal lengths of the individual lenses.

Answer 1

Step 1: Use the lens formula for convex lens.
The lens formula 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 (30 cm),
- \( u \) is the object distance (-15 cm, since the object is on the opposite side of the light sourc. For the convex lens: \[ \frac{1}{f_{\text{convex}}} = \frac{1}{30} - \frac{1}{(-15)} \] \[ \frac{1}{f_{\text{convex}}} = \frac{1}{30} + \frac{1}{15} = \frac{1}{10} \] So, \( f_{\text{convex}} = 10 \, \text{cm} \).
Step 2: Use the lens formula for concave lens.
When a concave lens is placed in contact with the convex lens, the total focal length of the system changes. The distance of the image shifts further by 30 cm. So the new image distance is 30 cm + 30 cm = 60 cm. The total effective focal length \( f_{\text{total}} \) of two lenses in contact is given by: \[ \frac{1}{f_{\text{total}}} = \frac{1}{f_{\text{convex}}} + \frac{1}{f_{\text{concave}}} \] Let the focal length of the concave lens be \( f_{\text{concave}} \).
For the new image distance: \[ \frac{1}{f_{\text{total}}} = \frac{1}{60} \quad \text{and} \quad \frac{1}{f_{\text{convex}}} = \frac{1}{10} \] \[ \frac{1}{60} = \frac{1}{10} + \frac{1}{f_{\text{concave}}} \] \[ \frac{1}{f_{\text{concave}}} = \frac{1}{60} - \frac{1}{10} = \frac{1}{60} - \frac{6}{60} = -\frac{5}{60} \] Thus, \( f_{\text{concave}} = -12 \, \text{cm} \).
Final Answer:
The focal length of the convex lens is \( \boxed{10 \, \text{cm}} \), and the focal length of the concave lens is \( \boxed{-12 \, \text{cm}} \).