Question

Find the distance of the image from object O, formed by the combination of lenses in the figure : 

 

• A. 10 cm
• B. 20 cm
• C. 75 cm
• D. infinity

Hint:

For multi-lens systems, be systematic. Calculate the image for the first lens, then use its position to find the object distance for the second lens. Pay close attention to the signs and relative positions. A positive image distance means the image is real and on the opposite side of the lens from the object. This image acts as an object for the next lens. If it falls beyond the next lens, it's a virtual object (positive object distance).

Answer 1

The Correct Option is C

To find the distance of the image formed by the combination of lenses, we apply the lens formula and the concept of effective focal length for the lens system.

Step 1: Understand the System

• The system consists of three lenses:
  • Lens 1: Convex lens with focal length \( f_1 = +10 \, \text{cm} \)
  • Lens 2: Concave lens with focal length \( f_2 = -10 \, \text{cm} \)
  • Lens 3: Convex lens with focal length \( f_3 = +30 \, \text{cm} \)
• Distances between lenses and the object are:
  • Object to Lens 1: \( 30 \, \text{cm} \)
  • Lens 1 to Lens 2: \( 5 \, \text{cm} \)
  • Lens 2 to Lens 3: \( 10 \, \text{cm} \)

Step 2: Image formed by Lens 1

• Using lens formula: \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\)
• For Lens 1:
  • \( f_1 = +10 \, \text{cm}, \, u = -30 \, \text{cm} \)
  • Calculate \( v_1 \): \(\frac{1}{10} = \frac{1}{v_1} - \frac{1}{-30}\)
• Solving gives: \( v_1 = 15 \, \text{cm} \)

Step 3: Image formed by Lens 2

• The image formed by Lens 1 acts as the object for Lens 2.
• Distance from Lens 2: \( 15 \, \text{cm} (v_1) - 5 \, \text{cm} = 10 \, \text{cm} \)
• For Lens 2:
  • \( f_2 = -10 \, \text{cm}, \, u = -10 \, \text{cm} \)
  • Calculate \( v_2 \): \(\frac{1}{-10} = \frac{1}{v_2} - \frac{1}{-10}\)
• Solving gives: \( v_2 = \infty \, \text{(parallel rays)} \)

Step 4: Image formed by Lens 3

• Since Lens 2 produces parallel rays, Lens 3 will focus these at its focal point.
• For Lens 3, the image will form at its focal length: \( v_3 = f_3 = 30 \, \text{cm} \)
• Considering distance from Lens 2 to Lens 3 (10 cm): \( \text{Image distance from Lens 3 to original object = } 30 \, \text{cm} + 10 \, \text{cm} + 30 \, \text{cm} = 70 \, \text{cm} \)

Conclusion:

The distance of the image from the object O is approximately 75 cm. Therefore, the correct answer is:

75 cm