Question

An object placed in front of a convex lens, forms an image of same size on a screen. Moving the object 12 cm closer to the lens results in the formation of a real image which is three times the size of the object. Calculate the focal length of the lens.

Hint:

The condition "image of same size" for a convex lens is a strong hint that the object is at 2f. This is often the starting point for solving such problems.

Answer 1

Step 1: Understanding the Concept:
A convex lens forms a real, inverted image of the same size when the object is at \( 2f \). Magnification \( m = v/u \).
Step 2: Key Formula or Approach:
Magnification for lens: \( m = \frac{f}{f+u} \) (Using sign convention)
Step 3: Detailed Explanation:
Case 1: Image same size as object.
This happens when object distance \( u_{1} = -2f \). Magnification \( m_{1} = -1 \).
Case 2: Object is moved 12 cm closer.
New object distance \( u_{2} = -(2f - 12) = -2f + 12 \).
New magnification \( m_{2} = -3 \) (Real image is inverted).
Using the formula \( m = \frac{f}{f+u} \):
\[ -3 = \frac{f}{f + (-2f + 12)} \]
\[ -3 = \frac{f}{12 - f} \]
\[ -3(12 - f) = f \]
\[ -36 + 3f = f \]
\[ 2f = 36 \implies f = 18 \text{ cm} \]
Step 4: Final Answer:
The focal length of the convex lens is 18 cm.