Question

If \(m_1\) and \(m_2\) (\(m_1 \geq m_2\)) are the magnification for two positions of the lens between the object and the screen, and \(d\) is the distance between the two positions of the lens, the focal length of the lens is:

• A. \(\frac{m_1 - m_2}{d}\)
• B. \(\frac{m_1 m_2}{d}\)
• C. \((m_1 - m_2)d\)
• D. \(\frac{d}{m_1 - m_2}\)

Hint:

When analyzing lens movement and magnification, understanding the relationship between object and image distances simplifies the calculation of focal lengths.

Answer 1

The Correct Option is D

Step 1: Analyze the magnification formula. \[ m = -\frac{v}{u} \] where \(v\) is the image distance and \(u\) is the object distance. 
Step 2: Apply the lens formula for different positions. Using the lens formula, \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\), and the changes in magnification, \[ f = \frac{d}{m_1 - m_2} \]

Answer 2

To solve this problem, we need to find the formula for the focal length of a lens based on the magnifications at two different positions and the distance between these positions.

1. Understanding the Concept:
The magnifications at two positions of the lens, \(m_1\) and \(m_2\) (with \(m_1 > m_2\)), are related to the object, the image, and the focal length of the lens. The distance between these two positions is given as \(d\), and we are asked to find the formula for the focal length \(f\) of the lens.

2. Formula Derivation:
In optics, the lens formula can be written as:

\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] where \(f\) is the focal length, \(v\) is the image distance, and \(u\) is the object distance. The magnification of the lens is related to these distances by: \[ m = -\frac{v}{u} \] For two different positions, the magnifications at these positions give us a relationship between the image and object distances. By applying the magnification and the given distance between the two positions, the formula for the focal length becomes: \[ f = \frac{d}{m_1 - m_2} \] where \(d\) is the distance between the two positions of the lens, and \(m_1\) and \(m_2\) are the magnifications at the two positions.

3. Conclusion:
Thus, the formula for the focal length is:

\[ f = \frac{d}{m_1 - m_2} \]

Final Answer:
The correct answer is Option D: \( \frac{d}{m_1 - m_2} \).