In question 118, if `m_(1)` and `m_(2)` are the magnifications for two positions of the lens, then

Options

A. `f=(d)/(m_(1)+m_(2))`
B. `f=(2d)/(m_(1)+m_(2))`
C. `f=(3d)/(m_(1)-m_(2))`
D. `f=(d)/(m_(1)-m_(2))`

Explanation

To solve the problem, we need to establish the relationship between the focal length of a lens and the magnifications (m1 and m2) for two different positions of the lens. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have a lens that can be moved between two positions while still forming an image on the same screen. The distance from the object to the lens in the first position is \( d_1 \) and in the second position is \( d_2 \). The distance between the lens positions is \( d \), and the total distance from the object to the screen is \( D \).

### Step 2: Use the Lens Formula
The focal length \( f \) of the lens can be related to the distances using the lens formula:
\[ \frac{1}{f} = \frac{1}{d_1} + \frac{1}{d_2} \]

### Step 3: Define the Magnifications
The magnifications for the two positions of the lens are given by:
\[ m_1 = \frac{h_i}{h_o} = \frac{D + d}{D - d} \]
\[ m_2 = \frac{h_i}{h_o} = \frac{D - d}{D + d} \]

### Step 4: Find the Difference in Magnifications
To find the relationship between the magnifications, we calculate \( m_1 - m_2 \):
\[ m_1 - m_2 = \frac{D + d}{D - d} - \frac{D - d}{D + d} \]

### Step 5: Simplify the Expression
To simplify \( m_1 - m_2 \), we find a common denominator:
\[ m_1 - m_2 = \frac{(D + d)^2 - (D - d)^2}{(D - d)(D + d)} \]
Expanding the numerator:
\[ (D + d)^2 - (D - d)^2 = D^2 + 2Dd + d^2 - (D^2 - 2Dd + d^2) = 4Dd \]
Thus, we have:
\[ m_1 - m_2 = \frac{4Dd}{(D - d)(D + d)} \]

### Step 6: Relate to Focal Length
From the lens formula, we can also express the focal length as:
\[ f = \frac{D^2 - d^2}{4D} \]
Now, substituting \( D^2 - d^2 \) in terms of \( f \):
\[ D^2 - d^2 = 4fD \]
Thus, we can relate \( D \) and \( d \) to \( f \):
\[ m_1 - m_2 = \frac{4Dd}{4fD} = \frac{d}{f} \]

### Step 7: Final Relationship
Rearranging gives us:
\[ f = \frac{d}{m_1 - m_2} \]

### Conclusion
The relationship between the focal length \( f \) and the magnifications \( m_1 \) and \( m_2 \) is:
\[ f = \frac{d}{m_1 - m_2} \]