In the displacement method a conves lens is placed in between an object and a screen. If the magnificaiton in the two position are `m_(1) and `m_(2)` (`m_(1)` gt m_(2)` ), and the distance between the two positions of the lens is x, the focal length of the lens is

Options

A. `x/(m_(1) + m_(2))`
B. `x/(m_(1)-m_(2))`
C. `x/(m_(1)-m_(2))^(2)`
D. `x/(m_(1)+m_(2))^(2)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Magnification Formula
The magnification (m) produced by a lens is given by the formula:
\[ m = \frac{V}{U} \]
where \( V \) is the image distance and \( U \) is the object distance.

### Step 2: Write the Magnification for Two Positions
For the two positions of the lens, we have:
- For the first position:
\[ m_1 = \frac{V_1}{U_1} \]
- For the second position:
\[ m_2 = \frac{U_2}{V_2} \]

### Step 3: Relate the Distances
Since the distance between the two positions of the lens is \( x \), we can write:
\[ V_1 - U_1 = x \]
This implies that:
\[ V_1 = U_1 + x \]

### Step 4: Express Magnifications in Terms of Distances
From the magnifications:
- \( m_1 = \frac{V_1}{U_1} \)
- \( m_2 = \frac{U_2}{V_2} \)

### Step 5: Use the Displacement Method
Using the displacement method, we can express the difference in magnifications:
\[ m_1 - m_2 = \frac{V_1}{U_1} - \frac{U_2}{V_2} \]

### Step 6: Substitute Values
Substituting the expressions for \( V_1 \) and \( U_2 \):
\[ m_1 - m_2 = \frac{(U_1 + x)}{U_1} - \frac{(U_1 + x)}{(U_1 + x)} \]

### Step 7: Simplify the Equation
This simplifies to:
\[ m_1 - m_2 = \frac{(U_1 + x) - U_2}{U_1} \]

### Step 8: Use the Lens Formula
Using the lens formula:
\[ \frac{1}{f} = \frac{1}{V} - \frac{1}{U} \]
For the first position, we have:
\[ \frac{1}{f} = \frac{1}{V_1} + \frac{1}{U_1} \]

### Step 9: Relate the Focal Length to the Magnifications
From the previous steps, we can relate the focal length \( f \) to the magnifications:
\[ \frac{1}{f} = \frac{m_1 - m_2}{x} \]

### Step 10: Solve for the Focal Length
Rearranging gives us:
\[ f = \frac{x}{m_1 - m_2} \]

### Final Result
The focal length of the lens is:
\[ f = \frac{x}{m_1 - m_2} \]

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