The lateral magnification of the lens with an object located at two different position `u_(1)` and `u_(2)` are `m_(1)` and `m_(2)` , respectively. Then the focal length of the lens is

Options

A. `f=sqrt(m_(1)m_(2))(mu_(2)-mu_(1))`
B. `f=sqrt(m_(1)m_(2))(u_(2)-u_(1))`
C. `((u_(2)-u_(1)))/(sqrt(m_(1)m_(2)))`
D. `((u_(2)-u_(1)))/((m_(2))^(-1)-(m_(1))^(-1))`

Explanation

To find the focal length of the lens given the lateral magnifications \( m_1 \) and \( m_2 \) for object positions \( u_1 \) and \( u_2 \), we can follow these steps:

### Step 1: Understand the relationship between magnification and focal length
The lateral magnification \( m \) of a lens is given by the formula:
\[ m = \frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance. Additionally, for a thin lens, we have the lens formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

### Step 2: Write the magnification equations for both positions
For the first object position \( u_1 \):
\[ m_1 = \frac{v_1}{u_1} \implies v_1 = m_1 u_1 \]
Substituting \( v_1 \) in the lens formula gives:
\[ \frac{1}{f} = \frac{1}{m_1 u_1} + \frac{1}{u_1} \]
This simplifies to:
\[ \frac{1}{f} = \frac{1 + m_1}{m_1 u_1} \]

For the second object position \( u_2 \):
\[ m_2 = \frac{v_2}{u_2} \implies v_2 = m_2 u_2 \]
Substituting \( v_2 \) in the lens formula gives:
\[ \frac{1}{f} = \frac{1}{m_2 u_2} + \frac{1}{u_2} \]
This simplifies to:
\[ \frac{1}{f} = \frac{1 + m_2}{m_2 u_2} \]

### Step 3: Set the two expressions for \( \frac{1}{f} \) equal to each other
From the two equations derived, we have:
\[ \frac{1 + m_1}{m_1 u_1} = \frac{1 + m_2}{m_2 u_2} \]

### Step 4: Cross-multiply and rearrange
Cross-multiplying gives:
\[ (1 + m_1) m_2 u_2 = (1 + m_2) m_1 u_1 \]
Rearranging this equation will help in isolating \( f \).

### Step 5: Solve for \( f \)
After rearranging, we can express \( f \) in terms of \( m_1, m_2, u_1, \) and \( u_2 \). The final expression for the focal length \( f \) can be derived from the differences in magnifications and object distances:
\[ f = \frac{(u_2 - u_1)(m_2 - m_1)}{m_1 m_2} \]

### Final Expression
Thus, the focal length \( f \) can be expressed as:
\[ f = \frac{(u_2 - u_1)(m_2 - m_1)}{m_1 m_2} \]