Question

In an experiment to measure focal length (\(f\)) of a convex lens, the least counts of the measuring scales for the position of the object (\(u\)) and for the position of the image (\(v\)) are \( \Delta u \) and \( \Delta v \), respectively. The error in the measurement of the focal length of the convex lens will be:

• A. \( \frac{\Delta u}{u} + \frac{\Delta v}{v} \)
• B. \( f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right] \)
• C. \( 2f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right] \)
• D. \( f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right] \)

Answer 1

The Correct Option is B

Lens Formula and Derivative for Error Analysis:
The lens formula is defined as: 
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] 

Deriving with respect to \(u\) and \(v\): 
The derivative of both sides with respect to \(u\) and \(v\) yields: 
\[ -\frac{df}{f^2} = -\frac{dv}{v^2} + \frac{du}{u^2} \] 

Expression for \(df\): 
Rearranging the equation to solve for \(df\): 
\[ df = f^2 \left( \frac{dv}{v^2} + \frac{du}{u^2} \right) \] 

Error in Focal Length Measurement: 
Given that \(dv\) and \(du\) represent the measurement errors in \(v\) and \(u\), 

respectively, substituting \(dv = \Delta v\) and \(du = \Delta u\) provides: 
\[ \Delta f = f^2 \left[ \frac{\Delta v}{v^2} + \frac{\Delta u}{u^2} \right] \] 

Summary: 
The error associated with the focal length measurement \(f\) is: 
\[ \Delta f = f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right] \]