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 given by: 
$$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$ 

Taking the derivative of both sides with respect to $u$ and $v$, we get: 
$$-\frac{df}{f^2} = -\frac{dv}{v^2} + \frac{du}{u^2}$$ 

Rearranging for $df$: 
$$df = f^2 \left( \frac{dv}{v^2} + \frac{du}{u^2} \right)$$ 

Error in Measurement of Focal Length: 
Since $dv$ and $du$ represent the measurement errors in $v$ and $u$, 

respectively, we can substitute $dv = \Delta v$ and $du = \Delta u$: 
$$\Delta f = f^2 \left[ \frac{\Delta v}{v^2} + \frac{\Delta u}{u^2} \right]$$ 

Conclusion: 
The error in the measurement of the focal length $f$ is: 
$$\Delta f = f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right]$$