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:
- \( \frac{\Delta u}{u} + \frac{\Delta v}{v} \)
- \( f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right] \)
- \( 2f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right] \)
- \( f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right] \)
The Correct Option is B
Solution and Explanation
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:
\[ \Delta f = f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right] \]
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