Question:

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

Updated On: Mar 28, 2025
  • Δuu+Δvv \frac{\Delta u}{u} + \frac{\Delta v}{v}
  • f2[Δuu2+Δvv2] f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right]
  • 2f[Δuu+Δvv] 2f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right]
  • f[Δuu+Δvv] f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right]
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The Correct Option is B

Solution and Explanation

Lens Formula and Derivative for Error Analysis: 
The lens formula is given by: 
1f=1v1u \frac{1}{f} = \frac{1}{v} - \frac{1}{u}  

Taking the derivative of both sides with respect to uu and vv, we get: 
dff2=dvv2+duu2 -\frac{df}{f^2} = -\frac{dv}{v^2} + \frac{du}{u^2}  

Rearranging for dfdf
df=f2(dvv2+duu2) df = f^2 \left( \frac{dv}{v^2} + \frac{du}{u^2} \right)  

Error in Measurement of Focal Length: 
Since dvdv and dudu represent the measurement errors in vv and uu

respectively, we can substitute dv=Δvdv = \Delta v and du=Δudu = \Delta u
Δf=f2[Δvv2+Δuu2] \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 ff is: 
Δf=f2[Δuu2+Δvv2] \Delta f = f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right]

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