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

<p>First, let's understand the relationship between the object distance ($u$), the image distance ($v$), and the focal length ($f$) of a convex lens, which is given by the lens formula:</p> $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ <p>Now, to find the error in the focal length ($\Delta f$) due to the errors in the measurements of $u$ and $v$ ($\Delta u$ and $\Delta v$, respectively), we have to differentiate the lens formula with respect to $u$ and $v$, keeping in mind the propagation of error.</p> <p>By differentiating both sides of the lens formula with respect to $v$ and $u$, and also considering the negative reciprocal relation (given $f$ is a constant for a specific lens), we have:</p> $\frac{\Delta f}{f^2} = \frac{\Delta v}{v^2} + \frac{\Delta u}{u^2}$ <p>Rearranging this equation to find $\Delta f$, we get:</p> $\Delta f = f^2 \left( \frac{\Delta v}{v^2} + \frac{\Delta u}{u^2} \right)$ <p>Therefore, the correct option showing the error in the measurement of the focal length of the convex lens, taking into account the least counts ($\Delta u$ and $\Delta v$) of the measuring scales for the position of the object ($u$) and for the position of the image ($v$), is:</p> <p>Option C: $$f^2\left[\frac{\Delta \mathrm{u}}{\mathrm{u}^2}+\frac{\Delta \mathrm{v}}{\mathrm{v}^2}\right]$$</p>