A 2 meter long scale with least count of $0.2 \mathrm{~cm}$ is used to measure the locations of objects on an optical bench. While measuring the focal length of a convex lens, the object pin and the convex lens are placed at $80 \mathrm{~cm}$ mark and $1 \mathrm{~m}$ mark, respectively. The image of the object pin on the other side of lens coincides with image pin that is kept at $180 \mathrm{~cm}$ mark. The $\%$ error in the estimation of focal length is:

<p>In this problem, you are asked to find the percentage error in the estimation of the focal length of a convex lens using a 2-meter long scale with a least count of 0.2 cm.</p> <p>First, let&#39;s determine the object distance (u), image distance (v), and focal length (f) of the lens.</p> <ol> <li><p>Object distance (u): It&#39;s the distance between the object pin and the convex lens. The object pin is at the 80 cm mark, and the convex lens is at the 1 m (100 cm) mark, so the object distance is $u = 100 - 80 = 20~cm$.</p> </li> <li><p>Image distance (v): It&#39;s the distance between the image pin and the convex lens. The image pin is at the 180 cm mark, and the convex lens is at the 1 m (100 cm) mark, so the image distance is $v = 180 - 100 = 80~cm$.</p> </li> <li><p>Focal length (f): Using the lens formula, we can calculate the focal length:<br/><br/> $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{80} + \frac{1}{20} = \frac{5}{80}$$ So, $f = \frac{80}{5} = 16~cm$.</p> </li> </ol> <p>Now, we will calculate the error in the focal length (df) using the given least count (0.2 cm). The error in the object distance and image distance will both be 0.2 cm.</p> <ol> <li>Error in the focal length (df): We can use the formula for the error in the focal length:<br/><br/> $\frac{df}{f^2} = \frac{0.2 \times 2}{6400} + \frac{0.2 \times 2}{400}$</li> </ol> <p>Solving for df:<br/><br/> $$df = \frac{16 \times 16 \times 0.2 \times 6800 \times 2}{6400 \times 400} = 0.136 \times 2$$</p> <ol> <li>Percentage error in the focal length: Finally, we will calculate the percentage error using the formula:<br/><br/> $\frac{df}{f} = \frac{0.0085 \times 2}{1} = 1.70$</li> </ol> <p>So, the percentage error in the estimation of the focal length is 1.70%.</p>