Question 148: Ray Optics

Question

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:

Accepted Answer

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.
First, let's determine the object distance (u), image distance (v), and focal length (f) of the lens.
Object distance (u): It'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 $$u = 100 - 80 = 20~cm$$.
Image distance (v): It'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 $$v = 180 - 100 = 80~cm$$.
Focal length (f): Using the lens formula, we can calculate the focal length:
$$\frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{80} + \frac{1}{20} = \frac{5}{80}$$
So, $$f = \frac{80}{5} = 16~cm$$.
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.
Error in the focal length (df): We can use the formula for the error in the focal length:
$$\frac{df}{f^2} = \frac{0.2 	imes 2}{6400} + \frac{0.2 	imes 2}{400}$$
Solving for df:
$$df = \frac{16 	imes 16 	imes 0.2 	imes 6800 	imes 2}{6400 	imes 400} = 0.136 	imes 2$$
Percentage error in the focal length: Finally, we will calculate the percentage error using the formula:
$$\frac{df}{f} = \frac{0.0085 	imes 2}{1} = 1.70$$
So, the percentage error in the estimation of the focal length is 1.70%.

Answer

1.70

Answer

0.51

Answer

1.02

Answer

0.85

Ray Optics

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