Question 133: Ray Optics

Question

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:

Accepted Answer

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:
$$
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
$$
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.
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:
$$
\frac{\Delta f}{f^2} = \frac{\Delta v}{v^2} + \frac{\Delta u}{u^2}
$$
Rearranging this equation to find $$\Delta f$$, we get:
$$
\Delta f = f^2 \left( \frac{\Delta v}{v^2} + \frac{\Delta u}{u^2} \right)
$$
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:
Option C: $$f^2\left[\frac{\Delta \mathrm{u}}{\mathrm{u}^2}+\frac{\Delta \mathrm{v}}{\mathrm{v}^2}\right]$$

Answer

$$2 f\left[\frac{\Delta \mathrm{u}}{\mathrm{u}}+\frac{\Delta \mathrm{v}}{\mathrm{v}}\right]$$

Answer

$$f\left[\frac{\Delta \mathrm{u}}{\mathrm{u}}+\frac{\Delta \mathrm{v}}{\mathrm{v}}\right]$$

Answer

$$f^2\left[\frac{\Delta \mathrm{u}}{\mathrm{u}^2}+\frac{\Delta \mathrm{v}}{\mathrm{v}^2}\right]$$

Answer

$$\frac{\Delta \mathrm{u}}{\mathrm{u}}+\frac{\Delta \mathrm{v}}{\mathrm{v}}$$

Ray Optics

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:

Choose the correct answer: