AIIMS 2003 Physics Mirror Formula MCQ Question

To analyze the relationship between the object distance $u$ and the image distance $v$ for a concave mirror, we use the mirror formula:

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

where $f$ is the focal length of the mirror, $u$ is the object distance (measured from the mirror along the principal axis), and $v$ is the image distance (also measured from the mirror along the principal axis). For a concave mirror, the focal length $f$ is considered negative according to the sign convention used in optics.

Understanding the Graph of Magnitudes of $u$ and $v$

When we plot a graph of the magnitudes of $u$ against $v$, we need to consider the signs of these distances. The mirror formula can be rearranged to express $v$ in terms of $u$:

$$v = \frac{fu}{u - f}$$

In this equation, as the object distance $u$ varies, we can observe the following:

1. When $u$ approaches $f$ from the right (greater than $f$), $v$ becomes very large (positively or negatively depending on the sign convention).
2. When $u$ is less than $f$, $v$ becomes negative, indicating that the image is virtual and located on the same side as the object.

Correct Answer and Explanation

Correct Answer: C - The graph will be a hyperbola. This is because the equation $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ can be manipulated to show that as $u$ approaches $f$, $v$ tends to infinity, and as $u$ increases beyond $f$, $v$ becomes positive and approaches a finite limit. This behavior is characteristic of a hyperbolic curve.

Why Other Options Are Incorrect

1. Option A: Linear Graph - A linear relationship would imply that $v$ changes directly in proportion to $u$. This does not hold true since the relationship is hyperbolic, not linear.

2. Option B: Parabolic Graph - A parabolic relationship suggests a quadratic dependency, which is not the case here. The relationship derived from the mirror formula is not quadratic in nature but follows the inverse relationship characteristic of hyperbolas.

3. Option D: Circular Graph - A circular or elliptical relationship implies a constant radius or a set of values that maintain a fixed distance. This is not applicable to the behavior of a concave mirror, where the relationship between $u$ and $v$ does not form a closed loop.

Summary

In conclusion, the relationship between the magnitudes of $u$ and $v$ for a concave mirror is hyperbolic due to the inverse relationship defined by the mirror formula. Thus, plotting the magnitudes of $u$ against $v$ results in a hyperbolic curve, confirming that the correct answer is option C.