Problem 93 - Master Resource Book in JEE Main Physics

Question

The following statements carefully to mark correct option are options given below (a) Statement 1 is true, Statement 2 is true. Statement 2 is a correct explanation for Statement 1 (b) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 (c) Statement 1 is true, Statement 2 is false (d) Statement 1 is false, Statement 2 is true Statement 1 The formula connecting \(u, v\) and \(f_{4}\) for a spherical mirror is valid only for mirrors whose sizes are very small compared to their radii of curvature. Statement 2 Laws of reflection are strictly valid for plane surface, but not for large spherical surfaces.

Step-by-Step Solution

Answer

The correct answer is (b) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

Step 1: Understanding Statement 1

Statement 1 suggests that the mirror formula, which relates object distance \(u\), image distance \(v\), and focal length \(f\), is only accurate for mirrors that are much smaller than their radius of curvature. This is true because spherical aberrations can occur in larger mirrors, affecting the validity of this formula.

Step 2: Understanding Statement 2

Statement 2 claims that the laws of reflection apply accurately only to plane surfaces, not to large spherical surfaces. This is true because, in the case of large spherical surfaces, the approximations used in deriving the mirror formula break down, resulting in deviations from the ideal reflective behavior.

Step 3: Comparing Statements 1 and 2

Both statements are individually true. Now, determine if Statement 2 correctly explains Statement 1. Statement 2 addresses the limitation of reflection laws on large spherical surfaces, while Statement 1 focuses on the formula's validity considering the size of mirrors. Although related, Statement 2 directly connects to the reflection aspect, not the size limitation itself.

Step 4: Choosing the Correct Option

When evaluating options: (a) Both statements are true, but the explanation in Statement 2 is not directly about the mirror formula's size limitations, so this is incorrect. (b) Both statements are true, and Statement 2 is related to but not the reason for Statement 1; this option works. (c) Does not fit since Statement 2 is true. (d) Is incorrect since both statements are true. Therefore, the best choice is (b).

Key Concepts

Mirror Formula

The mirror formula is a crucial equation in optics that connects three important variables: the object distance \(u\), the image distance \(v\), and the focal length \(f\) of a spherical mirror. This relationship is expressed mathematically as:

• \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

This formula is especially useful when dealing with thin mirrors whose size is small compared to the radius of curvature. These thin mirrors allow for precise predictions of where an image will form.
The reason for the small size requirement lies in spherical aberration, which leads to deviations from the ideal behavior predicted by the mirror formula when the mirror is large. Essentially, the light rays coming from an object point may not all converge at a single image point on a large mirror, causing blurring. Therefore, the mirror formula holds precisely only for small mirrors, ensuring more accurate imaging.

Spherical Aberration

Spherical aberration occurs when light rays entering a spherical mirror at different distances from the principal axis do not converge to a single focal point. This happens because of the curved nature of the mirror's surface.
In a perfectly curved spherical mirror, rays closer to the edge converge at different points than rays near the center. This discrepancy leads to blurred or distorted images, as the ideal convergence predicted by simple optics breaks down. The extent of spherical aberration increases with the size of the mirror. To minimize spherical aberration, two solutions are often employed:

• Using parabolic mirrors that ensure all incoming parallel rays reflect and meet at a single focal point.
• Restricting the aperture by using smaller portions of the mirror, hence the mirror formula applies correctly.

Understanding spherical aberration helps explain why the mirror formula only truly applies to small mirrors and relates to the principles of reflection.

Law of Reflection

The law of reflection is a fundamental principle in optics stating that the angle of incidence equals the angle of reflection. Mathematically, this can be expressed as:

• \[ \theta_i = \theta_r \]

where \(\theta_i\) is the angle of incidence, and \(\theta_r\) is the angle of reflection.
This law applies perfectly to plane (flat) mirrors where each light ray behaves predictably. However, in spherical mirrors, especially larger ones, this law encounters limitations, leading to effects like spherical aberration.
While the law of reflection holds for each point on a spherical mirror, the cumulative effect on various parts of a large mirror surface can cause deviations from expected behavior, as illustrated by spherical aberration. As such, the laws of reflection show their greatest accuracy with flat mirrors and very small spherical mirrors, due to minimized aberrative effects.