Question

The formula connecting \(u, v\) and \(f\) for a spherical mirror is valid only for mirrors whose sizes are very small compared to their radii of curvature. because STATEMENT-2 Laws of reflection are strictly valid for plane surfaces, but not for large spherical surfaces. (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

Short answer

(A) Both Statement-1 and Statement-2 are true; Statement-2 provides a correct explanation for Statement-1.

Step-by-step solution

Understanding Statement-1

Statement-1 mentions that the mirror equation relating object distance (u), image distance (v), and focal length (f), is only accurate when the mirror size is small compared to its radius of curvature. In optics, this is related to the paraxial approximation, which assumes that rays strike the mirror at angles very close to the principal axis and thus undergo minimal deviation.

Understanding Statement-2

Statement-2 says that the laws of reflection are strictly valid only for plane surfaces, but not for large spherical surfaces. This is because for a spherical surface, as the size increases relative to the radius of curvature, the deviation from the ideal reflection laws increases, leading to aberrations.

Relating Both Statements

The validity of Statement-1 is derived from the fact that reflection laws work best for smaller, paraxial regions of a mirror where the surface approximates a plane. As the statement-2 rightly points out, these reflections laws are not accurate for large spherical surfaces where the mirror no longer behaves like a plane surface and therefore do not hold true for large spherical mirrors. Thus, Statement-2 correctly explains why the mirror equation is only valid for small mirror sizes.

Choosing the Correct Option

Since both statements are true and Statement-2 offers a correct explanation for Statement-1, the correct answer is (A).

Key concepts

Paraxial Approximation

In the study of optics, especially when dealing with the behavior of light in lenses and mirrors, the term paraxial approximation frequently comes up. At its core, the paraxial approximation is an assumption that simplifies how we predict the paths of light rays near the optical axis of a system.

When light rays strike a mirror or pass through a lens, their behavior can be quite complex. However, if we only consider rays that are very close to the principal axis and make small angles with it, the equations describing the system become much simpler. This angle is so small that its sine, tangent, and angle itself (in radians) can all be approximated as equal. This means that the curvature of a lens or mirror only minimally affects these rays, and they behave almost as though they were reflecting from or refracting through a flat surface.

To apply this approximation in a real-world context, imagine aiming a flashlight towards a spherical mirror. The light rays at the edges of the beam, which strike the mirror at larger angles, will not adhere to the simple mirror equation. But those very close to the middle, the paraxial rays, will follow the equation much more closely. This explains why precision optical instruments tend to use small apertures; it's to harness the simplifying power of the paraxial approximation and ensure accurate focusing.

Laws of Reflection

The laws of reflection govern the direction in which light bounces off surfaces and are a fundamental aspect of optics. These are two-fold: First, the angle at which the light hits the surface (the angle of incidence) is equal to the angle at which it leaves (the angle of reflection). Second, the incident ray, the reflected ray, and the normal to the reflection surface at the point of incidence all lie in the same plane.

These laws are consistently accurate for flat, plane mirrors, leading to predictable and easily calculable results. For example, when you look into a bathroom mirror, the image you see is due to light reflecting off the mirror and obeying these laws.

However, the scenario changes with spherical mirrors. For small mirrors, or those parts close to the principal axis, the curvature is so subtle that the surface acts almost like a plane, and the laws hold true. But as you move away from the center towards the edges of a larger spherical mirror, the discrepancy between the spherical surface and an ideal plane becomes significant, and thus the reflection doesn't perfectly follow these laws, leading to errors known as aberrations. This is especially important in the design of optical devices like telescopes and cameras, where the quality of reflected images is paramount.

Optics

Optics is the branch of physics that's concerned with the study and manipulation of light and its interactions with matter. It's an extensive field that covers various phenomena, from the bending and reflection of light to the dispersion of light into its constituent colors.

Optics can be divided into several subfields, such as geometrical optics, where light is considered in terms of rays that travel in straight lines, and physical optics, which deals with the wave nature of light. Reflection and refraction are two of the key concepts in geometrical optics and are essential to understand the behavior of lenses and mirrors.

Geometrical optics is relevant to everyday life in many ways, from the corrective eyeglasses that correct our vision to the functioning of optical fibers that carry our internet signals. It also underpins the technology behind cameras, microscopes, and telescopes, allowing us to capture images from the microscopic world to the far edges of the universe. The principles of optics are applied in so many domains that its understanding is crucial for both developing new technologies and unraveling the mysteries of light.