why the newton rings are like circles ?please give the answer with proof and diagram ?

Newton's rings are fascinating optical phenomena that arise from the interference of light waves. They appear as concentric circles when viewed through a microscope or other optical device. To understand why these rings take on a circular shape, we need to delve into the principles of light interference and the geometry of the setup used to create them.

The Setup of Newton's Rings

Newton's rings are typically observed using a setup that consists of a plano-convex lens placed on a flat glass surface. The lens creates a thin air film between its curved surface and the flat glass. When light is shone onto this arrangement, some of the light reflects off the top surface of the glass, while some reflects off the bottom surface of the lens. The two reflected light waves can interfere with each other, leading to the formation of bright and dark rings.

Understanding Interference

The key to understanding the circular nature of Newton's rings lies in the concept of interference. When two waves meet, they can either reinforce each other (constructive interference) or cancel each other out (destructive interference). The conditions for these types of interference depend on the path difference between the two waves.

• Constructive Interference: Occurs when the path difference is an integer multiple of the wavelength (nλ).
• Destructive Interference: Happens when the path difference is a half-integer multiple of the wavelength ((n + 0.5)λ).

Why Circles?

The circular pattern of Newton's rings can be explained through the geometry of the lens and the nature of the air film. As you move away from the center of the lens, the thickness of the air film increases. This variation in thickness leads to different path lengths for the light waves reflecting off the two surfaces.

At the center, where the air film is thinnest, the path difference is minimal, leading to constructive interference and the formation of a bright ring. As you move outward, the thickness of the air film increases, creating regions where the path difference corresponds to both constructive and destructive interference, resulting in alternating bright and dark rings.

Mathematical Proof

The radius of the nth ring can be derived from the interference conditions. The formula for the radius (r) of the nth dark ring is given by:

r_n = √(nλR)

Where:

• n: The ring number (1, 2, 3,...)
• λ: The wavelength of the light used
• R: The radius of curvature of the lens

This equation shows that the radius of each ring increases with the square root of the ring number, which mathematically confirms that the rings are circular and spaced apart in a predictable manner.

Visual Representation

To visualize Newton's rings, imagine a series of concentric circles radiating from a central point. Each circle corresponds to a different order of interference, with the central bright spot representing the first order of constructive interference. The diagram below illustrates this concept:

In this diagram, the central bright spot is surrounded by alternating dark and bright rings, demonstrating the interference pattern created by the varying thickness of the air film.

Conclusion

In summary, Newton's rings appear as circles due to the interference of light waves reflecting off the surfaces of a lens and a glass plate. The circular pattern results from the geometry of the lens and the varying thickness of the air film, which leads to different conditions for constructive and destructive interference. This phenomenon beautifully illustrates the wave nature of light and the principles of optics.