A lens is placed on a flat plate of glass to test whether...

A lens is placed on a flat plate of glass to test whether its surface is spherical. (a) Show that the radius $r_{m}$ of the $m$ th dark ring should be $$ r_{m}=\sqrt{m \lambda R} $$ where $R$ is the radius of curvature of the lens surface facing the plate and the wavelength of the light used is $\lambda$. Assume that $r_{m} \ll R .$ [Hint: Start by finding the thickness $t$ of the air gap at a radius $r=R \sin \theta=R \theta$. Use small angle approximations.] (b) Are the dark fringes equally spaced? If not, do they get closer together or farther apart as you move out from the center?

A lens is placed on a flat plate of glass to test whether its surface is spherical. (a) Show that the radius $r_{m}$ of the $m$ th dark ring should be
$$
r_{m}=\sqrt{m \lambda R}
$$
where $R$ is the radius of curvature of the lens surface facing the plate and the wavelength of the light used is $\lambda$. Assume that $r_{m} \ll R .$ [Hint: Start by finding the thickness $t$ of the air gap at a radius $r=R \sin \theta=R \theta$. Use small angle approximations.] (b) Are the dark fringes equally spaced? If not, do they get closer together or farther apart as you move out from the center?

Key Concepts

Optical Path Difference and Phase Change

The optical path difference is the extra distance that light travels in one medium relative to another, which in this case is twice the thickness of the air gap between the lens and the plate. Additionally, the phase change upon reflection (typically a half-wavelength shift at an interface with a higher refractive index) plays a crucial role in determining the conditions for destructive interference, thereby leading to the formation of dark fringes.

Thin Film Interference

This concept explains how interference arises when light waves reflected from different interfaces of a thin film combine. In the context of Newton's rings, the interference is between light reflected from the bottom surface of the lens and the top surface of the flat glass plate, creating a pattern of alternating dark and bright fringes due to path difference and phase shifts.

Newton's Rings

Newton's rings are a specific manifestation of thin film interference that occurs when a convex lens is placed on a flat glass surface. The air film of varying thickness between the two surfaces causes a systematic change in the optical path length, thereby producing concentric circular fringes whose radii depend on the wavelength of light, the curvature of the lens, and the order of the fringe.

Small-Angle Approximation

The small-angle approximation is used when the angles involved in a problem are small enough that trigonometric functions can be approximated by their leading terms (for example, sin? ? ?). This approximation simplifies the mathematical treatment of the geometry of the air gap under the lens, allowing one to derive the relationship between the fringe radius, the radius of curvature of the lens, and the order number.

Fringe Spacing

Fringe spacing refers to the distance between successive dark or bright rings in an interference pattern. In the case of Newton's rings, the spacing is not uniform; as the radius increases, the rings become more closely spaced due to the square root dependence on the order of the fringe. This reflects the nonlinear relationship between the air gap thickness and the radial position.

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