A plano-convex lens has index of refraction n . The curved...

A plano-convex lens has index of refraction $n .$ The curved side of the lens has radius of curvature $R$ and rests on a flat glass surface of the same index of refraction, with a film of index $n_{\text {film }}$ between them, as shown in Figure $\mathrm{P} 36.42 .$ The lens is illuminated from above by light of wavelength $\lambda$. Show that the dark Newton's rings have radii given approximately by $$ r=\sqrt{\frac{m \lambda R}{n_{\text {film }}}} $$ where $r<<R$ and $m$ is an integer.

A plano-convex lens has index of refraction $n .$ The curved side of the lens has radius of curvature $R$ and rests on a flat glass surface of the same index of refraction, with a film of index $n_{\text {film }}$ between them, as shown in Figure $\mathrm{P} 36.42 .$ The lens is illuminated from above by light of wavelength $\lambda$. Show that the dark Newton's rings have radii given approximately by
$$
r=\sqrt{\frac{m \lambda R}{n_{\text {film }}}}
$$
where $r<<R$ and $m$ is an integer.

Key Concepts

Lens Geometry and Radius of Curvature

The geometry of the plano-convex lens, particularly its radius of curvature, crucially influences the air or film gap thickness between the lens and the flat surface at different radial distances from the point of contact. This variation in thickness introduces a relationship between the radius of the interference rings and the physical parameters of the lens. Understanding the lens geometry is essential to derive expressions that relate measurable quantities, such as the ring radii, to the wavelength of light and other optical properties.

Optical Path Difference and Phase Shift

Optical path difference (OPD) is the difference in the product of refractive index and physical path length traveled by two beams of light. In thin film interference, the OPD determines whether the interference is constructive or destructive. Additionally, phase shifts may be introduced upon reflection, especially when light reflects from a medium of higher refractive index, further affecting the interference conditions. Together, these factors determine the positions and radii of the interference fringes observed in experiments like Newton's Rings.

Newton's Rings

Newton's Rings refer to the circular interference pattern produced when a convex lens is placed on a flat glass surface. The pattern results from the interference of light waves reflecting from the top and bottom surfaces of the thin air or film gap between the lens and the plate. This phenomenon is a classic demonstration of wave interference and is commonly used to study the coherence and wavelength of light, as well as to measure the flatness of surfaces.

Thin Film Interference

Thin film interference occurs when light is reflected off the top and bottom boundaries of a thin layer, causing the reflected waves to interfere with each other. The interference can be constructive or destructive depending on the optical path difference between the two reflected beams and any phase shifts that occur upon reflection. This concept underlies the formation of colorful patterns in everyday objects like soap bubbles and oil films on water, and it is directly relevant in analyzing and predicting the fringe patterns in Newton's Rings experiments.

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