How are Newton's rings formed and why is the central spot dark in reflected light?

In one line

Newton's rings are concentric circular fringes formed by interference in the air film between a plano-convex lens and a flat glass plate, where the film thickness is constant along circles around the contact point.

Explained simply

Rest a slightly curved lens on a flat glass sheet. Where they touch the gap is zero; moving outward, the gap grows, slowly at first and then faster. Rings of light and dark appear because each ring is a circle of equal gap thickness, just as contour lines on a map show equal height.

How it works

1. Set-up: a plano-convex lens of large radius \(R\) is placed on an optically flat glass plate. Sodium light (589 nm) is sent in using a glass plate at 45°, so it falls normally, and the rings are viewed with a travelling microscope.
2. Light reflected from the bottom of the lens and the top of the plate interferes. The air film has \(\mu = 1\) and \(\cos r = 1\).
3. Path difference \(= 2t - \lambda/2\) (extra \(\lambda/2\) from reflection at the glass plate, a denser medium).
4. At the contact point \(t = 0\), so path difference \(= \lambda/2\): destructive interference, hence a dark centre in reflected light.
5. From the geometry of the circle, \(r^2 = (2R - t)t \approx 2Rt\), so \(t = r^2/2R\).
6. Dark rings: \(2t = n\lambda\) gives \(r_n^2 = n\lambda R\), so \(D_n^2 = 4n\lambda R\) (for a liquid film, divide by \(\mu\)).
7. Since \(D_n \propto \sqrt{n}\), rings get closer together as you move outward.
8. In transmitted light the pattern is complementary, with a bright centre.

Key formula

Dark ring: \(D_n^2 = \dfrac{4n\lambda R}{\mu}\); bright ring: \(D_n^2 = \dfrac{2(2n-1)\lambda R}{\mu}\). \(D_n\): diameter of \(n\)th ring (m), \(R\): radius of curvature of the lens (m), \(\lambda\): wavelength (m), \(\mu\): index of the film (1 for air).

Quick example

With \(R = 1\) m and \(\lambda = 589\) nm in air, the 4th dark ring has \(D_4 = \sqrt{4\times4\times589\times10^{-9}\times1} = \sqrt{9.42\times10^{-6}} \approx 3.07\) mm, and the 16th has twice that, about 6.14 mm, showing the \(\sqrt{n}\) spacing.

Common mistakes

• Saying the centre is bright in reflected light; it is dark due to the \(\lambda/2\) shift.
• Writing \(r_n^2 = n\lambda R\) for diameter instead of radius.
• Saying rings are equally spaced; they crowd together outward.

How MU asks it

Asked as "Explain the formation of Newton's rings and show that the diameters of dark rings are proportional to the square root of natural numbers" (5 marks). Draw the set-up and the lens-plate geometry, derive \(t = r^2/2R\), then \(D_n^2 = 4n\lambda R\), and comment on the dark centre.