Class 11 > Unit # 13: Physical Optics > Newton's Rings

Talha's Physics Academy

Newton’s Rings: Production, Working & Derivation

Video Lecture: Newton’s Rings

Watch the complete step-by-step video lecture explaining the production of Newton’s Rings and the derivation for bright and dark ring radii:

What are Newton’s Rings?

“Newton’s rings are circular interference fringes produced by the interference of light waves reflecting from a variable air film enclosed between a plano-convex lens and a plane glass plate.”

Production & Experimental Setup: A plano-convex lens of large radius of curvature is placed with its curved surface on a smooth, plane glass plate. A wedge-shaped air film of variable thickness is enclosed between them. The thickness of the air film is zero at the point of contact $A$ and increases gradually as we move outward.

Monochromatic light is reflected from a glass plate inclined at $45^\circ$ and falls normally on the lens. Interference occurs between:

• Ray 1: Reflected from the upper surface of the air film (lower surface of the lens).
• Ray 2: Reflected from the lower surface of the air film (upper surface of the glass plate), which suffers a phase change of $180^\circ$ ($\pi$ radians) due to reflection from an optically denser medium.

When viewed through reflected light, circular alternating bright and dark rings are observed with a dark center at the point of contact because the path difference is zero and destructive interference occurs due to the $180^\circ$ phase change.

Fig: Experimental arrangement for observing Newton’s rings with plano-convex lens and glass plate.

Geometrical Relation for Film Thickness ($t$)

According to the geometrical theorem of intersecting chords (where a chord of a circle of radius $R$ is intersected by a diameter divided into segments of length $t$ and $2R - t$, and the chord of radius $r$ forms perpendicular segments of length $r$ and $r$):

$r^2 = t(2R - t) = 2Rt - t^2$

Since the thickness of the air film $t$ is extremely small compared to the radius of curvature $R$, $t^2$ is negligible and can be omitted:

$r^2 \approx 2Rt \implies t = \frac{r^2}{2R} \quad \text{--- (i)}$

Derivation for Radii of Dark and Bright Rings

For thin-film interference with a single phase reversal ($180^\circ$) at the lower boundary, the standard conditions for constructive and destructive interference are interchanged.

1. For Constructive Interference (Bright Rings):

The condition for constructive interference in thin films with a phase change is:

$2nt = \left(m + \frac{1}{2}\right)\lambda \quad (m = 0, 1, 2, 3, \dots)$

Substituting $t = \frac{r^2}{2R}$ from equation (i) and setting refractive index $n = 1$ for air:

$2(1)\left(\frac{r^2}{2R}\right) = \left(m + \frac{1}{2}\right)\lambda \implies \frac{r^2}{R} = \left(m + \frac{1}{2}\right)\lambda$

$r^2 = \left(m + \frac{1}{2}\right)\lambda R \implies r_m = \sqrt{\left(m + \frac{1}{2}\right)\lambda R}$

For the $N^{\text{th}}$ bright ring, substituting $m = N - 1$ or expressing in order:

$r_N = \sqrt{\left(N - \frac{1}{2}\right)\lambda R}$

2. For Destructive Interference (Dark Rings):

The condition for destructive interference in thin films with a phase change is:

$2nt = m\lambda \quad (m = 0, 1, 2, 3, \dots)$

Substituting $t = \frac{r^2}{2R}$ and $n = 1$ for air:

$2\left(\frac{r^2}{2R}\right) = m\lambda \implies \frac{r^2}{R} = m\lambda$

$r^2 = m\lambda R \implies r_m = \sqrt{m\lambda R}$

For the $N^{\text{th}}$ dark ring (noting that for $m = 0$, $r = 0$ corresponding to the dark center, so for the $N^{\text{th}}$ dark ring $m = N$):

$r_N = \sqrt{N\lambda R} \quad (N = 0, 1, 2, 3, \dots)$

Thus, the radii of dark rings are proportional to the square root of whole numbers ($\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \dots$), while the radii of bright rings are proportional to the square root of odd integers ($\sqrt{1/2}, \sqrt{3/2}, \sqrt{5/2}, \dots$).

© 2026 Talha's Physics Academy. All rights reserved.