Q18Engineering Physics

Newton's rings constitute one of the most classic and elegant demonstrations of the phenomenon of interference of light by division of amplitude. When a plano-convex lens of a very large radius of curvature is placed with its convex surface resting on an optically flat glass plate, a thin wedge-shaped air film of varying thickness is formed between them. The thickness of this air film is precisely zero at the point of contact and gradually, symmetrically increases radially outward. When this arrangement is illuminated by a broad source of monochromatic light falling normally upon it, a beautifully striking pattern of concentric alternating bright and dark circular fringes is observed around the central point of contact. These are known as Newton's rings.

Mechanism of Interference

A ray of monochromatic light (say, from a sodium lamp) falls normally on the upper planar surface of the plano-convex lens. It travels through the glass of the lens and reaches the convex glass-air interface. At this boundary, a portion of the light is reflected back upwards, while the rest is transmitted down through the extremely thin air film. The transmitted ray strikes the top surface of the underlying flat glass plate and is reflected back upwards again.

These two reflected rays—one from the upper surface of the air film and one from the lower surface of the air film—originate from the exact same incident ray. They are therefore perfectly coherent. Because the second ray had to travel down through the air film and back up, it has traveled an extra geometric distance equal to $2t$, where $t$ is the precise thickness of the air film at that specific radial point.

Stokes' Phase Change: It is a fundamental principle of physical optics (Stokes' law) that when a light wave reflects off the surface of an optically denser medium (like the air-to-glass reflection at the bottom of the air film), it undergoes a sudden phase change of $\pi$ radians, which is mathematically equivalent to an extra path difference of $\frac{\lambda }{2}$. The reflection at the top of the air film (glass-to-air) occurs at a rarer medium boundary, so it suffers no such phase change.

Therefore, the total effective optical path difference ($\Delta$) between the two interfering rays is:

For normal incidence (angle of refraction $r=0$, so $\cos r=1$) and for an air film (refractive index $\mu =1$), the equation dramatically simplifies to:

Condition for Dark and Bright Rings

• Constructive Interference (Bright Rings): The total path difference must be an even multiple of $\frac{\lambda }{2}$. $2t+\frac{\lambda }{2}=(2n)\frac{\lambda }{2}⟹2t=(2n-1)\frac{\lambda }{2}\text{where }n=1,2,\mathrm{3...}$
• Destructive Interference (Dark Rings): The total path difference must be an odd multiple of $\frac{\lambda }{2}$. $2t+\frac{\lambda }{2}=(2n+1)\frac{\lambda }{2}⟹2t=n\lambda \text{where }n=0,1,2,\mathrm{3...}$

The Central Spot: At the exact point of contact, the air film thickness $t=0$. The path difference becomes exactly $\frac{\lambda }{2}$. This satisfies the condition for destructive interference. Therefore, the central spot of Newton's rings in reflected light is always perfectly dark.

Derivation of the Diameter of Dark Rings

We must geometrically relate the microscopic thickness of the air film ($t$) to the macroscopic, measurable radius ($r$) of the corresponding ring and the massive radius of curvature ($R$) of the lens.

Using the geometric theorem of intersecting chords for a circle, if a chord intersects a diameter, the products of their segments are equal. Let the diameter of the lens sphere be $2R$. At a radial distance $r$ from the center, the thickness is $t$. The chords give the relation:

Because the radius of curvature $R$ is very large (e.g., $100\text{ cm}$) and the film thickness $t$ is microscopic (on the order of wavelengths of light), the term $t^2$ is infinitesimally small and can be entirely neglected without losing accuracy.

Substitute this geometric relation into the condition for destructive interference ($2t=n\lambda$):

Since diameter $D_n=2r_n$, the square of the diameter is $D_n^2=4r_n^2$. Therefore:

This profoundly important equation proves that the square of the diameter of the $n$-th dark ring is directly proportional to the natural number $n$ ($D_n^2\propto n$). The rings get progressively closer to each other as we move outward from the center.

Proof of Constant Difference and Determination of Wavelength

If we consider the square of the diameter of the $n$-th dark ring:

And we consider the square of the diameter of the $(n+p)$-th dark ring (where $p$ is any integer, usually taken as 5 or 10):

By subtracting the first equation from the second, we find the difference between the squares of their diameters:

If we take consecutive rings (i.e., $p=1$), the difference is:

Conclusion of Proof: We have successfully proven that the difference between the squares of the diameters of any two consecutive dark rings is absolutely constant and relies only on the wavelength of light and the radius of curvature of the lens. It does not depend on the order number $n$.

Experimental Application (Finding $\lambda$): This constant difference is heavily exploited in the physics laboratory to find the unknown wavelength of a light source. By using a traveling microscope, an experimenter precisely measures the diameters of various rings (e.g., the 5th, 10th, 15th, and 20th rings). They then use the difference formula rearranged for $\lambda$:

Because the exact point of contact is often distorted by dust or immense physical pressure (making the absolute center difficult to locate), measuring the difference between two outer, well-defined rings completely eliminates any error associated with identifying the central point. A graph of $D_n^2$ on the y-axis against $n$ on the x-axis yields a perfectly straight line whose slope is $4\lambda R$. Knowing the radius of curvature $R$ (measured using a spherometer), the wavelength $\lambda$ is easily computed with very high precision.