How are Newton's rings used to find the wavelength of monochromatic light?

In one line

By measuring the diameters of two dark rings that are \(p\) rings apart and knowing the lens radius \(R\), the wavelength is found from \(\lambda = (D_{n+p}^2 - D_n^2)/(4pR)\).

Explained simply

The ring sizes depend on the wavelength of light: longer waves make bigger rings. So if you measure ring sizes accurately with a microscope, you can work backwards to find the wavelength. Using a difference of two rings cancels errors from dust or imperfect contact at the centre.

How it works

1. Set up Newton's rings with an air film (\(\mu = 1\)) and the given monochromatic source.
2. Using a travelling microscope, measure diameters \(D_n\) and \(D_{n+p}\) of the \(n\)th and \((n+p)\)th dark rings (take readings on both sides of centre and subtract).
3. From \(D_n^2 = 4n\lambda R\) and \(D_{n+p}^2 = 4(n+p)\lambda R\), subtract: \(D_{n+p}^2 - D_n^2 = 4p\lambda R\).
4. So \(\lambda = \dfrac{D_{n+p}^2 - D_n^2}{4pR}\).
5. Find \(R\) of the lens with a spherometer. Plotting \(D_n^2\) against \(n\) gives a straight line with slope \(4\lambda R\), which averages many readings.
6. The difference method is used because the exact centre may not be a perfect point (dust, lens pressure), which shifts all \(D_n^2\) by the same amount; subtracting removes this error.

Key formula

\(\lambda = \dfrac{D_{n+p}^2 - D_n^2}{4pR}\). \(D_{n+p}, D_n\): diameters of the \((n+p)\)th and \(n\)th dark rings (m), \(p\): number of rings between them, \(R\): radius of curvature (m), \(\lambda\) in m.

Quick example

Sample reading: \(D_{15} = 0.590\) cm, \(D_5 = 0.336\) cm, \(p = 10\), \(R = 100\) cm. \(D_{15}^2 - D_5^2 = 0.3481 - 0.1129 = 0.2352\) cm². \(\lambda = 0.2352/(4\times10\times100) = 5.88\times10^{-5}\) cm = 588 nm, close to sodium light.

Common mistakes

• Using radii in a formula written for diameters (or the reverse), giving an error of a factor of 4.
• Mixing cm and m within one calculation.
• Taking \(p\) as \(n+p\) instead of the difference in ring numbers.

How MU asks it

Asked as "Explain how Newton's rings are used to determine wavelength of light" (5 marks) and very often as a numerical (3 to 5 marks). Write the set-up briefly, derive \(D_{n+p}^2 - D_n^2 = 4p\lambda R\), and in numericals keep units consistent and give \(\lambda\) in nm or Å.