Newton's rings, refractive index of liquid

Newton's rings are circular, concentric interference fringes formed in the thin air film between a plano-convex lens of large radius of curvature R placed on a plane glass plate, when viewed in monochromatic light. They are fringes of equal thickness.

Theory

     source -> glass plate at 45° -> lens
              ____|____
             /         \       air film
 ____________\_________/_____ plate
              t at radius r

1. For a ring of radius r, film thickness t = r²/(2R) (since r² = 2Rt − t² ≈ 2Rt).
2. Reflection at the lower surface adds λ/2, so for a dark ring 2μt = nλ.
3. Hence r² = nλR/μ and diameter D² = 4nλR/μ.

For air (μ = 1): D_air² = 4nλR. The centre is dark.

Determination of refractive index of a liquid

1. Set up the apparatus, view with a travelling microscope, and measure the diameter D_n of the nth dark ring in air.
2. Fill the space between the lens and plate with the liquid (using a container), and measure the diameter D'_n of the same nth ring.
3. From D_n² = 4nλR and D'_n² = 4nλR/μ:

μ = D_n² / D'_n²

4. To reduce error, use a ring pair: D²_{n+p} − D²_n = 4pλR (air) and = 4pλR/μ (liquid), so

μ = (D²_{n+p} − D²_n)air / (D²{n+p} − D²_n)_liquid

This removes errors from the exact centre and the ring order. The liquid ring diameters are smaller than in air.