Number of bright Newton's rings

In reflected light, the nth bright Newton's ring has radius r_n² = (2n − 1)λR / (2μ), where μ is the refractive index of the film between lens and plate. The number of bright rings is the largest n for which r_n does not exceed the lens radius.

Given: R = 5 cm = 0.05 m, lens diameter 20 mm so r = 1 cm = 0.01 m, λ = 589 nm = 5.89 × 10⁻⁷ m.

(a) In air (μ = 1)

r² = (2n − 1) λR / 2
2n − 1 = 2r² / (λR)
       = 2 × (0.01)² / (5.89×10⁻⁷ × 0.05)
       = 2×10⁻⁴ / 2.945×10⁻⁸
       = 6791.2
n      = (6791.2 + 1) / 2 = 3396.1

Only whole rings count, so n = 3396.

Answer (a): about 3396 bright rings.

(b) In water (μ = 1.33)

The wavelength in the film is λ/μ, so r² = (2n − 1)λR / (2μ).

2n − 1 = 2μ r² / (λR)
       = 1.33 × 6791.2
       = 9032.3
n      = (9032.3 + 1) / 2 = 4516.6

Answer (b): about 4516 bright rings.

The rings crowd closer together in water, so more of them fit inside the same lens aperture (ratio 4516/3396 ≈ 1.33 = μ).

Note: the centre is dark in reflected light, and the ring count is only a theoretical limit; in practice the rings become too closely packed to resolve long before this number. (If R = 5 m, as in the common textbook version of this problem, the same method gives 34 bright rings in air and 45 in water.)