thickness of air film of fourth bright circle

Question

Newton’s rings are observed by reflected light of wavelength 589 nm. The central area is dark and is surrounded by light and dark circles. Find the thickness of the air film at the fourth bright circle.

 

Answer is supposed to be 1.03 x10^-6 m, please show me how to calculate.

Answer 1

Newton’s Rings (Reflected Light)

Given that Newton’s rings are observed using reflected light of wavelength

λ=589 nm=589×10−9 m\lambda = 589\,\text{nm} = 589 \times 10^{-9}\,\text{m}λ=589nm=589×10−9mand the central spot is dark.

Theory

For Newton’s rings observed in reflected light, the condition for the nᵗʰ bright ring is:

2t=(2n−1)λ22t = \frac{(2n-1)\lambda}{2}2t=2(2n−1)λ​or

t=(2n−1)λ4t = \frac{(2n-1)\lambda}{4}t=4(2n−1)λ​where ttt is the thickness of the air film.Calculation

For the 4th bright ring, n=4n = 4n=4:

t=(2×4−1)λ4t = \frac{(2\times4 - 1)\lambda}{4}t=4(2×4−1)λ​ t=7×589×10−94t = \frac{7 \times 589 \times 10^{-9}}{4}t=47×589×10−9​ t=1.03×10−6 mt = 1.03 \times 10^{-6}\,\text{m}t=1.03×10−6m Final Answer

t=1.03×10−6 m\boxed{t = 1.03 \times 10^{-6}\,\text{m}}t=1.03×10−6m​