Interference from Thin Films — Question 7

Question 7

Squared Newton-ring radii remove an unknown central gap

A curved glass surface above a glass plate leaves an air film $t(r)=t_0+r^2/(2R)$, with $t_0\geq0$. Normal reflected dark rings satisfy $2t=m\lambda_0$. At $\lambda_0=500\,\mathrm{nm}$, the known orders m=2 and m=5 have squared radii $0.80\,\mathrm{mm^2}$ and $2.30\,\mathrm{mm^2}$.

1. Derive the squared-radius difference relation and infer R.

2. Find the central gap and decide whether the center is a reflected minimum.

Question 7 - Solution

Squared Newton-ring radii remove an unknown central gap

(a) Method and calculation

$$r_m^2=R(m\lambda_0-2t_0)\Longrightarrow r_5^2-r_2^2=3R\lambda_0.$$$$\boxed{R=\frac{1.50\times10^{-6}\,\mathrm{m^2}}{3(500\times10^{-9}\,\mathrm m)}=1.00\,\mathrm m}.$$

(b) Interpretation and further calculation

$$2t_0=2\lambda_0-r_2^2/R=1.00\,\mu\mathrm m-0.80\,\mu\mathrm m\Longrightarrow\boxed{t_0=100\,\mathrm{nm}}.$$$$2t_0/\lambda_0=0.40\notin\mathbb Z\Longrightarrow\boxed{\text{the center is not a reflected minimum}}.$$

Physical check

An unknown gap shifts every squared radius by the same amount. Taking a difference eliminates that gap before solving for curvature.