When dust gets between the glass elements of a Newton€™s ring setup, it can cause an unknown shift in the film thickness Δd, and a corresponding change in the fringe pattern. The path difference is then 2(d + Δd) = mλ_Æ’, and because of the additional relative phase shift on reflection, this corresponds to a dark band. Prove that the radius of curvature of the lens (R) given by

can be determined in the lab (via adjacent dark fringes) independent of Δd.

, R = ( -1)Af 1 -1)As