$4$ Newton$$s rings $(5$ marks$)$

A plano$-$convex lens is placed with its curved surface down onto a flat block of glass made from the same

material, and is illuminated from above with red light of wavelength $.$ When the lens is viewed from

above, a concentric series of finely spaced bright and dark rings is observed.

$1.$ Explain the origin of the observed rings, and the reason why the centre of the fringe pattern is dark

rather than bright.

$2.$ Show that, if the curvature radius of the curved lens surface is R$,$ the radius rn of the nth dark

fringe will be approximately given by:

rn $$nR $(3\mathrm{.}3)$

$3.$ The radius of the tenth dark ring is measured to be $2\mathrm{.}13$ nm$.$ If the wavelength $=635$ nm$,$

determine the radius of curvature R$,$ and estimate the precision of your result.