With the help of proper diagram and necessary equation, explain how Newtons ring experiment is useful to determine the radius of curvature of plano convex lens. In a newton rings experiment the diameter of 5th dark ring is 0.336 cm and the diameter of 15th ring is 0.590 cm. Find the radius of curvature of plano convex lens if the wavelength of light used is 5890 A0.

#### Solution

Consider a planoconvex lens of radius ‘R’ as shown.

As radius is comparatively large, the space between lens and base can be considered as wedge shaped. Thus path difference is given by

Δ = 2μtcos`(r_e+alpha)+- λ/2.........(r_e=\text{angle of refraction})`

For almost normal incidence,

cos `(r_e+alpha)≈ 1`

Δ =2t+` λ/2`...........(i)

Now,

`R^2=(R-t)^2+r^2`

`R^2=R^2-2Rt+t^2+r^2`

`2t=t^2/R+r^2/R`

`R `>>`t^2,k^2/R`≈0

`2t=r^2/R=D^2/(4R)...............(ii)[D=2r]`

From (i) and (ii)

Δ =`D^2/(4R)+λ/2..........(iii)`

For Bright Rings:-

Δ = nλ

`D_n^2/(4R)+λ/2=nλ........[D_n=\text{dia.of bright ring}]`

`D_n^2/(4R)=(2n-1)λ/2`

`R=D_5^2/(2(2xx5-1)xx5890xx10^-8)-D_15^2/(2(2xx15-1)xx5890xx10^-8)`

`R=0.336^-3/(2(2xx5-1)xx5890xx10^-8)-0.590^2/(2(2xx15-1)xx5890xx10^-8)`

𝑅=4.59 𝑐𝑚

Thus radius of curvature of plano convex lens is 4.59 cm.