With the experimental set up of Newton’s ring, explain the determination of refractive index of a liquid.

Experimental set-up:

             travelling microscope
                     |
                     v
      S (sodium lamp)--> L (lens) --> G (glass plate at 45°)
                                         |  reflects light down
                                         v
                        ( plano-convex lens, radius R )
                        ----------------------------  glass plate P
                          liquid film (μ) in the gap

• Light from a sodium lamp S (λ = 5893 Å) is made parallel by the lens L. A glass plate G inclined at 45° reflects it normally onto a plano-convex lens of large radius R resting on a flat glass plate P.
• The air film between the lens and the plate has thickness \(t=r^2/2R\). Reflected light gives concentric dark and bright rings (Newton's rings), viewed through a travelling microscope.

Theory: For the n-th dark ring in reflected light, \(2\mu t=n\lambda\), so \(D_n^2=\dfrac{4n\lambda R}{\mu}\).

Procedure:

1. Air film (\(\mu=1\)): Measure the diameters of the (n+p)-th and n-th dark rings with the microscope. Then \(D_{n+p}^2-D_n^2=4p\lambda R\) ... (1)
2. Liquid film: Put a drop of the liquid between the lens and the plate, without disturbing the set-up, and measure the same rings again. Then \((D'_{n+p})^2-(D'_n)^2=\dfrac{4p\lambda R}{\mu}\) ... (2)
3. Dividing (1) by (2):

\[ \boxed{\mu=\frac{D_{n+p}^2-D_n^2}{(D'_{n+p})^2-(D'_n)^2}} \]

The rings shrink when the liquid is introduced (\(D'<D\)), so \(\mu>1\). Taking the difference of squares removes errors from dust or a poor contact at the centre. A graph of \(D^2\) against n for both cases also gives μ, as the ratio of the slopes. This method needs only a drop of the liquid.