91Ó°ÊÓ

Newton's rings is a phenomena in which light reflection between two surfaces—a spherical surface and an adjacent contacting flat surface—creates an interference pattern.

The interference pattern formed by waves reflected from the upper and lower surface of the air wedge. At the place of condition for the maximum intensity, the thickness of the wedge is .

The following figure shows the experimental setup of Newton’s ring.

Expression for the condition of constructive interference is,

$2d=\left(m+\frac{1}{2}\right)\mathrm{λ}$

Here, d is the thickness of the wedge, m is the order of the fringe pattern, and $\mathrm{λ}$ is the wavelength of light in air.

Rearrange the above expression for d.

$d=\left(2m+1\right)\frac{\mathrm{λ}}{4}$ ...(1)

Here, d thickness of the wedge.

Express the relation form the diagram.

$DC×ED=BD×DA$

Here, $DC,ED,BD$ and DA are lengths.

Substitute r for both DC and ED, d for BD, and R for DA find d.

$\begin{array}{l}r×r=d×R\\ R=\frac{r^2}{d}\end{array}$$\begin{array}{rcl}r×r& =& d×R\\ R& =& \frac{r^2}{d}\end{array}$ ...(2)

Here r is the radius of the Newton’s ring,

And R is the radius of curvature of the lens.

We use condition $r/R≤1$ to simplify the expression in equation since that

$2R-d≈2R$

Rearrange the expression in equation (2) for d.

$d=\frac{r^2}{2R}$ ...(3)

Here thickness of the wedge.

Solve equation (1) and (3) for d.

$\begin{array}{rcl}\frac{r^2}{2R}& =& \frac{\left(2m+1\right)\mathrm{λ}}{4}\\ r^2& =& \frac{\left(2m+1\right)}{2}\mathrm{λ}R\\ r& =& \sqrt{\frac{\left(2m+1\right)R\mathrm{λ}}{2}}\text{for m = 0,1,2,3,}.......\end{array}$

Therefore, the radii of the interference maxima is $\boxed{r=\sqrt{\left(2m+1\right)R\mathrm{λ}/2}}$.