Name: Constructive and Destructive Interference of Light Waves
Uploaded: 2019-07-19
Duration: 11 min 15 s
Description: If 2 balls rolling on a surface with a certain speed approach each other, then we expect that they will collide, and their velocities get altered after the c...

Question

In Newton's rings experiment, obtain an expression for the radius of the $n^{th}$ dark ring.

Accepted Answer

Let us consider the vertical section $SOP$ of the planoconvex lens through its centre of curvature C. Let $R$ be the radius of curvature of the planoconvex lens and $O$ be the point of contact of the lens with the plane surface. Let $t$ be the thickness of the air film at $S$ and $P$ . Draw $ST$ and $PQ$ perpendiculars to the plane surface of the glass plate. Then $ST = AO = PQ = t$ .
Let $r_{n}$ be the radius of the $n^{th}$ dark ring which passes through the points $S$ and $P$ .
Then $SA = AP = r_{n}$ .
If $ON$ is the vertical diameter of the circle, then by the law of segments.

$SA . AP = OA . AN$ .

$\Rightarrow {r_{n}}^{2} = t (2 R - t) \Rightarrow {r_{n}}^{2} = 2 Rt - t^{2} \Rightarrow {r_{n}}^{2} = 2 Rt$
(neglecting $t^{2}$ comparing with $2 R$ )

$\Rightarrow 2 t = \frac{{r_{n}}^{2}}{R}$
According to the condition for darkness
$2 t = nλ$

Therefore,

$\Rightarrow \frac{{r_{n}}^{2}}{R} = nλ \Rightarrow {r_{n}}^{2} = nλ R \Rightarrow r_{n} = \sqrt{nλ R}$
Since $R$ and $λ$ are constants, we find that the radius of the dark king is directly proportional to square root of its order i.e $r_{n} \propto \sqrt{n}$ .

In Newton's rings experiment, obtain an expression for the radius of the $n^{th}$ dark ring.

Important Questions on Optics

Learn and Prepare for any exam you want !

In Newton's rings experiment, obtain an expression for the radius of the nth dark ring.

Important Questions on Optics

Learn and Prepare for any exam you want !

In Newton's rings experiment, obtain an expression for the radius of the $n^{th}$ dark ring.