Question

Back surface of a glass (refractive index $n$ and thickness $t$) is polished to work as a mirror as shown below. A laser beam falls on it and is partially reflected and refracted at the air-glass interface and fully reflected at the mirror surface respectively. A pattern of discrete spots of light is observed on the screen.
The spacing between the spots on the screen will be

• A. $\frac{t\sin 2\theta }{\sqrt{n^2-{\sin }^2\theta }}$
• B. $\frac{2t\sin \theta }{\sqrt{n^2-{\sin }^2\theta }}$
• C. $\frac{2t\tan \theta }{\sqrt{n^2-{\sin }^2\theta }}$
• D. $\frac{2t\sin \theta }{\sqrt{1-\frac{{\sin }^2\theta }{n^2}}}$

Answer 1

The correct option is A $\frac{t\sin 2\theta }{\sqrt{n^2-{\sin }^2\theta }}$

$1\times sin\left(\theta \right)=nsin\left(r\right)$

from the diagram, $2x=2t\tan \left(r\right)=2t\frac{\frac{\sin \theta }{n}}{\sqrt{1-\frac{si{n}^2\theta }{n^2}}}$

$\Rightarrow$ spacing between spots on screen is $O^{\prime }A=2xsin(90-\theta )=2t\frac{sin\left(\theta \right)cos\left(\theta \right)}{\sqrt{n^2-si{n}^2\theta }}$