Question

A glass of refractive index $1.5$ is coated with a thin layer of thickness of $t$ of refractive index $1.5$ light of wavelength $\lambda$ travelling in air is incident normally on the layer. It is partly reflected at the upper and the lower surfaces of the layer and the two reflected rays interfere. If $\lambda =648\text{nm}$ obtain the least value of $t$ (in ${10}^{-8}$) for which the rays interfere consructively.

Answer 1

Path difference between rays reflected from upper and lower faces of layer $=2\mu t$ (for normal incidence).

But there is abrupt change in path of $\frac{\lambda }{2}$ of light at upper surface. So actual path difference is $2\mu t-\frac{\lambda }{2}$


For constructive interference $2\mu r-\frac{\lambda }{2}=n\lambda$

$t=\frac{(2n+1)\lambda }{4\mu }$. For least thickness $n=0$
$\therefore I_{min}=\frac{\lambda }{4\mu }=\frac{648}{4\times 1.8}\text{nm}=90\text{nm}$