ABC is a spherical wavefront centred at ‘O’ symmetric about BE is incident on slits S₁ and ${\mathrm{S}}_2.{\mathrm{BS}}_1=3\mathrm{\lambda };{\mathrm{S}}_1{\mathrm{S}}_2=4\mathrm{\lambda };\mathrm{BO}=6\mathrm{\lambda };$${\mathrm{S}}_1\mathrm{E}=128\mathrm{\lambda }\text{ and }\mathrm{\lambda }$ is the wavelength of incident light wave. A mica sheet of refractive index 1.5 is pasted on S₂ . Find the minimum value of thickness of mica sheet for which central fringe forms at E?

Let ${\mathrm{S}}_1^1\text{ and }{\mathrm{S}}_2^1$ are the points on the wave front where perpendiculars can be drawn from  $S_1\text{ and }{S}_2.$For central fringe formed ar E the path difference should be zero
$\begin{array}{l}{\mathrm{S}}_1^1+{\mathrm{S}}_2+(\mathrm{\mu }-1)\mathrm{t}+{\mathrm{S}}_2\mathrm{E}={\mathrm{S}}_1^1{\mathrm{S}}_1+{\mathrm{S}}_1\mathrm{E}\\ \Rightarrow \left({\mathrm{OS}}_2^1-{\mathrm{OS}}_2\right)+(\mathrm{\mu }-1)\mathrm{t}+\sqrt{{\mathrm{D}}^2+{\mathrm{d}}^2}-\left({\mathrm{OS}}_1^1-{\mathrm{OS}}_1\right)+\mathrm{D}\\ \Rightarrow (\mathrm{\mu }-1)\mathrm{t}=\left({\mathrm{OS}}_2-{\mathrm{OS}}_1\right)+\left(\sqrt{{\mathrm{D}}^2+{\mathrm{d}}^2}-\mathrm{D}\right)\\ =(\mathrm{\mu }-1)\mathrm{t}=2\mathrm{\lambda }-\left[\mathrm{D}{\left(1+\frac{{\mathrm{d}}^2}{{\mathrm{D}}^2}\right)}^{1/2}-\mathrm{D}\right]\\ =2\mathrm{\lambda }-\left[\mathrm{D}\left(1+\frac{{\mathrm{d}}^2}{2{\mathrm{D}}^2}\right)-\mathrm{D}\right]=2\mathrm{\lambda }-\left[\mathrm{D}+\frac{{\mathrm{d}}^2}{2\mathrm{D}}-\mathrm{D}\right]\\ (\mathrm{\mu }-1)\mathrm{t}=2\mathrm{\lambda }-\frac{{\mathrm{d}}^2}{2\mathrm{D}}\left(\frac{3}{2}-1\right)\mathrm{t}=2\times \mathrm{\lambda }-\frac{16{\mathrm{\lambda }}^2}{2\times 128\mathrm{\lambda }}\\ \frac{1}{2}\mathrm{t}=2\mathrm{\lambda }-\frac{\mathrm{\lambda }}{16}\frac{1}{2}\mathrm{t}=\frac{31\mathrm{\lambda }}{16}=\frac{31\mathrm{\lambda }}{8}\end{array}$
Thickness of mica sheet $\mathrm{t}=\frac{31\mathrm{\lambda }}{8}$