question_answer

If fringes width \[\lambda =5.89\times {{10}^{-5}}cm\] is 0.431 mm and shift of white central fringe on introducing a mica sheet in one path is 1.89 mm. Thickness of the mica sheet will be \[(\mu =1.59)\]

A)  \[4.38\times {{10}^{-6}}m\] 

B)  \[5.38\times {{10}^{-6}}m\]

C)  \[6.38\times {{10}^{-6}}m\]

D)  none of these

Correct Answer: A

Solution :

Key Idea: When mica sheet is introduced fringe width is not changed. When a mica sheet is introduced in the path of one of the two interfering beams, then entire fringe pattern is displaced towards the beam is the path of which plate is introduced, but fringe width is not changed.                 \[{{x}_{0}}=\frac{D}{d}\,(\mu -1)t\] ?. (i) Also fringe width is                 \[W=\frac{D\lambda }{d}\] \[\therefore \] \[\frac{W}{\lambda }=\frac{D}{d}\] ?. (ii) Using Eq. (ii) we get Eq. (i) as                 \[{{x}_{0}}=\frac{W}{\lambda }\,(\mu -1)\,t\] Given, \[{{x}_{0}}=1.89\times {{10}^{-3}}m,\,W=0.431\times {{10}^{-3}}m\], \[\mu =1.59,\,\lambda =5.89\times {{10}^{-7}}m\], \[1.89\times {{10}^{-3}}=\frac{0.431\times {{10}^{-3}}}{5.89\times {{10}^{-7}}}(1.59-1)\,t\] \[\Rightarrow \] \[t=\frac{5.89\times {{10}^{-7}}\times 1.89\times {{10}^{-3}}}{0.431\times 0.59\times {{10}^{-3}}}\] \[\Rightarrow \] \[t=4.38\times {{10}^{-6}}m\]