Question

A plate of thickness \[t\] made of material of refractive index\[\mu \] is placed in front of one of the slits in a double-slit experiment. What should be the minimum thickness \[t\] which will make the intensity at the center of the fringe pattern zero?
\[(A)\left( {\mu - 1} \right)\dfrac{\lambda }{2}\]
\[(B)\left( {\mu - 1} \right)\lambda \]
\[(C)\dfrac{\lambda }{{2\left( {\mu - 1} \right)}}\]
\[(D)\]\[\dfrac{\lambda }{{\left( {\mu - 1} \right)}}\]

Answer 1

Hint:Because the wavelength of glass is shorter than that of air, the phases of the two emerging waves will differ if one slit is covered by a glass plate (right).
The relative phase of the two waves determines interference. It also depends on how far apart they are on their respective paths.
The intensity at a given point is proportional to the resultant intensity at that location, the square of the generated electric field.

Complete step-by-step solution:
Given ,
The thickness of the plate\[ = {\text{ }}t\]
Refractive index \[ = \mu \]
Path difference = \[\left( {m + \dfrac{1}{2}} \right)\dfrac{\lambda }{2}\]
path difference = \[\left( {\mu - 1} \right)t\]
Now, for intensity to be zero ,
\[\left( {\mu - 1} \right)t\]=\[\left( {n + \dfrac{1}{2}} \right)\dfrac{\lambda }{2}\]
For minimum thickness \[n = 0\]
So,
\[t = \dfrac{\lambda }{{2\left( {\mu - 1} \right)}}\]
Hence, the right answer is in option(C).

Note:The two coherent sources of waves produced by double slits interfere. Because the slots are tiny, light spreads out (diffracts) from each one. These waves collide and interfere in both a constructive (bright lines) and destructive (dark lines) way (dark regions).
Waves have another attribute besides interference: diffraction, which is the bending of waves as they pass through objects or an aperture. Huygens' principle can be used to explain the phenomenon of diffraction.
The phase difference between the two waves is represented by the interference pattern, with maxima occurring when the phase difference is a multiple of two. The maxima are four times as bright as the individual beams, while the minima have no intensity if the two beams are of equal intensity.