Question

(a) Why are coherent sources necessary to produce a sustained interference pattern?
(b) In Young's double slit experiment using Monochromatic light of wavelength k, the intensity of light at a point on the screen where the path difference is \[\lambda \] is k units. Find out the intensity of light at a point where path difference is \[\dfrac{\lambda }{3}\].

Answer 1

Hint: Interference is the phenomena of light. It is found by two coherent sources when they emit light continuously with the same frequencies.

Complete step by step answer:
(a). 1. To produce a sustained interference, coherently they cannot emit light waves continuously.
2. In Interference phenomena, the coherent source should be independent, because independent sources emit the waves, which don't have the same phase or a constant phase difference.

(b) It is given by Young's double slit experiment the wavelength of monochromatic light is k.
Path difference \[=\lambda \]
In Interference pattern, the Intensity I at a
Point is given by,
\[\text{I}={{\text{I}}_{0}}{{\cos }^{2}}\left( \dfrac{\pi }{\lambda } \right)x\]
Where
\[\begin{align}
  & x=\text{Path Difference} \\
 & \lambda =\text{wave length} \\
 & {{\text{I}}_{0}}=\text{Intensity of coherent bright fringe} \\
 & \text{where x}=\lambda ,\text{I}=\text{k} \\
\end{align}\]
\[\Rightarrow \text{k}={{\text{I}}_{\text{0}}}\text{co}{{\text{s}}^{\text{2}}}\left( \dfrac{\pi }{\lambda } \right)={{\text{I}}_{\text{0}}}\text{co}{{\text{s}}^{\text{2}}}\text{ }\!\!\pi\!\!\text{ }\]
\[\Rightarrow \text{k}={{\text{I}}_{\text{0}}}\]
For \[x=\dfrac{\lambda }{3},\text{I}=\text{I }\!\!'\!\!\text{ }\]
\[\text{I}'={{\text{I}}_{0}}{{\cos }^{2}}\left( \dfrac{\pi }{\lambda } \right)x\]
\[\Rightarrow \text{I}'=k{{\cos }^{2}}\left( \dfrac{\pi }{\lambda } \right)\dfrac{\lambda }{3}=k{{\cos }^{2}}\left( \dfrac{\lambda }{3} \right)\]
\[\Rightarrow I'=k{{\left( \dfrac{1}{2} \right)}^{2}}=\dfrac{k}{4}\]

Note:
Coherent sources are mandatory for observable interference patterns because the rate of change of phase at a given point is constant for both the sources and hence maxima and minimum.