Question

The line-width of a bright fringe is sometimes defined as the separation between the points on the two sides of the central line where the intensity falls to half the maximum. Find the line-width of a bright fringe in a Young's double slit experiment in terms of \[\lambda,\] d and D where the symbols have their usual meanings.

Answer 1

Given:-

Separation between two slits = d

Wavelength of the light = \[\lambda\]

Distance of the screen = D

Let I_max be the maximum intensity and I be half the maximum intensity at a point at a distance y from the central point.

So, \[I =  a^2  +  a^2  + 2 a^2 \cos\phi\]

Here, \[\phi\] is the phase difference in the waves coming from the two slits.

So, \[I = 4 a^2  \cos^2 \left( \frac{\phi}{2} \right)\]

\[\Rightarrow \frac{I}{I_\max} = \frac{1}{2}\]

\[ \Rightarrow \frac{4 a^2 \cos^2 \left( \frac{\phi}{2} \right)}{4 a^2} = \frac{1}{2}\]

\[ \Rightarrow \cos^2 \left( \frac{\phi}{2} \right) = \frac{1}{2}\]

\[ \Rightarrow \cos\left( \frac{\phi}{2} \right) = \frac{1}{\sqrt{2}}\]

\[ \Rightarrow \frac{\phi}{2} = \frac{\pi}{4}\]

\[ \Rightarrow \phi = \frac{\pi}{2}\]

Corrosponding path difference, \[∆ x = \frac{1}{4}\]

\[ \Rightarrow y = \frac{∆ xD}{d} = \frac{\lambda D}{4d}\]

The line-width of a bright fringe is defined as the separation between the points on the two sides of the central line where the intensity falls to half the maximum.

So, line-width = 2y

\[= 2\frac{D\lambda}{4d} = \frac{D\lambda}{2d}\]

Thus, the required line width of the bright fringe is \[\frac{D\lambda}{2d}.\]