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, $I=a^2+a^2+2a^2\cos \varphi$
Here, $\varphi$ is the phase difference in the waves coming from the two slits.
So, $I=4a^2{\cos }^2\left(\frac{\varphi }{2}\right)$
$$\begin{gathered} \Rightarrow \frac{I}{I_{\max }}=\frac{1}{2} \\ \Rightarrow \frac{4a^2{\cos }^2\left({\displaystyle \frac{\varphi }{2}}\right)}{4a^2}=\frac{1}{2} \\ \Rightarrow {\cos }^2\left(\frac{\varphi }{2}\right)=\frac{1}{2} \\ \Rightarrow \cos \left(\frac{\varphi }{2}\right)=\frac{1}{\sqrt{2}} \\ \Rightarrow \frac{\mathrm{\varphi }}{2}=\frac{\mathrm{\pi }}{4} \\ \Rightarrow \varphi =\frac{\mathrm{\pi }}{2} \\ \mathrm{Corrosponding}\mathrm{path}\mathrm{difference},∆x=\frac{1}{4} \\ \Rightarrow y=\frac{∆xD}{d}=\frac{\lambda D}{4d} \end{gathered}$$
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 $\frac{D\lambda }{2d}$.