Question

A beam of light consisting of two wavelengths, $$650nm$$ and $$520nm$$,is used to obtain interference fringes in a Young double slit experiment.
(a). Find the distance of the third bright fringe on the screen from the central maximum for wavelength $$650nm$$.
(b).What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide? The distance between the two slits is $$0.28nm$$ and the screen is at a distance of $$1.4m$$ from the slits.

Answer 1

Hint:To solve this question we have to know about fringes. We know that fringes are the bright or dark bands which are caused by beams of light which are in phase or out of phase.Interference is what happens when two or more waves meet each other.Depending on the alignment of peaks and troughs of the overlapping waves, they might add up, or they can partially or entirely cancel each other.

Complete step by step answer:
We can say, the wavelength of the light is $${\lambda }_1=650nm$$.
Again, the wavelength of second light, $${\lambda }_2=520nm$$.
Distance between the slit and the screen according to this question is, $$1.4m$$.
Distance between the slits is $$0.28nm$$.

(a) We know that, the relation between the $$n^{th}$$ bright fringe and the width of fringe is, $$x=n{\lambda }_1{\displaystyle \frac{D}{d}}$$
Here n is the number of bright fringe
Here the value of n is $$3$$. So,
$$\begin{gathered} x=3\times 650{\displaystyle \frac{1.4}{0.28\times {10}^{-3}}} \\ \Rightarrow x=1950\times 5\times {10}^{3}nm \end{gathered}$$
$$\therefore x=9.75mm$$

(b) We can now consider that $$n^{th}$$ bright fringe of $${\lambda }_2$$ and the $$(n-1{)}^{th}$$ bright fringe of wavelength $${\lambda }_1$$ coincide with each other.
$$n{\lambda }_2=(n-1){\lambda }_1$$
After putting the values
$$\begin{gathered} 520n=650n-650 \\ \Rightarrow 650=130n \\ \Rightarrow n=5 \end{gathered}$$
Therefore we can say, the least distance from the central maximum can be obtained as.
$$\begin{gathered} x^{\prime }=n{\lambda }_2{\displaystyle \frac{D}{d}} \\ \Rightarrow x^{\prime }=5\times 520{\displaystyle \frac{D}{d}}=2600{\displaystyle \frac{1.4}{0.28\times {10}^{-3}}}nm \end{gathered}$$
$$\begin{gathered} \Rightarrow x^{\prime }=1.30\times {10}^{-2}m \\ \therefore x^{\prime }=1.3cm \end{gathered}$$
This is the right answer.

Note: We know that, for a large wavelength, a large path difference is needed to have a change of phase, then the distance between fringes gets larger. We know that, if the screen is farther for a fixed angle the distance between two fringes gets larger.