Question

White light is used in a Young's double slit experiment. Find the minimum order of the violet fringe $\left(\lambda =400\mathrm{nm}\right)$ which overlaps with a red fringe $\left(\lambda =700\mathrm{nm}\right)$.

Answer 1

Let the separation between the slits be d and distance between screen from the slits be D.
Suppose, the m^th bright fringe of violet light overlaps with the n^th bright fringe of red light.
Now, the position of the m^th bright fringe of violet light, y_v = $\frac{m{\lambda }_{v}D}{d}$
Position of the n^th bright fringe of red light, y_r = $\frac{n{\lambda }_{r}D}{d}$
For overlapping, y_{v =} y_r .
So, as per the question,
$$\begin{gathered} \frac{m\times 400\times \mathrm{D}}{d}=\frac{n\times 700\times \mathrm{D}}{d} \\ \Rightarrow \frac{m}{n}=\frac{7}{4} \end{gathered}$$
Therefore, the $7^{\mathrm{th}}$ bright fringe of violet light overlaps with the 4^th bright fringe of red light.
It can also be seen that the 14^th violet fringe will overlap with the 8^th red fringe.
Because, $\frac{m}{n}=\frac{7}{4}=\frac{14}{8}$