10.6 A beam of light consisting of two wavelengths, 650 nm, and 520 nm, is used to obtain interference fringes in Young’s double-slit experiment.

(a) Find the distance of the third bright fringe on the screen from the central maximum for wavelength 650 nm.

(b) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide?

(a)
Hint: \(x=n\lambda _{1}\left ( \frac{D}{d} \right )\)

Step 1: Find the distance of the third bright fringe on the screen from the central maximum.
The wavelength of the light beam,  = 650 nm
The wavelength of another light beam,  = 520 nm
The distance of the slits from the screen = D
Distance between the two slits = d
\(x=n\lambda _{1}\left ( \frac{D}{d} \right )\)
For the third bright fringe, n=3
\(\Rightarrow x=3\times 650\left ( \frac{D}{d} \right )=1950\left ( \frac{D}{d} \right )~nm.\)

(b)
Hint: \(x=n\lambda _{2}\left ( \frac{D}{d} \right )\)


Step 1: Equate the conditions for bright fringes.
\(n\lambda _{2}=\left ( n-1 \right )\lambda _{1}\)
\(\Rightarrow~520n=650n-650~\Rightarrow n=5 \)
Step 2: Find the least distance from the central maximum.
\(x=n\lambda _{2}\left ( \frac{D}{d} \right )=5\times520\left ( \frac{D}{d} \right )=2600\left ( \frac{D}{d} \right )~nm.\)