Question

White coherent light (400 nm-700 nm) is sent through the slits of a Young's double slit experiment (figure 17-E3). The separation between the slits is 0⋅5 mm and the screen is 50 cm away from the slits. There is a hole in the screen at a point 1⋅0 mm away (along the width of the fringes) from the central line. (a) Which wavelength(s) will be absent in the light coming from the hole? (b) Which wavelength(s) will have a strong intensity?

Figure

Answer 1

Given: Separation between two slits,
$d=0.5\mathrm{mm}=0.5\times {10}^{-3}\mathrm{m}$
Wavelength of the light, $\lambda =400\mathrm{nm}\mathrm{to}700\mathrm{nm}$
Distance of the screen from the slit, $D=50\mathrm{cm}=0.5\mathrm{m}$,
Position of hole on the screen, $y_n=1\mathrm{mm}=1\times {10}^{-3}\mathrm{m}$

(a) The wavelength(s) will be absent in the light coming from the hole, which will form a dark fringe at the position of hole.
$y_n=\frac{\left(2n+1\right){\lambda }_n}{2}\frac{D}{d}$ , where n = 0, 1, 2, ...

$$\begin{gathered} \Rightarrow {\lambda }_n=\frac{2}{\left(2n+1\right)}\frac{y_{n}d}{D} \\ =\frac{2}{\left(2n+1\right)}\times \frac{{10}^{-3}\times 0.05\times {10}^{-3}}{0.5} \\ =\frac{2}{\left(2n+1\right)}\times {10}^{-6}\mathrm{m} \\ =\frac{2}{\left(2n+1\right)}\times {10}^3\mathrm{nm} \end{gathered}$$

$$\begin{gathered} \mathrm{For}n=1, \\ {\lambda }_1=\left(\frac{2}{3}\right)\times 1000=667\mathrm{nm} \\ \mathrm{For}n=2, \\ {\lambda }_2=\left(\frac{2}{5}\right)\times 1000=400\mathrm{nm} \end{gathered}$$
Thus, the light waves of wavelength 400 nm and 667 nm will be absent from the light coming from the hole.

(b) The wavelength(s) will have a strong intensity, which will form a bright fringe at the position of the hole.
$$\begin{gathered} \mathrm{So},y_n=n{\lambda }_n\frac{D}{d} \\ \Rightarrow {\lambda }_n=y_n\frac{d}{nD} \\ \mathrm{For}n=1, \\ {\lambda }_1=y_n\frac{d}{D} \\ ={10}^{-3}\times \left(0.5\right)\times \frac{{10}^{-3}}{0.5} \\ ={10}^{-6}\mathrm{m}=1000\mathrm{nm}. \end{gathered}$$

But 1000 nm does not fall in the range 400 nm − 700 nm.

$$\begin{gathered} \mathrm{Again},\mathrm{for}n=2, \\ {\lambda }_2=y_n\frac{d}{2D}=500\mathrm{nm} \end{gathered}$$
So, the light of wavelength 500 nm will have strong intensity.