Question

A parallel beam of light of wavelength $500{\text{ }}nm$ falls on a narrow slit and the resulting diffraction pattern is observed on a screen $1{\text{ }}m$ away. It is observed that the first minimum is at a distance of $2.5mm$ from the centre of the screen. Find the width of the slit.

Answer 1

In order to solve this question study the interference and diffraction of the light wave through the single slit experiment. Take all the given data in SI units. Remember for the first minimum n is one.

Formula Used:
$n\lambda = \dfrac{{xd}}{D}$
Here, $n$ is the number of the minimum
$\lambda$ is the wavelength of the light
$x$ is the distance of the first minimum from the centre of the screen
$d$ is the width of the slit
$D$ is the distance Between the slit and the screen

Complete step by step answer:
It is given that the Wavelength of light beam is $500nm$ , $\lambda = 500nm$
Distance of the screen from the slit is given as $1m$ , $D = 1m$
For first minima, $n = 1$
Distance between the slits is d
Distance of the first minimum from the centre of the screen can be obtained as,
$x = 2.5mm \\\ \Rightarrow x = 2.5 \times {10^{ - 3}} \\\$

Now,
$n\lambda = \dfrac{{xd}}{D} \\\ \Rightarrow d = \dfrac{{n\lambda D}}{x} \\\ \Rightarrow d = \dfrac{{500 \times {{10}^{ - 9}}}}{{2.5 \times {{10}^{ - 3}}}} \\\ \therefore d = 0.2mm \\\$
Therefore, the width of the slits is $0.2mm$.

Additional Information: The difference between interference and diffraction has not been defined yet satisfactorily by anyone. It is just a question of usage, and no specific, important physical difference between them. The best we can do is, roughly speaking, is to say that when there are few sources, say two interfering sources, then the result is usually called interference, but if there is a large number of them, it seems that word diffraction is often used.

Note: Diffraction is a general characteristic exhibited by all types of waves, be it light waves, sound waves, or matter waves. Since the wavelength of light is much smaller than the dimensions of most obstacles; we do not encounter diffraction effects of light in everyday observations. However, the finite resolution of our eye or of optical instruments such as telescopes or microscopes is limited due to the phenomenon of diffraction. Indeed the colours seen when a CD is viewed is due to diffraction effects.