Question

Obtain the formula for the width of central maxima from the experiment of diffraction of light through a single slit and draw the diagram of the intensity distribution of the light obtained on the screen.

Hint:

In single slit diffraction, the angular width of the central maximum is inversely proportional to the slit width. For wider slits, the central maximum becomes narrower.

Answer 1

Diffraction through a Single Slit:
In the case of diffraction of light through a single slit, the central maximum is formed on the screen due to constructive and destructive interference. The angular width of the central maximum is given by the angle \( \theta \) for the first minimum. The condition for the first minimum is:
\[ a \sin \theta = m \lambda, m = \pm 1, \pm 2, \pm 3, \dots \] Where:
- \( a \) is the width of the slit,
- \( \lambda \) is the wavelength of the light,
- \( m \) is the order of the minima.
For the central maximum, we consider the case of \( m = \pm 1 \), and the angular width of the central maximum is the difference in angles between the first minima on either side of the central maximum:
\[ \theta_1 = \sin^{-1} \left( \frac{\lambda}{a} \right). \] The angular width of the central maximum is twice the angle for the first minimum:
\[ \Delta \theta = 2 \theta_1 = 2 \sin^{-1} \left( \frac{\lambda}{a} \right). \] For small angles (which is usually the case in most practical diffraction experiments), \( \sin \theta \approx \theta \), so the formula for the angular width becomes approximately:
\[ \Delta \theta \approx \frac{2\lambda}{a}. \] This formula gives the angular width of the central maximum. The linear width \( W \) on the screen at a distance \( L \) from the slit is given by:
\[ W = L \cdot \Delta \theta = \frac{2L \lambda}{a}. \] Thus, the width of the central maxima on the screen is \( W = \frac{2L \lambda}{a} \).
The intensity distribution of light through a single slit diffraction experiment shows a central bright fringe (central maximum) with diminishing intensity for subsequent maxima and minima.