Question

What is diffraction of light? Explain diffraction at a single slit and deduce the expression for the width of its central maxima.

Hint:

In a single-slit diffraction pattern, the width of the central maxima depends on the wavelength of the light and the width of the slit. Larger slits result in narrower central maxima.

Answer 1

Step 1: What is diffraction of light?
Diffraction of light refers to the bending or spreading of light waves around obstacles or through small openings. This phenomenon occurs when the wavelength of the light is comparable to the size of the obstacle or aperture. Diffraction causes the light to spread out and form patterns of dark and light regions, called interference patterns.
Step 2: Diffraction at a single slit.
When a monochromatic light source passes through a single narrow slit, the light diffracts and forms a pattern on a screen placed behind the slit. This diffraction pattern consists of a central bright fringe (central maxima) and a series of dimmer fringes on either side (secondary maxima and minima).
Step 3: Condition for minima.
The angular positions of the minima in the diffraction pattern are given by the condition: \[ a \sin \theta = n \lambda \] where:
- \( a \) is the width of the slit,
- \( \theta \) is the angle of diffraction,
- \( \lambda \) is the wavelength of the light,
- \( n \) is the order of the minima (for the first minima, \( n = 1 \), for the second minima, \( n = 2 \), and so on).
Step 4: Expression for the width of the central maxima.
The central maxima is the region between the first minima on both sides. The angular width \( \Delta \theta \) of the central maxima is given by: \[ \Delta \theta = 2 \theta_1 \] where \( \theta_1 \) is the angle of the first minima. Using the condition for minima, we have: \[ a \sin \theta_1 = \lambda \] For small angles, \( \sin \theta \approx \theta \), so: \[ a \theta_1 = \lambda \] Thus, the angular width of the central maxima is: \[ \Delta \theta = 2 \times \frac{\lambda}{a} \] Step 5: Conclusion.
The angular width of the central maxima in a single-slit diffraction pattern is \( \Delta \theta = \frac{2\lambda}{a} \), where \( \lambda \) is the wavelength of the light and \( a \) is the width of the slit.