Question

What is interference of light? Derive an expression for the fringe width in the Young’s double slit experiment.

Hint:

In Young's double slit experiment, the fringe width is directly proportional to the wavelength of the light and the distance between the slits and the screen, and inversely proportional to the separation between the slits.

Answer 1

Step 1: Understanding Interference of Light.
Interference is the phenomenon in which two or more light waves superpose to form a resultant wave of greater or lesser amplitude. This occurs when light from coherent sources (sources having a constant phase relationship) meets and combines.
There are two types of interference: - **Constructive interference**: When the crest of one wave coincides with the crest of another wave, resulting in a brighter light. - **Destructive interference**: When the crest of one wave coincides with the trough of another wave, resulting in darkness or no light.
Step 2: Young’s Double Slit Experiment.
In Young's double slit experiment, light from a monochromatic source is passed through two narrow slits. These slits act as coherent sources of light. The light waves emerging from the slits interfere with each other, producing a pattern of bright and dark fringes on a screen placed at some distance from the slits.
The fringe width (distance between two consecutive bright or dark fringes) is given by the formula: \[ \beta = \dfrac{\lambda D}{d} \] where: - \( \beta \) is the fringe width (distance between two consecutive bright or dark fringes), - \( \lambda \) is the wavelength of the light, - \( D \) is the distance between the slits and the screen, - \( d \) is the separation between the two slits.
Step 3: Derivation of Fringe Width.
Consider the two slits separated by a distance \( d \), and light of wavelength \( \lambda \) is incident on them. The condition for constructive interference (bright fringe) is: \[ \Delta \text{path} = m\lambda \] where \( m \) is an integer (order of the fringe). The condition for destructive interference (dark fringe) is: \[ \Delta \text{path} = (m + \frac{1}{2})\lambda \] The fringe width is the distance between two consecutive maxima (or minima), and it is calculated using the formula: \[ \beta = \dfrac{\lambda D}{d} \] Step 4: Conclusion.
The fringe width in Young’s double slit experiment is given by: \[ \beta = \dfrac{\lambda D}{d} \]