Fresnel's Biprism and Interference Fringe Width

In the fascinating world of wave optics, Fresnel's biprism is a clever device used to produce two coherent sources from a single light source, leading to an observable interference pattern. This experiment is conceptually similar to Young's Double Slit Experiment in terms of the factors affecting the interference fringes.

Understanding Fringe Width

The interference pattern observed consists of alternating bright and dark bands, known as fringes. The distance between two consecutive bright fringes or two consecutive dark fringes is called the fringe width. For an interference pattern produced by two coherent sources, the fringe width (often denoted by $\beta$) is given by the formula:

\[ \beta = \frac{\lambda D}{d} \]

Let's break down what each term in this important formula represents:

• \(\lambda\) (lambda): This is the wavelength of the light used in the experiment.
• \(D\): This represents the distance from the coherent sources (or effective slits in the case of Fresnel's biprism) to the screen where the interference pattern is observed.
• \(d\): This is the distance between the two coherent sources (or effective slits). In Fresnel's biprism, 'd' is the effective separation between the two virtual images of the original source formed by the biprism.

Slit Distance and Fringe Width Relationship

From the fringe width formula, \(\beta = \frac{\lambda D}{d}\), we can clearly see the relationship between fringe width and the distance between the sources. The fringe width \(\beta\) is inversely proportional to the distance between the coherent sources \(d\). This means:

• If \(d\) increases, \(\beta\) decreases.
• If \(d\) decreases, \(\beta\) increases.

The question states that the "distance between the slits is gradually increased." In the context of Fresnel's biprism experiment, this refers to increasing the effective separation \(d\) between the two coherent sources. As per the inverse proportionality, if \(d\) increases, the fringe width \(\beta\) must decrease.

Conclusion on Fringe Pattern

Therefore, when the distance between the slits (or coherent sources) in Fresnel's biprism experiment is gradually increased, the fringe width in the interference fringe pattern obtained will decrease. This results in the fringes appearing closer together on the screen.