What happens to the fringe width when we place a slab of refractive i​ndex of r?

When a slab of refractive index \( r \) is introduced into an interference pattern, such as that created by a double-slit experiment, it affects the fringe width in a specific way. To understand this, let's break down the concepts involved.

The Basics of Fringe Width

Fringe width, often denoted as \( \beta \), is the distance between two consecutive bright or dark fringes in an interference pattern. For a typical double-slit experiment, the fringe width can be calculated using the formula:

\( \beta = \frac{\lambda D}{d} \)

• \( \lambda \) is the wavelength of the light used.
• \( D \) is the distance from the slits to the screen.
• \( d \) is the distance between the two slits.

Effect of the Slab on Wavelength

When a slab of refractive index \( r \) is placed in the path of the light waves, it alters the effective wavelength of the light within the medium. The new wavelength \( \lambda' \) in the medium can be expressed as:

\( \lambda' = \frac{\lambda}{r} \)

This means that the wavelength decreases when the light travels through the slab, as the refractive index \( r \) is greater than 1 for any medium other than vacuum or air.

Calculating the New Fringe Width

With the new wavelength in mind, we can now find the new fringe width \( \beta' \) when the slab is introduced:

\( \beta' = \frac{\lambda' D}{d} = \frac{\left(\frac{\lambda}{r}\right) D}{d} = \frac{\lambda D}{rd} \)

From this equation, it is clear that the fringe width is inversely proportional to the refractive index \( r \). As \( r \) increases, the fringe width \( \beta' \) decreases.

Visualizing the Change

To visualize this, imagine a scenario where you have a series of alternating bright and dark lines on a screen. If you introduce a slab with a higher refractive index, the spacing between these lines becomes narrower. This is akin to compressing a spring: as you apply pressure (in this case, the refractive index), the coils come closer together, resulting in a smaller distance between them.

Summary of Effects

• Introducing a slab of refractive index \( r \) reduces the effective wavelength of light.
• The fringe width decreases as a result, following the relationship \( \beta' = \frac{\beta}{r} \).
• This change can be visualized as a compression of the interference pattern.

In essence, placing a slab with a refractive index \( r \) in the path of light alters the interference pattern by decreasing the fringe width, leading to a more closely spaced arrangement of bright and dark fringes on the screen.