Question

Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are \(\beta_{G}, \beta_{R}\) and \(\beta_{B}\), respectively. Then, (A) \(\beta_{G}>\beta_{B}>\beta_{R}\) (B) \(\beta_{B}>\beta_{G}>\beta_{R}\) (C) \(\beta_{R}>\beta_{B}>\beta_{G}\) (D) \(\beta_{R}>\beta_{G}>\beta_{B}\)

Short answer

\(\beta_{R} > \beta_{G} > \beta_{B}\)

Step-by-step solution

Understanding Fringe Width Dependency

The fringe width \(\beta\) in Young's double slit experiment is directly proportional to the wavelength \(\lambda\) of the light used. The relationship can be expressed as \(\beta = \frac{\lambda D}{d}\), where \(D\) is the distance between the slits and the screen, and \(d\) is the distance between the two slits. Given that \(D\) and \(d\) are constants for a particular setup, \(\beta\) depends only on the wavelength \(\lambda\). This implies that a longer wavelength will result in a larger fringe width, and a shorter wavelength will result in a smaller fringe width.

Comparing Light Wavelengths

The visible spectrum of light has the following order of wavelengths from longest to shortest: red, green, blue. This means that red light has the longest wavelength, and blue light has the shortest.

Applying Wavelength Order to Fringe Width

Given the order of wavelengths \(\lambda_{R} > \lambda_{G} > \lambda_{B}\), and using the dependency of \(\beta\) on \(\lambda\), we can infer the order of the fringe widths to be \(\beta_{R} > \beta_{G} > \beta_{B}\). Therefore, the fringe width for red light will be the largest, followed by green, and then blue with the smallest fringe width.

Key concepts

Understanding Fringe Width

The phenomenon of interference in Young's double slit experiment creates a pattern of bright and dark bands known as fringes. The fringe width, commonly denoted as \(\beta\), is the distance between two consecutive bright (or dark) fringes. It's essential to know that the fringe width is not just some arbitrary measurement; it has profound implications in understanding wave interference and the behavior of light.

Imagine holding a ruler up to this pattern on a screen; \(\beta\) would be the distance you measure from the center of one bright fringe to the center of the next. It’s a crucial parameter that can tell us a lot about the light we're working with. For instance, if we keep the setup constant—meaning the distance between slits \(d\) and the distance from the slits to the screen \(D\) are unchanged—then any differences in \(\beta\) are a direct result of the light's characteristics, particularly its wavelength.

It’s much like how waves behave in water; longer waves have a more spread out pattern, while shorter waves result in a tighter one. In the context of this experiment, if we were to switch the light's color without altering the experimental setup, we would see a visible change in the pattern’s spacing, validating the importance of fringe width understanding.

Wavelength Dependency in Fringe Formation

The concept of wavelength dependency is pivotal when analyzing the results of Young's double slit experiment. Light waves, much like the ripples in a pond, have a characteristic known as wavelength, symbolized as \(\lambda\). But here's the fascinating part: this wavelength is the key factor determining the spacing of the interference fringes. To put it simply, the equation \(\beta = \frac{\lambda D}{d}\) tells us that the fringe width \(\beta\) is directly proportionate to the wavelength.

This means that the different colors of light, which correspond to different wavelengths, will produce different fringe widths even if the experiment setup doesn't change. Red light, with its longer wavelength, will produce wider spaced fringes compared to blue light, which has a shorter wavelength. Think of it like tuning a guitar; each string represents a different wavelength and, thus, produces a different note, or in our case, a different fringe pattern.

Visible Spectrum and Fringe Width Variation

Delving into the visible spectrum of light is like exploring a rainbow; each color represents a different wavelength. In terms of our daily lives, this spectrum gifts us the plethora of colors we perceive. When it comes to Young's double slit experiment, the visible spectrum reveals something significant about how we perceive light's interference pattern.

The spectrum ranges from red, with the longest wavelength, down to blue, with the shortest, and encompasses all those colors we're familiar with: orange, yellow, green, indigo, and violet in between. Because we now understand that fringe width varies with wavelength, performing the experiment with each different color of light will yield unique fringe widths. Red light will result in the broadest fringes, green in moderately spaced ones, and blue in the most narrowly spaced patterns. This progression is not only crucial to understanding how light works but also demystifies why we get the results we do in the double slit experiment.