Problem 21 - Master Resource Book in JEE Main Physics

Question

A beam of electrons is used in a YDSE experiment to slit width is \(d\), when the velocity of electrons is increased, then (a) no interference is observed (b) fringe width increases (c) fringe width decreases (d) fringe width remains same

Step-by-Step Solution

Answer

(c) fringe width decreases.

Step 1: Understanding the YDSE Concept

In a Young's Double Slit Experiment (YDSE), when particles such as electrons pass through two slits, they create an interference pattern on a screen. This is characterized by a series of bright and dark fringes due to constructive and destructive interference of waves.

Step 2: Relating Velocity to Wavelength

The wavelength of particles like electrons is given by the de Broglie wavelength equation, \[\lambda = \frac{h}{mv}\]where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the electron's mass, and \( v \) is the velocity of the electrons. Increasing the velocity \( v \) leads to a decrease in \( \lambda \).

Step 3: Determining Fringe Width Change

In YDSE, the fringe width \( \beta \) of the interference pattern is given by,\[\beta = \frac{\lambda D}{d}\]where \( \beta \) is the fringe width, \( D \) is the distance to the screen, and \( d \) is the slit separation. A decrease in \( \lambda \) due to increased velocity results in a decrease in \( \beta \).

Step 4: Conclusion on Fringe Width

Since an increase in the velocity of electrons causes a decrease in their wavelength, and hence a decrease in fringe width, the correct option is (c) fringe width decreases.

Key Concepts

de Broglie Wavelength

The de Broglie wavelength is an essential concept in quantum mechanics that connects the behavior of particles with wave-like properties. It was proposed by physicist Louis de Broglie, who suggested that particles such as electrons exhibit wave characteristics. This dual nature allows particles to be described in terms of wavelength using the de Broglie wavelength equation:

• \( \lambda = \frac{h}{mv} \)
• where \( \lambda \) is the wavelength, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{m}^2 \text{kg/s}) \),
• \( m \) is the mass of the particle, and \( v \) is its velocity.

What makes this idea particularly interesting is how it unifies the properties of particles and waves. As the velocity of a particle increases, its de Broglie wavelength decreases. This has practical implications in experiments, such as electron diffraction, where the wave-like nature of electrons becomes evident.

Interference Pattern

Interference patterns occur when two or more waves overlap and combine, producing regions of constructive and destructive interference. In the context of the Young's Double Slit Experiment, this happens when particles pass through two closely spaced slits, leading to an observable pattern of alternating bright and dark lines (fringes) on the observation screen.

• Constructive interference occurs where the waves are in phase, leading to brighter fringes.
• Destructive interference appears where the waves are out of phase, causing darker fringes.

This pattern demonstrates the wave-like behavior of particles, confirming the wave theory of light and extending it to matter waves through experiments with electrons.

Fringe Width

Fringe width is a critical factor in analyzing interference patterns, as it denotes the distance between two consecutive bright or dark fringes. This feature depends on several variables:

• The wavelength \( \lambda \) of the waves (or particles).
• The distance \( D \) from the slits to the screen.
• The slit separation \( d \).

The formula for fringe width is: \[ \beta = \frac{\lambda D}{d} \], where \( \beta \) is the fringe width. As the de Broglie wavelength of particles decreases due to higher velocities, fringe width correspondingly decreases, leading to narrower spaces between fringes on the screen.

Electron Beam

An electron beam consists of a stream of electrons directed towards a target or through an arrangement, such as the one in Young's Double Slit Experiment. Electron beams are particularly useful in demonstrating wave-like properties of matter because electrons have small masses, allowing their de Broglie wavelengths to be significant. This wave property of electron beams is harnessed in technologies like electron microscopes, offering high-resolution imaging by exploiting the interference patterns created when electrons pass through specimens. When electrons pass through the two slits in a YDSE setup, they form an interference pattern akin to light waves, reinforcing the dual nature of particles like electrons.

Constructive and Destructive Interference

Interference patterns are composed of constructive and destructive interference. These phenomena occur when waves combine or cancel each other out.

• Constructive interference: Happens when waves are in phase, meaning their peaks align, leading to increased amplitude and brightness in fringes.
• Destructive interference: Occurs when waves are out of phase, with peaks aligning with troughs, cancelling amplitudes and creating dark fringes.

The regular alternation of these interferences in the Young's Double Slit Experiment effectively demonstrates the principle of superposition, where the net result at any point is the sum of individual wave effects. Understanding these principles is crucial in analyzing not just simple wave experiments, but also complex quantum systems.