Problem 101 - A Complete Resource Book in Chemistry for JEE Main

Question

A monoenergetic electron beam with a de Broglie wavelength of \(x \AA\) is accelerated till its wavelength is halved. By what factor is its kinetic energy changed? (a) 8 (b) 6 (c) 4 (d) 3

Step-by-Step Solution

Answer

The kinetic energy changes by a factor of 4.

Step 1: Understand the de Broglie Wavelength and Momentum Relationship

According to de Broglie's hypothesis, the wavelength \( \lambda \) is related to the momentum \( p \) of a particle by the formula: \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant. If the wavelength is halved, then the momentum becomes twice because \( p = \frac{h}{\lambda} \).

Step 2: Relate Momentum to Kinetic Energy

The kinetic energy \( K \) of a particle is given by \( K = \frac{p^2}{2m} \), where \( m \) is the mass of the particle and \( p \) is its momentum. Since the momentum has doubled (from Step 1), the kinetic energy changes to \( K' = \frac{(2p)^2}{2m} = \frac{4p^2}{2m} = 4 \times \frac{p^2}{2m} \).

Step 3: Calculate the Factor Change in Kinetic Energy

From Step 2, we found that the new kinetic energy \( K' = 4K \), meaning the kinetic energy has increased by a factor of 4 when the wavelength is halved.

Key Concepts

Understanding an Electron Beam

An electron beam is a stream of electrons that have been accelerated to high speeds, often through the application of an external electric field. When electrons move in a coherent and controlled direction, like in an electron beam, they can display properties similar to waves. This wave-particle duality is a fundamental concept in quantum mechanics and is best illustrated by de Broglie's hypothesis. According to this theory, every particle has a wavelength associated with it, determined by its momentum. In the case of an electron beam, this is particularly important because it helps us understand applications like electron microscopes.

• An electron beam is made up of electrons that move at high velocities.
• It displays wave properties, which leads to concepts like the de Broglie wavelength.
• Electron beams are used in technology and scientific research, including imaging techniques.

Electron beams reinforce the idea that particles can behave like waves under certain conditions, a cornerstone of quantum mechanics.

The Role of Kinetic Energy in Electron Beams

Kinetic energy refers to the energy that a particle, like an electron, possesses due to its motion. When an electron is part of an electron beam, it has kinetic energy that depends on its velocity and mass. In physics, kinetic energy is typically expressed by the formula \( K = \frac{1}{2}mv^2 \), but it can also be related to momentum using \( K = \frac{p^2}{2m} \). This reveals that as the momentum of an electron beam changes, so does its kinetic energy.

• The kinetic energy reflects how much energy the electrons have due to their movement.
• Accelerating an electron can increase its kinetic energy significantly.
• Understanding kinetic energy changes can help predict the effects of manipulating electron beams, like in the original exercise based on changing the wavelength.

In the context of the exercise, when the wavelength of the electron beam is halved, the electrons' momentum doubles, leading to an increase in kinetic energy by a factor of four.

Momentum and its Connection to Electron Beam Behavior

Momentum is a measure of the quantity of motion an object has, calculated as the product of mass and velocity. In quantum mechanics, it's a crucial concept because of its relationship with the de Broglie wavelength. A deeper dive into the relationship shows that as the de Broglie wavelength \( \lambda \) becomes smaller (or is halved), the momentum \( p \), which is inversely proportional (\( p = \frac{h}{\lambda} \)), of the particle increases.

• Momentum determines how much motion an electron carries through space.
• It's directly related to the wavelength associated with quantum particles.
• Changes in momentum affect kinetic energy, as highlighted by their quadratic relationship \( K = \frac{p^2}{2m} \).

The exercise illustrates that with the doubling of momentum, resulting from halving the wavelength, the kinetic energy increases significantly, showcasing the interplay between these fundamental concepts.