Problem 84 - A Complete Resource Book in Chemistry for JEE Main

Question

The mass of an electron is \(\mathrm{m}\). Its charge is e and it is accelerated from rest through a potential difference \(\mathrm{V}\). The velocity acquired by the electron will be (a) \(\sqrt{\mathrm{V} / \mathrm{m}}\) (b) \(\mathrm{VeV} / \mathrm{m}\) (c) \(\sqrt{2 \mathrm{e}} \mathrm{V} / \mathrm{m}\) (d) none

Step-by-Step Solution

Answer

Option (c): \( \sqrt{\frac{2eV}{m}} \) is the correct choice.

Step 1: Identify the Given Variables

The problem provides the mass of the electron, denoted by \( m \), its charge, \( e \), and the potential difference, \( V \). We are asked to find the velocity acquired by the electron when it is accelerated from rest through this potential difference.

Step 2: Use the Energy Conservation Principle

As the electron is accelerated from rest, the gain in its kinetic energy equals the work done by the electric field, which can be expressed by the formula: \( \text{Kinetic Energy} = \text{Potential Energy} \). In equations, this is \( \frac{1}{2} m v^2 = eV \).

Step 3: Solve for Velocity

Rearrange the energy conservation equation to solve for \( v \). \[ \frac{1}{2} m v^2 = eV \Rightarrow v^2 = \frac{2eV}{m} \Rightarrow v = \sqrt{\frac{2eV}{m}} \] This gives us the velocity of the electron.

Step 4: Check Answer Against Options

Compare the expression \( \sqrt{\frac{2eV}{m}} \) with the given options. Option (c): \( \sqrt{2e} V / m \) simplifies to \( \sqrt{\frac{2eV}{m}} \), which matches our derived expression.

Key Concepts

Energy Conservation

Energy conservation is a fundamental principle in physics. It states that the total energy in a closed system remains constant over time. When an electron accelerates through a potential difference, it illustrates this principle beautifully. Initially, the electron is at rest with potential energy due to the electric field created by the potential difference. As it moves, this potential energy is converted into kinetic energy.The law of energy conservation can be expressed as:

• Total Initial Energy = Total Final Energy
• Potential Energy = Kinetic Energy

In mathematical terms, when an electron is accelerated from rest, the equation becomes:\[ \frac{1}{2}mv^2 = eV \]This tells us that the work done on the electron (represented as potential energy) transforms into kinetic energy. This concept helps us calculate the velocity of the electron. Understanding how one form of energy changes to another is critical in solving physics problems.

Kinetic Energy

Kinetic energy is the energy a body possesses due to its motion. For an electron that has gained speed while moving through an electric potential, kinetic energy can be computed using:\[ \text{Kinetic Energy} = \frac{1}{2}mv^2 \]Here, \( m \) is the mass of the electron, and \( v \) is its velocity. When the electron accelerates, it starts transforming its potential energy into kinetic energy. The more energy it gains, the faster it moves.The relationship between kinetic energy and velocity is quadratic. This means that even a small increase in the electron's velocity can lead to a significant increase in its kinetic energy. This quadratic relationship is why kinetic energy becomes an essential consideration when analyzing moving particles like electrons.

Potential Energy

Potential energy in the context of an electron in an electric field is the energy held by the electron due to its position. As the electron is accelerated from rest, its potential energy is directly related to the potential difference it experiences. This potential difference, \( V \), can be thought of as the energy per unit charge available for acceleration.The expression for potential energy for a charge \( e \) within an electric field is:\[ \text{Potential Energy} = eV \]This relationship shows that the potential energy depends on both the electron's charge and the electric potential difference. This energy will eventually transform into kinetic energy as the electron speeds up. Thus, when understanding electric fields, potential energy is a vital factor that represents the potential work that can be used to move the charge.

Mass of Electron

The mass of an electron is a fundamental property that affects its motion under the influence of forces. Characteristically, electrons have a very small mass, approximately \( 9.11 \times 10^{-31} \) kilograms. Despite its small magnitude, an electron's mass is crucial in determining how it will respond to forces, such as when accelerated by a potential difference.To find the velocity of an electron, the mass is used in calculations involving kinetic energy:\[ \frac{1}{2}mv^2 = eV \]Here, \( m \) represents the electron's mass and influences how much kinetic energy is gained for a given amount of work done on it. The smaller the mass, the greater the acceleration it can achieve for the same energy input. This principle is key in many applications within electronic and atomic physics, where electrons play a central role.