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Chapter 19: Problem 41

Assertion A charged particle moves perpendicular to a magnetic field. Its kinetic energy remains constant, but momentum changes. Reason Force acts on the moving charged particles in the magnetic field.

Short Answer

Expert verified
Assertion and reason are correct; the reason explains the assertion.

Step by step solution

01

Understanding the Assertion

The assertion states that a charged particle moving perpendicular to a magnetic field experiences constant kinetic energy despite changes in momentum. To evaluate this, we consider kinetic energy: \( KE = \frac{1}{2}mv^2 \) and momentum which is \( p = mv \). As the particle moves in a circular path due to the magnetic force, its speed (and hence kinetic energy) does not change, but the direction of velocity changes, indicating a change in momentum.
02

Analyzing the Reason

The reason provided states that a force acts on the moving charged particle within the magnetic field. This force is given by the Lorentz force, \( \vec{F} = q\vec{v} \times \vec{B} \), where \( q \) is the charge, \( \vec{v} \) the velocity, and \( \vec{B} \) the magnetic field. Since the force is perpendicular to the velocity, it alters the particle's direction without doing work on it, thus the speed remains constant.
03

Linking Assertion and Reason

The assertion is true because when a charged particle moves perpendicular to a magnetic field, the magnetic force doesn't change the particle's speed, keeping kinetic energy constant. The reason supports this by explaining the presence of a perpendicular force, which changes momentum direction but not magnitude. Hence, both assertion and reason are correct and the reason correctly explains the assertion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy
Kinetic energy is a fundamental concept in physics that describes the energy a particle possesses due to its motion. For a charged particle:
  • The kinetic energy is calculated as \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the particle.
  • This indicates that kinetic energy depends solely on the speed of the particle.
  • In a magnetic field, if a charged particle moves perpendicular to it, the speed remains constant.
Hence, the kinetic energy is unaffected even though the particle's path might change direction. This means that magnetic fields do not perform work on the charged particle, allowing its speed and thus kinetic energy to remain unchanged despite changes in the trajectory.
Understanding this principle helps clarify why particles in magnetic fields can maintain their energy levels unless other external forces are involved.
Lorentz Force
The Lorentz force is central to the behavior of charged particles in magnetic fields. This force is described by the equation \( \vec{F} = q\vec{v} \times \vec{B} \):
  • Here, \( q \) represents the charge of the particle, \( \vec{v} \) the velocity vector, and \( \vec{B} \) the magnetic field vector.
  • The cross product signifies that the force is always perpendicular to both the velocity and the magnetic field.
  • This perpendicular force results in a change of direction for the particle's velocity, but not its speed.
Due to this perpendicular nature, the Lorentz force causes the particle to move in a circular path in the plane perpendicular to the magnetic field.
Thus, while the direction is constantly changing, the magnitude of the velocity 鈥� and therefore the kinetic energy 鈥� remains unchanged. This elucidates why the Lorentz force alters momentum direction without affecting the overall speed or kinetic energy.
Momentum Change
Momentum change occurs in charged particles moving through magnetic fields, even when kinetic energy remains constant. Momentum, defined as \( p = mv \), is a vector quantity, meaning it has both magnitude and direction:
  • In a magnetic field, a charged particle moving perpendicular to the field experiences a continuous change in direction due to the Lorentz force.
  • While the speed (magnitude of velocity) doesn't change, the alteration in direction modifies the particle's momentum.
Thus, the particle's path becomes curved, and it travels in a circular trajectory.
This continuous direction change illustrates how momentum can vary without altering the kinetic energy. The insight into momentum and its alteration despite energy constancy is crucial for understanding the dynamics of particles in a magnetic environment.

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Most popular questions from this chapter

Assertion Out of galvanometer, ammeter and voltmeter, resistance of ammeter is lowest and resistance of voltmeter is highest. Reason An ammeter is connected in series and a voltmeter is connected in parallel, in a circuit.

In a chamber, a uniform magnetic field of \(6.5 \mathrm{G}\) \(\left(1 \mathrm{G}=10^{-4} \mathrm{~T}\right)\) is maintained. An electron is shot into the field with a speed of \(4.8 \times 10^{6} \mathrm{~m} / \mathrm{s}\) normal to the field explain why the path of the electron is a circle. If \(\left(e=1.6 \times 10^{-19} \mathrm{C}, m_{e}=9.1 \times 10^{-31} \mathrm{~kg}\right)\), then obtain the frequency of revolution of the electron in its circular orbit. (a) \(6 \times 10^{6} \mathrm{~Hz}\) (b) \(18.18 \times 10^{6} \mathrm{~Hz}\) (c) \(10.10 \times 10^{6} \mathrm{~Hz}\) (d) \(12.10 \times 10^{6} \mathrm{~Hz}\)

Current \(i_{0}\) is passes through a solenoid of length \(l\) having number of turns \(N\) when it is connected to a DC source. A charged particle with charge \(q\) is projected along the axis of the solenoid with a speed \(v_{0} .\) The velocity of the particle in the solenoid (a) increases (b) decreases (c) remain same (d) becomes zero

An element, \(d l=d x\) \hat{ i } (where \(d x=1 \mathrm{~cm}\) ) is placed at the origin and carries a large current \(i=10 \mathrm{~A}\). What is the magnetic field on the \(y\)-axis at a distance of \(0.5 \mathrm{~m} ?\) (a) \(2 \times 10^{-8} \hat{\mathrm{k}} \mathrm{T}\) (b) \(4 \times 10^{-8} \hat{\mathrm{k}} \mathrm{T}\) (c) \(-2 \times 10^{-8} \hat{\mathrm{k}} \mathrm{T}\) (d) \(-4 \times 10^{-8} \hat{\mathrm{k}} \mathrm{T}\)

An electron is projected with uniform velocity along the axis of a current carrying long solenoid. Which of the following is true? (a) The electron will be accelerated along the axis (b) The electron path will be circular about the axis (c) The electron will experience a force at \(45^{\circ}\) to the axis and hence execute a helical path (d) The electron will continue to move with uniform velocity along the axis of the solenoid
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