12.6 Motion of a Charged Particle in a Uniform Magnetic Field

Introduction

When a charged particle enters a uniform magnetic field with its velocity perpendicular to the field lines, it is subjected to a constant magnetic force. This force always acts perpendicular to the particle's direction of motion and serves as a centripetal force, compelling the particle to follow a uniform circular path.

1. Equating Magnetic and Centripetal Forces

The motion of the particle is governed by the balance between the magnetic force and the centripetal force required for circular motion.

The maximum magnetic force ($F_{B(\max )}$) is exerted when the velocity is perpendicular to the magnetic field: $F_{B(\max )}=qvB$

The centripetal force ($F_c$) required to keep an object of mass $m$ in a circular path of radius $r$ at velocity $v$ is: $F_c=\frac{m{v}^2}{r}$

For the particle to move in a circle, the magnetic force provides the centripetal force: $F_{B(\max )}=F_c$

2. Radius of the Circular Path

By equating the expressions for the two forces, we can derive the radius of the particle's path. $qvB=\frac{m{v}^2}{r}$

Solving for the radius $r$, we get: $r=\frac{mv}{qB}$

This equation shows that the radius of the orbit is directly proportional to the particle's momentum ($mv$) and inversely proportional to its charge ($q$) and the magnetic field strength ($B$).

3. Angular Frequency, Period, and Cyclotron Frequency

These quantities describe the rotational motion of the particle.

Angular Frequency ($\omega$): This is the rate at which the particle rotates. It is derived from the relation $v=r\omega$. $\omega =\frac{v}{r}=\frac{v}{mv/qB}=\frac{qB}{m}$

Notably, the angular frequency is independent of the particle's velocity and the radius of its orbit.

Time Period ($T$): The time taken to complete one full revolution. $T=\frac{2\pi }{\omega }=\frac{2\pi m}{qB}$

Cyclotron Frequency ($f$): The number of revolutions per second, which is the reciprocal of the time period. $f=\frac{1}{T}=\frac{\omega }{2\pi }=\frac{qB}{2\pi m}$

This frequency is fundamental to the operation of particle accelerators called cyclotrons.

4. Energy Consideration

Since the magnetic force always acts perpendicular to the velocity of the particle, it does no work on the particle. Therefore, the kinetic energy of the particle remains constant throughout its motion in the magnetic field.

5. Helical Motion (Velocity Not Perpendicular to Field)

When a charged particle's velocity has components both parallel and perpendicular to the magnetic field:

• The parallel component ($v_{\parallel }$) experiences no magnetic force ($F=q{v}_{\parallel }B\sin 0^{\circ }=0$) and produces uniform linear motion along the field direction.
• The perpendicular component ($v_{\perp }$) experiences a magnetic force and produces circular motion.

The combination of these two motions results in a helical (spiral) path along the direction of the magnetic field.

6. Example Problem

Problem: A proton is moving in a circular orbit of radius $12\text{cm}$ in a uniform magnetic field of $0.40\text{T}$. Find its speed.

Given:

• Radius, $r=12\text{cm}=0.12\text{m}$
• Magnetic field, $B=0.40\text{T}$
• Charge of a proton, $e=1.60\times 10^{-19}\text{C}$
• Mass of a proton, $m_p=1.67\times 10^{-27}\text{kg}$

Solution:

From the radius formula, $r=\frac{m_{p}v}{eB}$, we can rearrange to solve for the speed $v$: $v=\frac{eBr}{m_p}$

Substituting the given values: $v=\frac{(1.60\times 10^{-19}\text{C})(0.40\text{T})(0.12\text{m})}{1.67\times 10^{-27}\text{kg}}$

$v=4.59\times 10^6\text{m/s}$

Possible Questions and Answers

Q: What happens to the radius of the circular path if the particle's velocity is doubled?

A: Since the radius $r$ is directly proportional to the velocity $v$ ($r=\frac{mv}{qB}$), doubling the velocity will also double the radius of the path.

Q: Does the time period of the particle's revolution depend on its speed?

A: No. The formula $T=\frac{2\pi m}{qB}$ shows that the time period is independent of the particle's velocity and depends only on its mass-to-charge ratio and the magnetic field strength.

Significance: The principles governing the motion of charged particles in magnetic fields are fundamental to the operation of important scientific instruments like cyclotrons (particle accelerators) and mass spectrometers (which separate ions based on their mass-to-charge ratio), and for explaining natural phenomena such as the auroras.