Motion in a Magnetic Field
Analyzes the trajectory (circular, helical) of a charged particle in a uniform magnetic field, including the principle of the cyclotron.
What is Motion in a Magnetic Field?
The force experienced by a charged particle moving in an electromagnetic field, comprising both electric and magnetic forces.
Key formula / rule: Lorentz Force
Key points
- Understand the nature of the magnetic force on a moving charge.
- Determine the trajectory (circular or helical) of a charged particle in a uniform magnetic field.
- Calculate the radius, time period, frequency, and pitch of the motion.
- Relate the motion to practical applications like cyclotrons.
Common exam trap
Confusing the direction of the magnetic force with the direction of velocity or magnetic field.
Definitions
- Term
Lorentz Force
- Meaning
The force experienced by a charged particle moving in an electromagnetic field, comprising both electric and magnetic forces.
- Term
Cyclotron Frequency
- Meaning
The frequency of revolution of a charged particle in a uniform magnetic field, given by f = qB/2πm.
- Term
Pitch
- Meaning
In helical motion, the distance traveled by the particle along the axis of the helix during one complete revolution.
Learning objectives
Understand the nature of the magnetic force on a moving charge.
Determine the trajectory (circular or helical) of a charged particle in a uniform magnetic field.
Calculate the radius, time period, frequency, and pitch of the motion.
Relate the motion to practical applications like cyclotrons.
Formulae
- Name
Lorentz Force
- Note
Vector form. Magnitude is F = |q|vBsinθ, where θ is the angle between v and B.
- Expression
F = q(v \× B)
- Name
Radius of Circular Path
- Note
When velocity is perpendicular to the magnetic field, v_⊥ = v.
- Expression
r = \frac{mv\perp}{qB}
- Name
Time Period of Circular Motion
- Note
Independent of velocity and radius.
- Expression
T = \frac{2\π m}{qB}
- Name
Frequency of Circular Motion (Cyclotron Frequency)
- Note
Also known as cyclotron frequency.
- Expression
f = \frac{qB}{2\π m}
- Name
Pitch of Helical Path
- Note
Distance covered along the magnetic field in one revolution.
- Expression
p = v\parallel T = \frac{2\π m v\parallel}{qB}
Prerequisites
Vectors (cross product)
Uniform Circular Motion (centripetal force)
Basic understanding of magnetic fields and magnetic force
Common mistakes
Confusing the direction of the magnetic force with the direction of velocity or magnetic field.
Assuming the magnetic force can change the speed or kinetic energy of the particle.
Incorrectly applying the right-hand/left-hand rule for charge direction.
Forgetting to resolve velocity into components when the initial velocity is not perpendicular to the magnetic field.
Keywords
Lorentz Force
Magnetic Field
Charged Particle
Circular Motion
Helical Motion
Centripetal Force
Cyclotron Frequency
Pitch
Practice preview
In a cyclotron, the magnetic field is directed perpendicular to the plane of the dees. What is the role of the electric field?…
medium
A charged particle is moving in a uniform magnetic field B. Which of the following statements is INCORRECT?…
hard
A charged particle of charge q and mass m moves with velocity v in a uniform magnetic field B. If the velocity is perpendicular to the magnetic field, what is the radius of the circular path traced by the particle?…
medium