Magnetic Forces and Fields in Systems
3. Circular motion in magnetic fields
Learning outcomes
- I can explain why charged particles move in circular paths within magnetic fields.
- I can identify the force responsible for circular motion.
- I can calculate the radius of circular motion.
- I can determine how particle speed affects the radius.
- I can analyze magnetic confinement of charged particles.
Circular Motion in Magnetic Fields
Why Charged Particles Can Move in Circles
When a charged particle moves through a magnetic field, it can experience a magnetic force.
For a particle moving perpendicular to the magnetic field:
F_B = |q|vB
where:
- F_B = magnetic force (N)
- |q| = magnitude of particle charge (C)
- v = particle speed (m/s)
- B = magnetic flux density (T)
An important feature of this force is that it acts perpendicular to the particle's velocity.
Because the force continually changes the direction of the velocity without changing its magnitude, the particle can follow a circular path.
Magnetic Force Provides the Centripetal Force
An object moving in a circle requires an inward force called the:
centripetal force
For a charged particle moving perpendicular to a uniform magnetic field, the magnetic force provides this centripetal force.
Therefore:
magnetic force = centripetal force
Magnetic force:
F_B = |q|vB
Centripetal force:
F_c = mv²/r
Therefore:
|q|vB = mv²/r
This relationship explains why a charged particle can travel in a circular path.
Deriving the Radius Equation
Start with:
|q|vB = mv²/r
Cancel one v:
|q|B = mv/r
Multiply by r:
r|q|B = mv
Rearrange:
r = mv/(|q|B)
This is one of the most important equations for charged-particle motion in magnetic fields.
What Determines the Radius?
From:
r = mv/(|q|B)
the radius depends on four quantities:
- mass, m
- speed, v
- charge magnitude, |q|
- magnetic flux density, B
The relationships are:
r ∝ m
r ∝ v
r ∝ 1/|q|
r ∝ 1/B
Effect of Particle Mass
From:
r = mv/(|q|B)
if v, q and B remain constant:
r ∝ m
Therefore:
greater mass → larger radius
A more massive particle is harder to deflect from its original path.
For example, if two particles have the same speed and charge magnitude:
- lighter particle → tighter circular path
- heavier particle → wider circular path
Effect of Particle Speed
If mass, charge and magnetic field remain constant:
r ∝ v
Therefore:
greater speed → larger radius
If the speed doubles:
radius doubles
If the speed triples:
radius triples
This may initially seem surprising because the magnetic force also increases with speed. However, the centripetal force required for a given radius increases with v², so the faster particle follows a larger-radius path.
Example: Changing Speed
A charged particle travels in a circular path with:
r = 0.15 m
Its speed is then doubled while:
- mass remains constant
- charge remains constant
- magnetic field remains constant
Since:
r ∝ v
the new radius is:
r = 2(0.15)
r = 0.30 m
The faster particle follows a wider circle.
Effect of Magnetic Flux Density
From:
r = mv/(|q|B)
we see:
r ∝ 1/B
Therefore:
stronger magnetic field → smaller radius
If B doubles:
radius halves
If B triples:
radius becomes one-third as large
A stronger magnetic field bends the particle's path more sharply.
Effect of Charge
The radius depends on the magnitude of the charge:
r ∝ 1/|q|
Therefore:
greater charge magnitude → smaller radius
Suppose two particles have identical mass and speed.
Particle A:
|q_A| = e
Particle B:
|q_B| = 2e
Then:
r_B = ½r_A
The particle with twice the charge magnitude curves twice as strongly.
Positive and Negative Charges
The radius equation contains:
|q|
because the magnitude of charge determines the size of the radius.
The sign of the charge determines the direction of curvature.
Positive and negative particles moving in the same direction through the same magnetic field curve in:
opposite directions
Therefore:
charge magnitude → radius
charge sign → direction of curvature
Determining the Direction of Curvature
For a positive charge, use the right-hand rule.
For example:
velocity → right
magnetic field → into page ⊗
The magnetic force points:
upward
The particle therefore begins curving upward.
For a negative particle, reverse the right-hand-rule result.
It would curve:
downward
Force Always Points Toward the Centre
During circular motion, the velocity is always tangent to the circular path.
The magnetic force is perpendicular to the velocity.
Therefore, the force continually points toward the centre of the circle.
At one instant:
velocity →
force ↑
Later:
velocity ↗
force ↖
Later still:
velocity ↑
force ←
The direction of the magnetic force changes continuously as the particle moves.
This inward magnetic force keeps the particle moving in a circle.
Magnetic Force Does Not Increase Speed
The magnetic force is perpendicular to the particle's motion.
Therefore, it does no work on the particle.
So:
magnetic field changes direction of velocity
but:
magnetic field does not change speed
when no other forces are acting.
Therefore, during ideal circular motion:
- speed remains constant
- kinetic energy remains constant
- direction changes continuously
- velocity changes because velocity includes direction
Speed vs Velocity
This distinction is important.
During uniform circular motion:
speed is constant
but:
velocity is changing
because velocity has both:
- magnitude
- direction
The continuously changing direction means the particle is accelerating even though its speed remains constant.
Centripetal Acceleration
The inward acceleration of a particle moving in a circle is:
a_c = v²/r
Since the magnetic force provides the centripetal force:
F = ma
Therefore:
a_c = F_B/m
For perpendicular motion:
a_c = |q|vB/m
A stronger magnetic field therefore produces greater centripetal acceleration for the same particle and speed.
Example: Calculating Radius
A proton travels perpendicular to a magnetic field.
Given:
m = 1.67 × 10⁻²⁷ kg
q = 1.60 × 10⁻¹⁹ C
v = 3.0 × 10⁶ m/s
B = 0.40 T
Use:
r = mv/(|q|B)
r = [(1.67 × 10⁻²⁷)(3.0 × 10⁶)] / [(1.60 × 10⁻¹⁹)(0.40)]
r = (5.01 × 10⁻²¹)/(6.40 × 10⁻²⁰)
r ≈ 0.0783 m
Therefore:
r ≈ 0.078 m
or approximately:
7.8 cm
Example: Finding Speed
A particle has:
m = 4.0 × 10⁻²⁷ kg
|q| = 1.6 × 10⁻¹⁹ C
B = 0.50 T
r = 0.10 m
Use:
r = mv/(|q|B)
Rearrange:
v = r|q|B/m
Substitute:
v = [(0.10)(1.6 × 10⁻¹⁹)(0.50)]/(4.0 × 10⁻²⁷)
v = 2.0 × 10⁶ m/s
Therefore:
v = 2.0 × 10⁶ m/s
Example: Finding Magnetic Flux Density
A charged particle has:
m = 2.0 × 10⁻²⁶ kg
v = 5.0 × 10⁵ m/s
|q| = 1.6 × 10⁻¹⁹ C
r = 0.25 m
Rearrange:
B = mv/(r|q|)
Substitute:
B = [(2.0 × 10⁻²⁶)(5.0 × 10⁵)] / [(0.25)(1.6 × 10⁻¹⁹)]
B = (1.0 × 10⁻²⁰)/(4.0 × 10⁻²⁰)
B = 0.25 T
Example: Comparing Two Particles
Particle A and Particle B have identical:
- mass
- charge magnitude
- speed
Particle A moves through:
B_A = 0.20 T
Particle B moves through:
B_B = 0.60 T
Since:
r ∝ 1/B
the second magnetic field is three times stronger.
Therefore:
r_B = r_A/3
Particle B follows a much tighter circular path.
Comparing an Electron and Proton
An electron and proton have the same magnitude of charge:
|q_e| = |q_p| = 1.60 × 10⁻¹⁹ C
However:
m_p ≈ 1836m_e
If they enter the same magnetic field with the same speed:
r ∝ m
Therefore, the proton follows a much larger-radius path.
The electron follows a much tighter path.
They also curve in opposite directions because they have opposite charges.
Momentum and Circular Radius
For non-relativistic motion:
p = mv
The radius equation is:
r = mv/(|q|B)
Therefore:
r = p/(|q|B)
or:
p = |q|Br
This is extremely useful because measuring the radius of a particle's path can provide information about its momentum.
At very high speeds, the relativistic momentum must be used instead of simply mv.
Why Particle Tracks Are Useful
Suppose charged particles pass through a magnetic field and leave visible tracks.
Scientists can examine:
- direction of curvature
- radius of curvature
- particle energy
- charge
The curvature direction provides information about:
charge sign
The radius provides information about:
momentum-to-charge ratio
This technique has played an important role in particle physics.
What Happens if Velocity Is Parallel to the Field?
Circular motion occurs only when the particle has velocity perpendicular to the magnetic field.
If:
v ∥ B
then:
θ = 0°
and:
F = |q|vB sin 0°
Therefore:
F = 0
The particle continues in a straight line if no other forces act.
What Happens if Velocity Is at an Angle?
Suppose the particle enters the magnetic field at an angle between 0° and 90°.
Its velocity can be separated into:
v∥ — parallel component
and:
v⊥ — perpendicular component
The parallel component experiences no magnetic force.
The perpendicular component produces circular motion.
Combining the two produces a:
helical path
Radius of Helical Motion
For a particle moving at an angle to the magnetic field, use the perpendicular velocity component:
r = mv⊥/(|q|B)
where:
v⊥ = v sin θ
Therefore:
r = mv sin θ/(|q|B)
The parallel component:
v∥ = v cos θ
moves the particle along the magnetic-field direction.
Together these produce the helix.
Pitch of the Helix
The pitch is the distance the particle travels along the magnetic-field direction during one complete revolution.
If the period is T:
pitch = v∥T
Therefore, the shape of a helical trajectory depends on both:
- perpendicular velocity
- parallel velocity
A larger parallel component produces a more stretched-out helix.
Period of Circular Motion
The time required for one complete revolution is called the:
period
The circumference is:
2πr
Therefore:
T = 2πr/v
Using:
r = mv/(|q|B)
gives:
T = 2π[mv/(|q|B)]/v
The speed cancels:
T = 2πm/(|q|B)
An Important Result
For non-relativistic particles in a uniform magnetic field:
T = 2πm/(|q|B)
The period does not depend on the particle's speed.
A faster particle travels around a larger circle, but it also travels faster.
These effects cancel.
Therefore, particles with the same:
- mass
- charge magnitude
- magnetic field
have the same orbital period in this ideal model.
Cyclotron Frequency
Frequency is:
f = 1/T
Therefore:
f = |q|B/(2πm)
This is called the:
cyclotron frequency
It depends on:
- charge magnitude
- magnetic flux density
- particle mass
A stronger magnetic field produces a higher cyclotron frequency.
A more massive particle produces a lower frequency.
The Cyclotron
A cyclotron is a particle accelerator that uses magnetic fields to bend charged particles into curved paths.
The magnetic field bends the particle's path.
An electric field accelerates the particle each time it crosses the gap between electrodes.
As its speed increases:
r = mv/(|q|B)
so the radius increases.
The particle therefore follows an outward spiral.
Electric and Magnetic Fields Have Different Jobs
In a particle accelerator:
electric field → changes particle energy and speed
magnetic field → changes particle direction
This distinction is important.
A magnetic field alone does not increase the particle's kinetic energy.
Instead, it bends the trajectory.
Magnetic Confinement
Charged particles can be guided and confined using magnetic fields.
This is especially important for plasmas.
A plasma contains large numbers of:
- electrons
- positive ions
Because these particles are charged, magnetic fields can influence their motion.
Instead of moving freely across magnetic-field lines, particles can spiral around them.
Why Magnetic Confinement Is Useful
Extremely hot plasmas cannot simply be allowed to touch ordinary container walls.
For example, fusion plasmas can reach temperatures of millions of degrees.
No ordinary solid container could directly contain such plasma without severe interaction with the walls.
Magnetic fields can help keep much of the plasma away from the walls by controlling charged-particle motion.
Magnetic Confinement in Fusion
Magnetic-confinement fusion devices use strong magnetic fields to guide charged particles.
One well-known design is the:
tokamak
A tokamak uses magnetic fields arranged around a toroidal, or doughnut-shaped, chamber.
Charged particles spiral around magnetic-field lines while the overall field geometry helps confine the plasma.
Why One Simple Magnetic Field Is Not Enough
A simple uniform magnetic field can cause charged particles to spiral around field lines.
However, the particles can still move freely along those field lines.
Therefore, practical magnetic confinement requires more complicated magnetic-field geometries.
Engineers design magnetic fields to reduce particle losses and improve confinement.
Magnetic Mirrors
A magnetic mirror uses a magnetic field that becomes stronger toward the ends of a confinement region.
Under suitable conditions, charged particles spiralling along the field can be reflected away from the stronger-field regions.
This can help trap particles between regions of strong magnetic field.
The physics of real magnetic mirrors is more advanced than simple circular motion, but it builds on the same principle:
magnetic fields control charged-particle trajectories.
Earth's Magnetic Field as Natural Confinement
Earth provides a natural example of magnetic control of charged particles.
Energetic charged particles can become trapped in Earth's magnetic field.
They spiral around magnetic field lines and can also move along them.
This contributes to the formation of the:
Van Allen radiation belts
Auroras and Particle Motion
Charged particles from Earth's space environment can be guided along magnetic field lines toward high-latitude regions.
When energetic particles interact with Earth's upper atmosphere, they can excite atmospheric atoms and molecules.
As those particles return to lower energy states, they emit light.
This produces:
auroras
Therefore, the circular and helical motion of charged particles has important natural consequences.
Magnetic Confinement Is Not Perfect
Magnetic fields can strongly influence charged particles, but confinement is not automatically perfect.
Real plasmas contain enormous numbers of interacting particles.
Their motion can be affected by:
- non-uniform magnetic fields
- electric fields
- collisions
- plasma instabilities
- particle-particle interactions
Modern fusion research therefore involves sophisticated magnetic-field designs and control systems.
Applications of Circular Charged-Particle Motion
Circular motion in magnetic fields is used or observed in:
- particle accelerators
- cyclotrons
- mass spectrometers
- particle detectors
- plasma confinement
- fusion research
- Earth's magnetosphere
- radiation belts
- space physics
- medical particle-beam technologies
Understanding circular magnetic motion connects fundamental physics to major scientific and technological applications.
Mass Spectrometers
A mass spectrometer can use magnetic fields to separate charged particles.
For particles entering with the same speed:
r = mv/(|q|B)
Particles with different mass-to-charge ratios follow different radii.
Therefore, measuring the radius can help identify different ions or isotopes.
Example: Two Isotopes
Suppose two singly charged ions enter the same magnetic field with the same speed.
Ion A has mass:
m
Ion B has mass:
2m
Since:
r ∝ m
Ion B follows a path with:
twice the radius
This allows magnetic fields to separate isotopes with different masses.
Example: Effect of Doubling Several Variables
Start with:
r = mv/(|q|B)
Suppose:
- speed doubles
- magnetic field doubles
Then:
r_new = m(2v)/(|q|(2B))
The factors of 2 cancel.
Therefore:
r_new = r_original
The particle follows the same-radius path despite moving faster because the stronger magnetic field compensates for the increased speed.
Example: Finding Charge Magnitude
A particle moves in a circular path with:
m = 5.0 × 10⁻²⁷ kg
v = 4.0 × 10⁵ m/s
B = 0.25 T
r = 0.050 m
Use:
r = mv/(|q|B)
Rearrange:
|q| = mv/(rB)
Substitute:
|q| = [(5.0 × 10⁻²⁷)(4.0 × 10⁵)] / [(0.050)(0.25)]
|q| = (2.0 × 10⁻²¹)/(0.0125)
|q| = 1.6 × 10⁻¹⁹ C
This is the magnitude of the elementary charge.
Equation Summary
For magnetic force:
F_B = |q|vB sin θ
For perpendicular motion:
F_B = |q|vB
For centripetal force:
F_c = mv²/r
For circular radius:
r = mv/(|q|B)
For circular period:
T = 2πm/(|q|B)
For cyclotron frequency:
f = |q|B/(2πm)
For helical radius:
r = mv⊥/(|q|B)
For non-relativistic momentum:
p = |q|Br
Common Mistakes
Mistake: Saying the magnetic force pushes the particle forward.
The magnetic force is perpendicular to the velocity.
Mistake: Saying magnetic force increases the particle's speed.
A magnetic field acting alone changes direction, not speed.
Mistake: Assuming every charged particle in a magnetic field follows a circle.
Circular motion requires a velocity component perpendicular to the field.
Mistake: Forgetting the absolute value of charge in radius calculations.
Charge magnitude determines the radius; charge sign determines the direction of curvature.
Mistake: Thinking a faster particle curves more sharply.
For the same m, q and B, a faster particle has a larger radius.
Mistake: Thinking a stronger magnetic field produces a larger radius.
A stronger field produces a smaller radius.
Mistake: Assuming a particle moving parallel to B curves.
It experiences zero magnetic force.
Mistake: Confusing circular and helical motion.
Pure perpendicular motion produces a circle; combined parallel and perpendicular motion produces a helix.
Mistake: Saying magnetic confinement prevents all particle motion.
Particles can still move along magnetic field lines and real confinement systems are more complicated.
Key Terms
Circular motion — motion along a circular path.
Magnetic force — force exerted on a moving charged particle by a magnetic field.
Centripetal force — inward force required to maintain circular motion.
Centripetal acceleration — acceleration directed toward the centre of a circular path.
Magnetic flux density (B) — measure of magnetic-field strength, measured in teslas.
Radius of curvature — radius of the circular path followed by a charged particle.
Helical motion — spiral motion produced when a charged particle has both parallel and perpendicular velocity components relative to a magnetic field.
Cyclotron frequency — frequency of circular motion of a non-relativistic charged particle in a uniform magnetic field.
Cyclotron — particle accelerator that uses magnetic fields to bend charged particles and electric fields to accelerate them.
Magnetic confinement — use of magnetic fields to control and restrict the motion of charged particles.
Plasma — ionized gas containing free charged particles.
Tokamak — toroidal device designed to confine hot plasma using magnetic fields.
Magnetic mirror — magnetic configuration that can reflect some charged particles away from regions of stronger magnetic field.
Momentum — quantity related to mass and velocity; for non-relativistic motion, p = mv.
Key Takeaways
- A moving charged particle can experience a magnetic force.
- For perpendicular motion, F_B = |q|vB.
- Magnetic force acts perpendicular to particle velocity.
- The magnetic force can provide the centripetal force required for circular motion.
- Equating magnetic and centripetal forces gives r = mv/(|q|B).
- Greater particle mass produces a larger radius.
- Greater particle speed produces a larger radius.
- Greater charge magnitude produces a smaller radius.
- Greater magnetic flux density produces a smaller radius.
- Charge sign determines the direction of curvature.
- Positive and negative particles curve in opposite directions.
- Magnetic force changes direction but not speed when acting alone.
- A particle moving parallel to B experiences no magnetic force.
- A particle moving perpendicular to B can follow a circular path.
- A particle with both parallel and perpendicular velocity components can follow a helical path.
- The period of ideal non-relativistic circular motion is T = 2πm/(|q|B).
- Circular particle motion is used in cyclotrons, mass spectrometers and particle detectors.
- Magnetic fields can help confine charged particles in plasmas.
- Tokamaks use complex magnetic fields to help confine extremely hot fusion plasmas.
- Earth's magnetic field naturally influences and traps some energetic charged particles.
Check Your Understanding
- Explain why a charged particle moving perpendicular to a magnetic field can follow a circular path.
- What force provides the centripetal force?
- State the magnetic-force equation for a particle moving perpendicular to a magnetic field.
- State the equation for centripetal force.
- Derive r = mv/(|q|B).
- How does increasing particle mass affect the radius?
- How does increasing particle speed affect the radius?
- How does increasing magnetic flux density affect the radius?
- How does increasing charge magnitude affect the radius?
- What determines whether a particle curves clockwise or counterclockwise?
- Explain why a magnetic field does not change the speed of a particle when it is the only force acting.
- A proton travels at 2.0 × 10⁶ m/s perpendicular to a 0.50 T field. Calculate its radius using m = 1.67 × 10⁻²⁷ kg and q = 1.60 × 10⁻¹⁹ C.
- A particle's speed doubles while all other quantities remain constant. What happens to its radius?
- The magnetic flux density doubles while all other quantities remain constant. What happens to the radius?
- Describe the motion of a charged particle travelling exactly parallel to a uniform magnetic field.
- Explain why a particle entering a magnetic field at an angle can follow a helical path.
- Two singly charged ions travel at the same speed through the same magnetic field. One ion has twice the mass of the other. Compare their circular radii.
- Explain how circular motion in a magnetic field can be used to separate ions in a mass spectrometer.
- Explain why magnetic fields are useful for confining extremely hot plasmas.
- A particle's speed and the magnetic flux density are both doubled while its mass and charge remain unchanged. Determine what happens to the radius of its circular path.