Charged Particle Motion

Charged Particles in Electric and Magnetic Fields

Charged particles respond to electric fields and magnetic fields, but the two fields affect particle motion in different ways.

An electric field can exert a force on a charged particle whether it is:

• stationary
• moving

A magnetic field exerts a magnetic force only when the particle is moving with a component of velocity perpendicular to the magnetic field.

Understanding these differences allows us to predict and control the paths of:

• electrons
• protons
• ions
• other charged particles

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Motion in an Electric Field

The force on a charge in an electric field is:

F_E = qE

where:

• F_E = electric force (N)
• q = charge (C)
• E = electric field strength (N/C or V/m)

The magnitude of the force is:

F_E = |q|E

The direction depends on the sign of the charge.

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Positive Charges in Electric Fields

Electric-field direction is defined as the direction of force on a positive test charge.

Therefore, a positive charge experiences force:

in the direction of E

For example:

E →

then:

positive charge force →

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Negative Charges in Electric Fields

A negative charge experiences force in the direction opposite to the electric field.

For example:

E →

then:

electron force ←

Therefore:

positive charge → force along E

negative charge → force opposite E

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Electric Force Produces Acceleration

Using Newton's Second Law:

F = ma

and:

F_E = qE

we obtain, for magnitude:

a = |q|E/m

Therefore, acceleration in a uniform electric field depends on:

• charge magnitude
• electric field strength
• particle mass

Greater |q| → greater acceleration

Greater E → greater acceleration

Greater m → smaller acceleration

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Electric Fields Can Change Particle Speed

An electric force can act partly or completely in the same or opposite direction as a particle's motion.

Therefore, an electric field can:

• increase speed
• decrease speed
• change direction
• change kinetic energy

Electric fields can therefore accelerate charged particles and transfer energy to or from them.

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Example: Proton in an Electric Field

A proton is placed in a uniform electric field:

E = 2.0 × 10⁴ N/C

The proton charge is:

q = 1.60 × 10⁻¹⁹ C

Calculate the force.

Use:

F = qE

F = (1.60 × 10⁻¹⁹)(2.0 × 10⁴)

F = 3.2 × 10⁻¹⁵ N

The force is in the direction of the electric field.

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Example: Electron in the Same Field

An electron has:

q = −1.60 × 10⁻¹⁹ C

The magnitude of the force is:

F = |q|E

F = (1.60 × 10⁻¹⁹)(2.0 × 10⁴)

F = 3.2 × 10⁻¹⁵ N

The magnitude is the same as for the proton.

However, the force direction is:

opposite to the electric field

because the electron is negatively charged.

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Motion Between Parallel Plates

Two oppositely charged parallel plates can create an approximately uniform electric field.

A charged particle entering sideways between the plates experiences an approximately constant force.

Therefore, it has:

• approximately constant horizontal velocity, if no horizontal force acts
• constant vertical acceleration

This is similar to projectile motion under gravity.

The resulting trajectory can be:

parabolic

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Electric Field and Projectile Motion

Suppose a positive particle enters horizontally into a downward electric field.

The particle experiences:

• horizontal motion at approximately constant velocity
• downward acceleration due to the electric force

Its path curves downward.

For a negative particle, the force is upward, so the trajectory curves upward.

This resembles a projectile except that the force is electrical rather than gravitational.

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Motion in a Magnetic Field

The magnetic force on a moving charge is:

F_B = |q|vB sin θ

where:

• F_B = magnetic force (N)
• |q| = charge magnitude (C)
• v = particle speed (m/s)
• B = magnetic flux density (T)
• θ = angle between v and B

The magnetic force acts perpendicular to both:

• velocity
• magnetic field

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Motion Perpendicular to a Magnetic Field

If:

v ⟂ B

then:

θ = 90°

and:

sin 90° = 1

Therefore:

F_B = |q|vB

The force is always perpendicular to the velocity.

As the particle turns, the force direction also changes.

This can produce:

uniform circular motion

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Radius of Circular Motion

The magnetic force provides the centripetal force:

|q|vB = mv²/r

Rearranging:

r = mv/(|q|B)

Therefore:

greater m → larger radius

greater v → larger radius

greater |q| → smaller radius

greater B → smaller radius

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Motion Parallel to a Magnetic Field

If a charged particle travels parallel to the magnetic field:

θ = 0°

Therefore:

sin 0° = 0

and:

F_B = 0

The magnetic field does not deflect the particle.

If no other forces act, it continues moving in a straight line at constant velocity.

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Motion at an Angle to a Magnetic Field

If a particle enters at an angle to the magnetic field, its velocity can be separated into:

v∥ — component parallel to B

v⊥ — component perpendicular to B

The parallel component is unaffected.

The perpendicular component produces circular motion.

Together these produce:

helical motion

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Electric vs Magnetic Fields

This distinction is fundamental to charged-particle physics.

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Why Magnetic Fields Do No Work

Work is:

W = Fd cos θ

The magnetic force is perpendicular to the particle's instantaneous displacement.

Therefore:

θ = 90°

and:

cos 90° = 0

So:

W = 0

A magnetic field acting alone changes the direction of motion without changing the particle's kinetic energy.

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Electric Fields Can Do Work

Electric fields can do work on charged particles.

The change in electric potential energy is:

ΔEₚ = qΔV

The work done by the electric field is:

W_field = −ΔEₚ

A charged particle accelerated through a potential difference can therefore gain kinetic energy.

For a particle gaining kinetic energy through a potential difference of magnitude V:

ΔE_k = |q|V

when the signs and direction correspond to acceleration.

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Accelerating a Particle Through a Voltage

If a particle starts from rest and gains kinetic energy:

|q|V = ½mv²

Therefore:

v = √(2|q|V/m)

This equation can be used to calculate the speed of a charged particle after acceleration through a known potential difference.

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Example: Proton Accelerated Through a Voltage

A proton starts from rest and is accelerated through:

V = 1000 V

Use:

|q|V = ½mv²

Given:

q = 1.60 × 10⁻¹⁹ C

m = 1.67 × 10⁻²⁷ kg

Then:

v = √[(2)(1.60 × 10⁻¹⁹)(1000)/(1.67 × 10⁻²⁷)]

v ≈ 4.38 × 10⁵ m/s

Therefore:

v ≈ 4.4 × 10⁵ m/s

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Combined Electric and Magnetic Fields

A particle can experience electric and magnetic forces at the same time.

The total electromagnetic force is called the:

Lorentz force

It can be written:

F = q(E + v × B)

This means the total force is the vector combination of:

• electric force
• magnetic force

Combined fields can be arranged so the forces:

• reinforce one another
• oppose one another
• act in different directions

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Crossed Electric and Magnetic Fields

An especially important arrangement uses:

E ⟂ B

These are called crossed fields.

A charged particle travelling through crossed electric and magnetic fields can experience two forces in opposite directions.

Under the correct conditions, these forces balance.

The particle then travels:

undeflected in a straight line

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The Velocity Selector

A velocity selector uses crossed electric and magnetic fields to select charged particles with a particular speed.

Electric force magnitude:

F_E = |q|E

Magnetic force magnitude for perpendicular motion:

F_B = |q|vB

For the particle to travel straight:

F_E = F_B

Therefore:

|q|E = |q|vB

Cancel |q|:

E = vB

Therefore:

v = E/B

Only particles with this selected speed travel undeflected through the device.

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Why Charge Cancels

Notice that:

|q|E = |q|vB

contains |q| on both sides.

It cancels.

Therefore:

v = E/B

does not depend directly on:

• particle charge magnitude
• particle mass

Any charged particle travelling with the selected speed can, under the appropriate geometry, pass through undeflected.

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Particles Moving Too Slowly

The electric force is:

F_E = |q|E

The magnetic force is:

F_B = |q|vB

Suppose:

v < E/B

Then:

F_B < F_E

Therefore, the electric force dominates.

The particle deflects in the direction determined by the net electric-force dominance.

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Particles Moving Too Fast

Suppose:

v > E/B

Then:

F_B > F_E

The magnetic force dominates.

The particle deflects in the direction of the magnetic force.

Therefore:

too slow → electric force dominates

correct speed → forces balance

too fast → magnetic force dominates

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Example: Velocity Selector

A velocity selector has:

E = 6.0 × 10⁴ N/C

B = 0.20 T

Calculate the selected speed.

Use:

v = E/B

v = (6.0 × 10⁴)/(0.20)

v = 3.0 × 10⁵ m/s

Particles travelling at this speed pass through undeflected.

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Example: Finding Electric Field Strength

Particles with speed:

v = 4.0 × 10⁶ m/s

must pass undeflected through:

B = 0.15 T

Use:

E = vB

E = (4.0 × 10⁶)(0.15)

E = 6.0 × 10⁵ N/C

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Example: Finding Magnetic Flux Density

A velocity selector uses:

E = 2.4 × 10⁵ N/C

to select particles travelling at:

v = 8.0 × 10⁵ m/s

Use:

B = E/v

B = (2.4 × 10⁵)/(8.0 × 10⁵)

B = 0.30 T

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Determining Directions in a Velocity Selector

Suppose a positive particle moves:

→

The electric field points:

↓

Therefore, the electric force on the positive particle points:

↓

To pass undeflected, the magnetic force must point:

↑

The magnetic field must therefore be arranged so that:

q(v × B)

points upward.

For a negative particle, both electric and magnetic force directions reverse. They can still oppose one another and balance at the same selected speed.

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From Velocity Selector to Circular Motion

A common particle-analysis system first uses a velocity selector.

Stage 1:

crossed E and B fields select v

Stage 2:

magnetic field bends particles into circular paths

This allows scientists to control the particle speed before analyzing its trajectory.

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Combining the Equations

The velocity selector gives:

v = E/B

The circular-motion equation gives:

r = mv/(|q|B)

Substitute v = E/B:

r = m(E/B)/(|q|B)

Therefore:

r = mE/(|q|B²)

This shows how the radius depends on mass-to-charge ratio when particles have been selected using crossed fields.

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Mass Spectrometry

Magnetic deflection is used in instruments that analyze charged particles.

A simplified system may involve:

1. ions are produced
2. ions are accelerated
3. a velocity selector chooses ions with a particular speed
4. ions enter a magnetic field
5. ions follow curved paths
6. different ions reach different positions

Because:

r = mv/(|q|B)

particles with different mass-to-charge ratios can be separated.

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Separating Isotopes

Isotopes of an element have the same number of protons but different numbers of neutrons.

If they form ions with the same charge, their masses are different.

Suppose two singly charged isotopes have the same speed in the same magnetic field.

Since:

r ∝ m

the heavier isotope follows a:

larger-radius path

The lighter isotope follows a:

smaller-radius path

This allows isotopes to be separated and identified.

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Example: Comparing Isotopes

Two singly charged ions pass through the same velocity selector and then enter the same magnetic field.

Ion A has mass:

20 u

Ion B has mass:

22 u

Because they have the same:

• speed
• charge magnitude
• magnetic field

we have:

r_B/r_A = m_B/m_A

r_B/r_A = 22/20

r_B/r_A = 1.10

Therefore:

Ion B's radius is 10% larger.

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Electric Deflection vs Magnetic Deflection

Suppose a particle enters a field region horizontally.

Electric field perpendicular to motion

The particle experiences approximately constant force in a fixed direction.

This can produce a:

parabolic trajectory

Magnetic field perpendicular to motion

The force continually remains perpendicular to velocity.

This can produce a:

circular trajectory

This difference is useful when identifying whether an observed particle path is caused primarily by an electric or magnetic field.

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Predicting Trajectories

When analyzing a charged particle, ask:

Is the particle charged?

If q = 0, electric and magnetic fields do not directly exert these forces.

Is there an electric field?

Use:

F_E = qE

Is there a magnetic field?

Use:

F_B = |q|vB sin θ

What is the particle's charge sign?

Positive and negative charges experience opposite force directions.

What is the angle between v and B?

This determines the magnetic-force magnitude.

Are both fields present?

Add the forces as vectors.

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Example: Particle Enters a Magnetic Field

A proton enters a uniform magnetic field perpendicular to the field.

Given:

m = 1.67 × 10⁻²⁷ kg

q = 1.60 × 10⁻¹⁹ C

v = 5.0 × 10⁶ m/s

B = 0.25 T

Calculate the radius.

Use:

r = mv/(|q|B)

r = [(1.67 × 10⁻²⁷)(5.0 × 10⁶)] / [(1.60 × 10⁻¹⁹)(0.25)]

r ≈ 0.209 m

Therefore:

r ≈ 0.21 m

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Multi-Step Problem

A singly charged positive ion passes undeflected through a velocity selector.

Given:

E = 4.0 × 10⁴ N/C

B₁ = 0.20 T

It then enters a second magnetic field:

B₂ = 0.50 T

The ion has:

m = 3.2 × 10⁻²⁶ kg

q = 1.60 × 10⁻¹⁹ C

Step 1: Find the speed

v = E/B₁

v = (4.0 × 10⁴)/(0.20)

v = 2.0 × 10⁵ m/s

Step 2: Find the circular radius

r = mv/(|q|B₂)

r = [(3.2 × 10⁻²⁶)(2.0 × 10⁵)] / [(1.60 × 10⁻¹⁹)(0.50)]

r = 0.080 m

Therefore:

r = 8.0 cm

This is a typical example of using electric and magnetic fields together to analyze particles.

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Particle Accelerators

Charged-particle accelerators use electric and magnetic fields for different purposes.

Electric fields

Used primarily to:

increase or decrease particle energy

Magnetic fields

Used primarily to:

bend and guide particle trajectories

Large accelerators use powerful magnets to keep high-energy charged particles travelling along designed paths.

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Magnetic Confinement

Magnetic fields can also confine charged particles.

Particles with perpendicular velocity components spiral around magnetic field lines.

This is important in:

• plasma physics
• fusion research
• Earth's magnetosphere
• radiation belts

Devices such as tokamaks use carefully designed magnetic fields to help confine extremely hot plasma.

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Earth's Magnetic Field

Charged particles from space interact with Earth's magnetic field.

Some particles are:

• deflected
• trapped
• guided along field lines

These interactions contribute to:

• Van Allen radiation belts
• auroras
• space-weather phenomena

Charged-particle motion is therefore important both in laboratories and on planetary scales.

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Particle Detectors

Magnetic fields are used in particle detectors because charged particles follow curved trajectories.

The direction of curvature can indicate:

charge sign

The radius can provide information about:

momentum

For perpendicular motion:

p = |q|Br

for the appropriate momentum definition.

This allows physicists to reconstruct properties of particles from their tracks.

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Solving Electric-Field Problems

Useful equations include:

F = |q|E

a = |q|E/m

ΔEₚ = qΔV

For acceleration from rest through an appropriate potential difference:

|q|V = ½mv²

Choose the equation according to the information provided.

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Solving Magnetic-Field Problems

Useful equations include:

F = |q|vB sin θ

For perpendicular motion:

F = |q|vB

For circular motion:

r = mv/(|q|B)

For momentum:

p = |q|Br

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Solving Velocity-Selector Problems

For undeflected particles:

F_E = F_B

Therefore:

|q|E = |q|vB

and:

v = E/B

Common rearrangements are:

E = vB

and:

B = E/v

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Choosing the Correct Equation

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Common Mistakes

Mistake: Saying magnetic fields accelerate stationary charges.
A stationary charge experiences no magnetic force.

Mistake: Saying electric fields only affect moving charges.
Electric fields affect stationary and moving charges.

Mistake: Saying a magnetic field increases particle speed.
A magnetic force acting alone changes direction but not speed.

Mistake: Assuming electric and magnetic forces always point in the same direction.
Their directions depend on the field geometry, velocity and charge sign.

Mistake: Forgetting to reverse force direction for a negative charge.
Always consider the sign of q.

Mistake: Using F = |q|vB when v is not perpendicular to B.
Use F = |q|vB sin θ.

Mistake: Assuming every magnetic trajectory is circular.
Parallel motion is straight; angled motion can be helical.

Mistake: Adding electric and magnetic force magnitudes without considering direction.
Forces are vectors.

Mistake: Forgetting that velocity-selector forces must oppose each other.
Equal forces in the same direction would not produce an undeflected particle.

Mistake: Including charge in the final velocity-selector equation.
Charge cancels, giving v = E/B.

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Key Terms

Charged particle — particle or object possessing net electric charge.

Electric force — force on a charge due to an electric field.

Magnetic force — force on a moving charge due to a magnetic field.

Lorentz force — combined electric and magnetic force on a charged particle.

Electric field strength (E) — force per unit positive charge.

Magnetic flux density (B) — measure of magnetic-field strength and direction.

Circular motion — motion along a circular path produced when magnetic force acts as centripetal force.

Helical motion — spiral trajectory produced when a charged particle has both parallel and perpendicular velocity components relative to B.

Velocity selector — device using crossed electric and magnetic fields to select particles of a particular speed.

Crossed fields — electric and magnetic fields arranged perpendicular to one another.

Mass spectrometer — instrument that separates and analyzes ions according to their mass-to-charge ratios.

Particle accelerator — device that uses electromagnetic fields to accelerate and control charged particles.

Magnetic confinement — use of magnetic fields to control charged-particle motion.

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Key Takeaways

• Electric and magnetic fields affect charged particles differently.
• Electric force is described by F_E = qE.
• Electric fields can act on stationary or moving charges.
• Electric fields can change particle speed and kinetic energy.
• Magnetic force is described by F_B = |q|vB sin θ for magnitude.
• Magnetic fields exert no magnetic force on stationary charges.
• Magnetic force is perpendicular to velocity and magnetic field.
• A magnetic field acting alone does not change particle speed or kinetic energy.
• Perpendicular magnetic motion can produce circular trajectories.
• The circular radius is r = mv/(|q|B).
• Motion at an angle to B can produce a helical trajectory.
• Electric fields can produce parabolic trajectories when particles enter perpendicular to a uniform field.
• Combined electric and magnetic fields can control particle trajectories.
• A velocity selector uses opposing electric and magnetic forces.
• For an undeflected particle in a velocity selector, v = E/B.
• Particles moving too slowly are dominated by the electric force in the standard selector arrangement.
• Particles moving too quickly are dominated by the magnetic force.
• Velocity selectors can be combined with magnetic deflection to separate ions.
• Electric fields are commonly used to change particle energy.
• Magnetic fields are commonly used to bend, guide and confine charged particles.

Check Your Understanding

1. State the equation for the force on a charge in an electric field.
2. Explain how the force directions differ for positive and negative charges in an electric field.
3. Can an electric field exert a force on a stationary charge? Explain.
4. Can a magnetic field exert a magnetic force on a stationary charge? Explain.
5. State the equation for the magnetic force on a moving charge.
6. Explain why a magnetic field acting alone does not change a particle's kinetic energy.
7. Compare the trajectory of a charged particle entering perpendicular to a uniform electric field with one entering perpendicular to a uniform magnetic field.
8. State the equation for the radius of circular motion in a magnetic field.
9. Explain how increasing particle speed affects this radius.
10. What trajectory can occur when a particle enters a magnetic field at an angle between 0° and 90°?
11. What are crossed electric and magnetic fields?
12. Explain how a velocity selector works.
13. Derive v = E/B for an undeflected particle.
14. A velocity selector has E = 8.0 × 10⁴ N/C and B = 0.20 T. Calculate the selected speed.
15. A particle travels at 5.0 × 10⁵ m/s through a velocity selector containing a 0.30 T magnetic field. Calculate the required electric field strength.
16. What happens to a particle travelling more slowly than the selected velocity in a standard velocity selector?
17. Explain why particle charge magnitude does not appear in v = E/B.
18. Explain how a velocity selector and a second magnetic field can be combined to separate ions.
19. A singly charged ion of mass 4.0 × 10⁻²⁶ kg travels at 3.0 × 10⁵ m/s perpendicular to a 0.50 T magnetic field. Calculate its circular radius using |q| = 1.60 × 10⁻¹⁹ C.
20. Compare the main roles of electric and magnetic fields in a particle accelerator.