Force on Moving Charges

Magnetic Fields and Moving Charges

A magnetic field can exert a force on an electric charge that is moving through the field.

The size of the magnetic force depends on:

• magnetic flux density, B
• charge, q
• speed of the particle, v
• angle between the particle's velocity and the magnetic field, θ

The magnitude of the magnetic force is:

F = |q|vB sin θ

where:

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

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When Is the Magnetic Force Greatest?

From:

F = |q|vB sin θ

the magnetic force depends on the angle between the particle's velocity and the magnetic field.

The force is greatest when:

θ = 90°

because:

sin 90° = 1

Therefore:

F_max = |q|vB

A particle moving perpendicular to the magnetic field experiences the maximum magnetic force.

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When Is the Magnetic Force Zero?

If the particle moves parallel to the magnetic field:

θ = 0°

sin 0° = 0

Therefore:

F = 0

The same is true if the particle moves directly opposite the magnetic field:

θ = 180°

sin 180° = 0

Therefore:

F = 0

So:

v parallel to B → no magnetic force

v perpendicular to B → maximum magnetic force

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Stationary Charges

A stationary electric charge has:

v = 0

Using:

F = |q|vB sin θ

we obtain:

F = |q|(0)B sin θ

Therefore:

F = 0

A stationary charge experiences no magnetic force from a magnetic field alone.

This is an important difference between electric and magnetic fields.

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Electric Fields vs Magnetic Fields

An electric field can exert a force on a charge whether the charge is:

• stationary
• moving

The electric force is:

F = qE

A magnetic field requires the charge to be moving:

F = |q|vB sin θ

Therefore:

electric field → can act on stationary or moving charges

magnetic field → magnetic force acts on moving charges

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Direction of Magnetic Force

The magnetic force acts perpendicular to both:

• the particle's velocity
• the magnetic field

This means the three directions form a three-dimensional relationship.

For a positive charge, the force direction is given by:

F = q(v × B)

The symbol × represents the vector cross product.

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Right-Hand Rule for a Positive Charge

For a positive charge, use your right hand.

One useful version is:

• point your fingers in the direction of velocity v
• curl them toward the direction of magnetic field B
• your thumb points in the direction of the magnetic force F

Another common classroom version uses three mutually perpendicular fingers:

• index finger → velocity
• middle finger → magnetic field
• thumb → force

The exact hand positioning may vary with the convention being taught, but the physics is always based on:

v × B

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Positive Charges

For a positive particle:

F = q(v × B)

The right-hand rule gives the force direction directly.

For example, suppose:

velocity → right

magnetic field → into the page

Then the magnetic force on a positive charge points:

upward

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Negative Charges

For a negative charge, first use the right-hand rule as though the charge were positive.

Then:

reverse the direction

because q is negative.

Therefore, a positive and negative charge moving with the same velocity through the same magnetic field experience forces in opposite directions.

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Into and Out of the Page

Magnetic-field diagrams often represent directions perpendicular to the page.

⊙ = out of the page

Imagine seeing the point of an arrow coming toward you.

⊗ = into the page

Imagine seeing the crossed tail of an arrow moving away from you.

For example:

⊗ ⊗ ⊗ ⊗

⊗ ⊗ ⊗ ⊗

represents a magnetic field directed:

into the page

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Example: Direction of Force

A proton moves to the right through a magnetic field directed into the page.

The proton is positively charged.

Using the right-hand rule:

v → right

B → into page

Therefore:

F → upward

If an electron moved in exactly the same direction through the same field, its force would be:

downward

because the electron has negative charge.

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Example: Another Direction

A positive particle moves upward while the magnetic field points to the right.

Use the right-hand rule.

Velocity:

↑

Magnetic field:

→

The force points:

into the page ⊗

For a negative particle, the force would point:

out of the page ⊙

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Calculating Magnetic Force

For a particle moving perpendicular to the field:

F = |q|vB

Suppose:

q = 2.0 × 10⁻⁶ C

v = 500 m/s

B = 0.40 T

Then:

F = (2.0 × 10⁻⁶)(500)(0.40)

F = 4.0 × 10⁻⁴ N

Therefore:

F = 4.0 × 10⁻⁴ N

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Example: Proton in a Magnetic Field

A proton moves at:

v = 3.0 × 10⁶ m/s

through a magnetic field:

B = 0.20 T

The motion is perpendicular to the field.

The charge of a proton is:

q = 1.60 × 10⁻¹⁹ C

Use:

F = |q|vB

F = (1.60 × 10⁻¹⁹)(3.0 × 10⁶)(0.20)

F = 9.6 × 10⁻¹⁴ N

Although this force looks extremely small, the proton's mass is also extremely small, so its motion can change significantly.

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Force at an Angle

Suppose a particle does not move perpendicular to the field.

Use:

F = |q|vB sin θ

For example:

q = 4.0 × 10⁻⁶ C

v = 200 m/s

B = 0.50 T

θ = 30°

Since:

sin 30° = 0.5

F = (4.0 × 10⁻⁶)(200)(0.50)(0.5)

F = 2.0 × 10⁻⁴ N

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Why the Angle Matters

Only the component of velocity perpendicular to the magnetic field contributes to the magnetic force.

The velocity can be separated into:

• v∥ = component parallel to B
• v⊥ = component perpendicular to B

The magnetic force depends on:

v⊥

Therefore:

F = |q|v⊥B

Since:

v⊥ = v sin θ

we obtain:

F = |q|vB sin θ

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Magnetic Force Is Perpendicular to Motion

One of the most important properties of magnetic force is that it acts perpendicular to the particle's instantaneous velocity.

Therefore, the force does not push directly forward or backward along the direction of motion.

Instead, it tends to:

change the direction of motion

This leads to curved particle paths.

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Does a Magnetic Field Do Work?

For an isolated charged particle acted on only by a magnetic field, the magnetic force is perpendicular to the particle's displacement at each instant.

Therefore:

W = 0

for the magnetic force.

A magnetic field can change the particle's:

direction

but not its:

speed

or:

kinetic energy

when no other forces are involved.

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Magnetic Fields Bend Paths

Imagine a charged particle entering a uniform magnetic field perpendicular to the field.

The magnetic force acts sideways.

The particle begins to turn.

As its direction changes, the magnetic force also changes direction so that it remains perpendicular to the velocity.

The result is:

circular motion

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Magnetic Force as Centripetal Force

For a charged particle moving perpendicular to a uniform magnetic field, the magnetic force provides the centripetal force.

Magnetic force:

F_B = |q|vB

Centripetal force:

F_c = mv²/r

Therefore:

|q|vB = mv²/r

Cancel one v:

|q|B = mv/r

Rearrange:

r = mv/(|q|B)

This equation gives the radius of the circular path.

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Radius of the Circular Path

From:

r = mv/(|q|B)

we can see several important relationships.

Increasing mass

larger m → larger r

A more massive particle bends less.

Increasing speed

larger v → larger r

A faster particle follows a larger circular path, in the non-relativistic case.

Increasing charge magnitude

larger |q| → smaller r

A more highly charged particle bends more strongly.

Increasing magnetic field

larger B → smaller r

A stronger magnetic field bends the particle more strongly.

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Example: Radius of a Proton's Path

A proton has:

m = 1.67 × 10⁻²⁷ kg

q = 1.60 × 10⁻¹⁹ C

v = 2.0 × 10⁶ m/s

B = 0.50 T

Use:

r = mv/(|q|B)

r = [(1.67 × 10⁻²⁷)(2.0 × 10⁶)] / [(1.60 × 10⁻¹⁹)(0.50)]

r ≈ 0.0418 m

Therefore:

r ≈ 4.2 cm

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Positive and Negative Particles Curve Opposite Ways

Suppose a positive and negative particle have:

• equal mass
• equal speed
• equal charge magnitude

and enter the same magnetic field.

The magnitude of their magnetic forces will be equal.

However, their force directions will be opposite.

Therefore, their paths curve in:

opposite directions

This allows magnetic fields to help distinguish particles by the sign of their charge.

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Comparing Proton and Electron Paths

A proton and electron have equal charge magnitudes:

|q_p| = |q_e|

but very different masses.

The proton is much more massive than the electron.

From:

r = mv/(|q|B)

if both have the same speed in the same field:

proton → larger radius

electron → smaller radius

They also curve in opposite directions because their charges have opposite signs.

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What If the Particle Moves Parallel to the Field?

Suppose:

v ∥ B

Then:

θ = 0°

and:

F = 0

The particle continues moving in a straight line at constant velocity if no other forces act.

Therefore, a magnetic field does not necessarily cause every moving charged particle to curve.

The direction of motion matters.

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What If the Particle Enters at an Angle?

Suppose the particle has both:

• velocity parallel to the field
• velocity perpendicular to the field

The parallel component is unaffected.

The perpendicular component produces circular motion.

Combining these motions creates a:

helical or spiral path

The particle spirals around the magnetic field direction while continuing to move along it.

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Understanding Helical Motion

Break the velocity into:

v∥ — parallel to the field

and:

v⊥ — perpendicular to the field

The parallel component produces forward motion.

The perpendicular component produces circular motion.

Together:

straight motion + circular motion = helical motion

This behavior is common in plasmas and space physics.

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Charged Particles Near Earth

Charged particles from the Sun interact with Earth's magnetic field.

Many follow curved or spiral trajectories around magnetic field lines.

This helps explain phenomena such as:

• Van Allen radiation belts
• charged-particle trapping
• auroras
• space weather

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Period of Circular Motion

For a particle moving perpendicular to a uniform magnetic field:

r = mv/(|q|B)

The circumference of the circular path is:

2πr

The period is:

T = distance/speed

T = 2πr/v

Substituting for r gives:

T = 2πm/(|q|B)

Notice that the speed cancels.

In the non-relativistic case, the period depends on:

• particle mass
• charge magnitude
• magnetic flux density

but not on speed.

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Cyclotron Frequency

The frequency of circular motion is:

f = 1/T

Therefore:

f = |q|B/(2πm)

This is called the cyclotron frequency.

A stronger magnetic field produces a greater cyclotron frequency.

A more massive particle has a lower cyclotron frequency for the same charge magnitude and field strength.

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Velocity Selectors

Electric and magnetic forces can be combined to select particles with a particular speed.

Suppose the electric and magnetic forces act in opposite directions.

Electric force:

F_E = |q|E

Magnetic force:

F_B = |q|vB

For a 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 speed pass through without being deflected.

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Mass Spectrometers

Magnetic forces are extremely useful for analyzing charged particles.

A mass spectrometer can separate ions according to their mass-to-charge ratio.

Since:

r = mv/(|q|B)

particles with different:

• masses
• charges
• speeds

can follow different paths.

This allows scientists to identify:

• isotopes
• atoms
• molecules

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Particle Accelerators

Magnetic fields are also used in particle accelerators.

Magnets can:

• bend particle beams
• focus charged particles
• guide particles around circular tracks

Because magnetic forces can change direction without directly changing kinetic energy, they are especially useful for steering particle beams.

Electric fields are generally used to increase the particles' energy.

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Bubble Chambers and Particle Tracks

Charged particles moving through magnetic fields can leave visible tracks in certain particle detectors.

The direction of curvature can reveal:

sign of charge

The radius of curvature can provide information about:

momentum

This has been extremely important in particle physics.

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Magnetic Force and Momentum

For perpendicular motion:

r = mv/(|q|B)

For non-relativistic motion:

p = mv

Therefore:

r = p/(|q|B)

or:

p = |q|Br

This means that measuring the radius of a charged particle's curved path can provide information about its momentum.

At very high speeds, relativistic momentum must be used.

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Example: Comparing Two Particles

Particle A and particle B have the same:

• mass
• speed
• charge magnitude

Particle A enters:

B = 0.20 T

Particle B enters:

B = 0.40 T

From:

r = mv/(|q|B)

doubling B halves r.

Therefore:

r_B = ½r_A

Particle B follows the tighter circular path.

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Example: Changing Speed

A particle follows a circular path of radius:

r = 0.10 m

If its speed doubles while:

• mass remains constant
• charge remains constant
• B remains constant

then:

r ∝ v

Therefore:

new radius = 0.20 m

The faster particle curves less sharply.

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Example: Changing Charge

Suppose two particles have identical mass and speed.

Particle A has charge magnitude:

|q|

Particle B has charge magnitude:

2|q|

Since:

r ∝ 1/|q|

Particle B has:

half the radius

Therefore, a more highly charged particle bends more strongly in the same magnetic field.

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Direction Problem Strategy

For magnetic-force direction questions:

Step 1: Identify the direction of velocity.

Step 2: Identify the direction of magnetic field.

Step 3: Use the right-hand rule for a positive charge.

Step 4: Determine the force direction.

Step 5: If the particle is negatively charged, reverse your answer.

This final step is particularly important for electrons.

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Calculation Problem Strategy

For force calculations:

Identify the quantities

q, v, B and θ

Select the equation

F = |q|vB sin θ

Check the angle

If perpendicular:

sin 90° = 1

so:

F = |q|vB

Substitute carefully

Pay particular attention to scientific notation.

Include units

Force must be expressed in:

newtons (N)

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Circular Motion Problem Strategy

If the particle moves perpendicular to a uniform magnetic field and follows a circular path, use:

r = mv/(|q|B)

You can rearrange this equation.

For mass:

m = r|q|B/v

For speed:

v = r|q|B/m

For magnetic flux density:

B = mv/(r|q|)

For charge magnitude:

|q| = mv/(rB)

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Multi-Step Example

An ion has:

q = +3.2 × 10⁻¹⁹ C

m = 6.6 × 10⁻²⁷ kg

v = 4.0 × 10⁵ m/s

B = 0.25 T

The velocity is perpendicular to the field.

Calculate the magnetic force

F = |q|vB

F = (3.2 × 10⁻¹⁹)(4.0 × 10⁵)(0.25)

F = 3.2 × 10⁻¹⁴ N

Calculate the radius

r = mv/(|q|B)

r = [(6.6 × 10⁻²⁷)(4.0 × 10⁵)] / [(3.2 × 10⁻¹⁹)(0.25)]

r = 0.033 m

r ≈ 3.3 cm

The magnetic force continually changes the direction of the particle's velocity, producing circular motion.

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Common Mistakes

Mistake: Saying a magnetic field exerts a magnetic force on a stationary charge.
If v = 0, the magnetic force is zero.

Mistake: Using F = qvB in every situation.
The full magnitude equation is F = |q|vB sin θ.

Mistake: Forgetting that force is zero for motion parallel to the field.
If θ = 0° or 180°, sin θ = 0.

Mistake: Using the right-hand-rule answer directly for an electron.
Use the rule for a positive charge and reverse the direction for a negative charge.

Mistake: Saying magnetic force increases the particle's speed.
A magnetic force acting alone changes direction, not speed.

Mistake: Assuming all moving charges follow circular paths.
Circular motion occurs when velocity is perpendicular to a uniform magnetic field.

Mistake: Assuming a charge moving parallel to B follows a circular path.
It experiences no magnetic force.

Mistake: Confusing ⊙ and ⊗.
⊙ = out of page; ⊗ = into page.

Mistake: Forgetting the particle's charge sign when analyzing curvature.
Positive and negative particles curve in opposite directions.

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Key Terms

Magnetic force — force experienced by a moving electric charge in a magnetic field.

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

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

Charge (q) — electrical property of a particle, measured in coulombs.

Velocity (v) — speed and direction of particle motion.

Right-hand rule — method used to determine magnetic-force direction for a positive charge.

Cross product — vector operation describing the directional relationship in F = q(v × B).

Perpendicular — meeting at an angle of 90°.

Centripetal force — inward force required for circular motion.

Circular motion — motion along a circular path.

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

Cyclotron frequency — frequency at which a non-relativistic charged particle circles in a uniform magnetic field.

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

Mass spectrometer — instrument that separates and identifies ions using their mass-to-charge ratios.

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Key Takeaways

• Magnetic fields exert forces on moving electric charges.
• A stationary charge experiences no magnetic force from a magnetic field alone.
• The magnetic-force magnitude is F = |q|vB sin θ.
• The force is greatest when velocity and magnetic field are perpendicular.
• The force is zero when velocity and magnetic field are parallel or antiparallel.
• Magnetic force is perpendicular to both velocity and magnetic field.
• The right-hand rule gives the force direction for a positive charge.
• The force direction is reversed for a negative charge.
• A magnetic force acting alone changes a particle's direction but not its speed or kinetic energy.
• A particle moving perpendicular to a uniform magnetic field can follow a circular path.
• The magnetic force provides the centripetal force.
• The circular-path radius is r = mv/(|q|B).
• Greater mass produces a larger radius.
• Greater speed produces a larger radius in the non-relativistic case.
• Greater charge magnitude produces a smaller radius.
• Greater magnetic flux density produces a smaller radius.
• Positive and negative particles curve in opposite directions.
• A particle moving parallel to the magnetic field experiences no magnetic force.
• A particle entering at an angle can follow a helical path.
• Magnetic deflection is used in mass spectrometers, particle accelerators and many other technologies.

Check Your Understanding

1. State the equation for the magnetic force on a moving charge.
2. What four factors determine the magnitude of the magnetic force?
3. When is the magnetic force on a moving charge greatest?
4. When is the magnetic force zero?
5. Explain why a stationary charge is not affected by a magnetic field alone.
6. Describe how to use the right-hand rule to determine the force on a positive charge.
7. How is the force direction determined for a negative charge?
8. What do ⊙ and ⊗ represent?
9. A charge of 3.0 × 10⁻⁶ C moves at 400 m/s perpendicular to a 0.50 T magnetic field. Calculate the magnetic force.
10. A proton moves at 2.0 × 10⁶ m/s perpendicular to a 0.30 T field. Calculate the magnetic force using q = 1.60 × 10⁻¹⁹ C.
11. Why does a magnetic force acting alone not change the speed of a charged particle?
12. Explain why a charged particle can follow a circular path in a uniform magnetic field.
13. Derive r = mv/(|q|B) from the equations for magnetic and centripetal force.
14. How does increasing particle speed affect the radius of its circular path?
15. How does increasing magnetic flux density affect the radius?
16. Why do positive and negative particles curve in opposite directions?
17. Describe the path of a charged particle moving exactly parallel to a uniform magnetic field.
18. Describe the path of a charged particle entering a magnetic field at an angle between 0° and 90°.
19. Explain how a magnetic field can be used to separate particles with different mass-to-charge ratios.
20. A particle's circular path becomes half as large while its mass, speed and charge remain unchanged. What happened to the magnetic flux density?