Force on Current-Carrying Wires

Why Does a Current-Carrying Wire Experience a Force?

When a wire carrying an electric current is placed in a magnetic field, the wire can experience a magnetic force.

Electric current consists of moving electric charges. Because magnetic fields exert forces on moving charges, the charges moving through the wire experience magnetic forces.

These forces are transferred to the conductor as a whole, causing the wire to experience a force.

This effect is sometimes called the motor effect because it is the basic principle behind electric motors.

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From Moving Charges to Force on a Wire

For an individual moving charge:

F = |q|vB sin θ

In a current-carrying conductor, enormous numbers of charged particles are moving.

Their combined magnetic interactions result in a force on the conductor.

For a straight conductor in a uniform magnetic field:

F = BIL sin θ

where:

• F = magnetic force (N)
• B = magnetic flux density (T)
• I = current (A)
• L = length of conductor in the magnetic field (m)
• θ = angle between the current and magnetic field

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The Simplified Force Equation

If the wire is perpendicular to the magnetic field:

θ = 90°

Since:

sin 90° = 1

the equation becomes:

F = BIL

This is the form commonly used in introductory magnetic-force calculations.

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Meaning of the Equation

From:

F = BIL sin θ

we can see that magnetic force depends on:

• magnetic flux density
• current
• length of wire inside the magnetic field
• angle between current and magnetic field

Therefore:

stronger field → greater force

larger current → greater force

longer wire in field → greater force

The angle also affects the force.

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Maximum Magnetic Force

The magnetic force is greatest when the wire is perpendicular to the magnetic field.

When:

θ = 90°

then:

sin θ = 1

Therefore:

F_max = BIL

This means a conductor experiences its maximum magnetic force when the current flows at right angles to the magnetic field.

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Zero Magnetic Force

If the current flows parallel to the magnetic field:

θ = 0°

Since:

sin 0° = 0

then:

F = 0

The same occurs when the current is directly opposite the magnetic-field direction:

θ = 180°

Therefore:

current parallel to field → zero magnetic force

current perpendicular to field → maximum magnetic force

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Direction of the Magnetic Force

The magnetic force acts perpendicular to both:

• current direction
• magnetic field direction

Therefore, determining the force direction requires thinking in three dimensions.

A convenient method is:

Fleming's Left-Hand Rule

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Fleming's Left-Hand Rule

Hold the thumb, first finger, and second finger of your left hand mutually perpendicular.

Each represents a different direction.

First finger → Field

The first finger points in the direction of the magnetic field.

Second finger → Current

The second finger points in the direction of conventional current.

Thumb → Force

The thumb points in the direction of the force or motion.

A useful memory aid is:

First finger = Field

seCond finger = Current

thuMb = Motion

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Conventional Current

When using Fleming's Left-Hand Rule, use the direction of:

conventional current

Conventional current is defined as the direction positive charge would move.

Therefore, conventional current flows:

from positive toward negative through the external circuit

Electrons in a metal move in the opposite direction.

Do not use electron-flow direction when applying Fleming's Left-Hand Rule unless you deliberately reverse it to obtain conventional current.

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Example: Finding the Force Direction

Suppose:

• magnetic field points from left to right
• current flows upward

Using Fleming's Left-Hand Rule:

First finger → right

Second finger → up

Thumb then indicates the direction of force.

The force is perpendicular to both current and field.

This may point into or out of the page depending on their exact orientation.

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Into and Out of the Page

As with moving-charge problems, diagrams often use symbols for directions perpendicular to the page.

⊙ = out of the page

Think of seeing the pointed tip of an arrow coming toward you.

⊗ = into the page

Think of 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: Field Into the Page

Suppose:

current → right

magnetic field → into page ⊗

Using Fleming's Left-Hand Rule:

Field → into page

Current → right

Force → upward

Therefore:

F ↑

If the current is reversed:

F ↓

Reversing current reverses the force.

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Reversing the Current

The force direction depends on current direction.

If:

I reverses

then:

F reverses

provided the magnetic field remains unchanged.

This is extremely important in electric motors.

Changing the current direction changes the direction in which the conductor is pushed.

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Reversing the Magnetic Field

The force also depends on magnetic-field direction.

If the magnetic poles are reversed:

B reverses

therefore:

F reverses

provided the current remains unchanged.

So either of the following reverses the force:

• reversing current
• reversing magnetic field

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Reversing Both

Suppose both the current and magnetic field are reversed simultaneously.

Current reverses once.

Field reverses once.

These two reversals cancel in terms of force direction.

Therefore:

reverse I and B together → force remains in the original direction

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Calculating the Magnetic Force

Suppose a wire has:

B = 0.40 T

I = 3.0 A

L = 0.20 m

and is perpendicular to the magnetic field.

Use:

F = BIL

F = (0.40)(3.0)(0.20)

F = 0.24 N

Therefore:

F = 0.24 N

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Example: Finding Magnetic Flux Density

A wire experiences a force:

F = 0.60 N

when:

I = 5.0 A

and:

L = 0.30 m

The wire is perpendicular to the field.

Use:

F = BIL

Rearrange:

B = F/(IL)

B = 0.60/(5.0 × 0.30)

B = 0.60/1.50

B = 0.40 T

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Example: Finding Current

A conductor of length:

L = 0.25 m

is perpendicular to a field:

B = 0.80 T

The force is:

F = 0.50 N

Use:

F = BIL

Rearrange:

I = F/(BL)

I = 0.50/(0.80 × 0.25)

I = 0.50/0.20

I = 2.5 A

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Example: Finding Length

A wire carries:

I = 4.0 A

through a field:

B = 0.50 T

The magnetic force is:

F = 0.80 N

The wire is perpendicular to the field.

Use:

F = BIL

Rearrange:

L = F/(BI)

L = 0.80/(0.50 × 4.0)

L = 0.80/2.0

L = 0.40 m

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Force at an Angle

If the conductor is not perpendicular to the magnetic field, use:

F = BIL sin θ

Suppose:

B = 0.50 T

I = 4.0 A

L = 0.30 m

θ = 30°

Since:

sin 30° = 0.5

F = (0.50)(4.0)(0.30)(0.5)

F = 0.30 N

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Why the Angle Matters

Only the component of the wire/current direction perpendicular to the magnetic field contributes fully to the magnetic force.

Therefore:

F = BIL sin θ

At 90°:

maximum force

At 45°:

intermediate force

At 0°:

zero force

This relationship can be summarized as:

0° → F = 0

90° → F = maximum

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Effect of Magnetic Flux Density

From:

F = BIL

when I and L remain constant:

F ∝ B

Therefore, doubling the magnetic flux density doubles the force.

For example:

B = 0.20 T → F = 0.40 N

If B doubles:

B = 0.40 T → F = 0.80 N

provided all other factors remain unchanged.

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Effect of Current

From:

F = BIL

when B and L remain constant:

F ∝ I

Therefore:

double current → double force

triple current → triple force

Increasing current is one way of increasing the force produced by an electric motor.

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Effect of Conductor Length

The equation uses the length of conductor that is actually inside the magnetic field.

From:

F = BIL

when B and I remain constant:

F ∝ L

Therefore:

greater length in field → greater force

If only 5 cm of a 30 cm wire lies within the magnetic field, use:

L = 0.05 m

not:

L = 0.30 m.

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Unit Conversions

Length must normally be expressed in:

metres (m)

For example:

25 cm = 0.25 m

8 cm = 0.08 m

50 mm = 0.050 m

Magnetic flux density should be in:

teslas (T)

For example:

200 mT = 0.200 T

Current should be in:

amperes (A)

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Example with Unit Conversion

A 12 cm wire carries a current of 5.0 A perpendicular to a magnetic field of 300 mT.

Convert:

L = 12 cm = 0.12 m

B = 300 mT = 0.300 T

Use:

F = BIL

F = (0.300)(5.0)(0.12)

F = 0.18 N

Unit conversion should be completed before substitution.

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Why the Wire Moves

The moving charge carriers inside the wire experience magnetic forces.

In a metal, the mobile charge carriers are electrons.

Their interactions with the conductor's structure transfer force to the wire.

As a result, the entire conductor can experience a mechanical force.

This is a direct link between:

electricity → magnetism → mechanical motion

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The Motor Effect

The force on a current-carrying conductor in a magnetic field is called the:

motor effect

The motor effect allows electrical energy to be converted into mechanical motion.

It is used in many technologies, including:

• electric motors
• loudspeakers
• headphones
• moving-coil meters
• actuators

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Force on a Current-Carrying Coil

Consider a rectangular coil placed in a magnetic field.

Current flows around the coil.

Opposite sides of the coil carry current in opposite directions.

The magnetic forces on these sides therefore point in opposite directions.

Instead of simply moving the whole coil sideways, these forces can create a:

turning effect

or:

torque

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Torque on a Coil

The forces on opposite sides of a coil form a couple.

A couple produces rotation.

For a coil, the turning effect depends on factors including:

• magnetic flux density
• current
• number of turns
• coil area
• orientation of the coil

For an ideal coil:

τ = NBIA sin θ

where:

• τ = torque (N·m)
• N = number of turns
• B = magnetic flux density (T)
• I = current (A)
• A = area of coil (m²)
• θ = appropriate angle between the coil's magnetic moment and the magnetic field

This relationship becomes important when studying electric motors in greater detail.

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Electric Motors

An electric motor uses the motor effect to produce rotational motion.

A simple DC motor contains:

• magnetic field
• current-carrying coil
• axle
• brushes
• split-ring commutator
• electrical supply

The magnetic field exerts forces on opposite sides of the current-carrying coil.

These forces create a turning effect.

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How a Simple DC Motor Works

The sequence is:

electric current enters coil

↓

coil is inside magnetic field

↓

opposite sides experience magnetic forces

↓

forces create torque

↓

coil rotates

↓

commutator reverses current every half-turn

↓

rotation continues in the same overall direction

The motor therefore converts:

electrical energy → mechanical energy

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Why the Current Must Reverse

After half a rotation, the sides of the coil have swapped positions.

Without reversing the current, the torque would reverse and tend to rotate the coil back.

The split-ring commutator reverses the current every half-turn.

This keeps the torque acting in the appropriate rotational direction.

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Increasing the Force in a Motor

The force on each conductor can be increased by:

Increasing magnetic flux density

Larger B → larger F

Increasing current

Larger I → larger F

Increasing conductor length in the field

Larger L → larger F

Motor torque can also be increased by:

• using more turns in the coil
• increasing coil area
• optimizing the magnetic-field geometry

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Loudspeakers

A loudspeaker also uses the motor effect.

A loudspeaker contains:

• permanent magnet
• voice coil
• flexible cone or diaphragm

An alternating electrical current flows through the coil.

The current changes direction repeatedly.

Therefore, the magnetic force also changes direction.

The coil moves back and forth.

The attached cone vibrates and produces:

sound waves

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Why Speakers Need Alternating Current

Sound requires the speaker cone to vibrate back and forth.

If the current changes direction:

current reverses → magnetic force reverses

Therefore, an electrical audio signal produces corresponding motion of the speaker cone.

The changing electrical signal is converted into:

mechanical vibration → sound

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Headphones

Headphones operate using a similar principle.

A current-carrying coil interacts with the magnetic field of a permanent magnet.

The changing force moves a small diaphragm.

The vibrating diaphragm produces pressure variations in the air that we hear as sound.

This allows an electrical audio signal to be converted into sound.

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Moving-Coil Meters

Traditional analogue electrical meters can use the motor effect.

A current passes through a coil located in a magnetic field.

The resulting torque rotates the coil and moves a pointer.

A larger current produces a larger torque, causing a greater pointer deflection.

This principle has been used in:

• ammeters
• voltmeters
• galvanometers

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Electromagnetic Actuators

A magnetic force can also be used to produce controlled linear or rotational movement.

Such systems are used in:

• relays
• valves
• robotic mechanisms
• automotive systems
• industrial machinery

Electrical signals can therefore control physical motion through magnetic forces.

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Force Between Parallel Current-Carrying Wires

Current-carrying wires produce magnetic fields.

Therefore, two nearby current-carrying wires can exert forces on each other.

For two parallel wires:

currents in the same direction → attraction

currents in opposite directions → repulsion

This occurs because each wire is located in the magnetic field produced by the other.

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Magnetic Force Is a Vector

Force has both:

• magnitude
• direction

Calculating:

F = BIL sin θ

gives the magnitude.

Fleming's Left-Hand Rule gives the direction.

A complete solution to many magnetic-force problems therefore requires both:

1. calculate the force
2. determine its direction

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Example: Magnitude and Direction

A wire carries:

I = 6.0 A

toward the right.

It lies perpendicular to a magnetic field:

B = 0.25 T

directed into the page.

The length inside the field is:

L = 0.40 m

Calculate the magnitude

F = BIL

F = (0.25)(6.0)(0.40)

F = 0.60 N

Determine the direction

Using Fleming's Left-Hand Rule:

Field → into page

Current → right

Force → upward

Therefore:

F = 0.60 N upward

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Multi-Step Example

A 15 cm wire carries a current of 8.0 A through a uniform magnetic field of 0.50 T.

The current is perpendicular to the field.

Convert length

15 cm = 0.15 m

Calculate force

F = BIL

F = (0.50)(8.0)(0.15)

F = 0.60 N

The current is then reduced to:

4.0 A

The current has been halved.

Since:

F ∝ I

the force is also halved:

F = 0.30 N

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Example with an Angle

A wire has:

B = 0.60 T

I = 5.0 A

L = 0.40 m

θ = 30°

Use:

F = BIL sin θ

F = (0.60)(5.0)(0.40)(sin 30°)

F = (0.60)(5.0)(0.40)(0.5)

F = 0.60 N

If the wire were perpendicular instead:

F = (0.60)(5.0)(0.40)

F = 1.20 N

The angled conductor therefore experiences a smaller force.

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Problem-Solving Strategy

For magnetic-force problems involving wires:

Identify the quantities

Look for:

• B
• I
• L
• θ

Convert units

Especially:

• cm → m
• mm → m
• mT → T

Choose the equation

If perpendicular:

F = BIL

Otherwise:

F = BIL sin θ

Calculate the magnitude

Substitute carefully.

Determine direction

Use:

Fleming's Left-Hand Rule

State a complete answer

Include:

• numerical value
• unit
• direction when required

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Common Mistakes

Mistake: Using the total wire length instead of the length inside the magnetic field.
Use only the conductor length exposed to the relevant field.

Mistake: Forgetting to convert centimetres into metres.
For example, 20 cm = 0.20 m.

Mistake: Forgetting to convert milliteslas into teslas.
For example, 250 mT = 0.250 T.

Mistake: Using F = BIL when the wire is not perpendicular to the field.
Use F = BIL sin θ.

Mistake: Saying force is greatest when the current is parallel to the field.
The force is zero when they are parallel and maximum when perpendicular.

Mistake: Using electron-flow direction in Fleming's Left-Hand Rule.
Use conventional current.

Mistake: Using Fleming's Right-Hand Rule for the motor effect.
The force on a current-carrying wire is found using the Left-Hand Rule.

Mistake: Forgetting that reversing current reverses the force.

Mistake: Assuming an electric motor simply pulls a coil toward a magnet.
Forces on different sides of the coil create a torque that produces rotation.

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Key Terms

Motor effect — force experienced by a current-carrying conductor in a magnetic field.

Current-carrying conductor — conductor through which electric current flows.

Magnetic force — force produced by the interaction between moving charges/current and a magnetic field.

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

Tesla (T) — SI unit of magnetic flux density.

Fleming's Left-Hand Rule — rule used to determine the directions of field, conventional current and force in the motor effect.

Conventional current — defined direction of positive-charge flow in a circuit.

Torque — turning effect of a force.

Couple — pair of forces that produces rotation.

Motor — device that converts electrical energy into mechanical energy.

Commutator — component in a simple DC motor that reverses current in the coil every half-turn.

Brushes — electrical contacts that transfer current to the rotating part of a simple DC motor.

Voice coil — current-carrying coil in a loudspeaker that moves within a magnetic field.

Actuator — device that converts an electrical control signal into physical movement.

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Key Takeaways

• A current-carrying wire can experience a force when placed in a magnetic field.
• Current consists of moving charges, which interact with magnetic fields.
• The effect is known as the motor effect.
• Magnetic-force magnitude is given by F = BIL sin θ.
• For a perpendicular conductor, F = BIL.
• Magnetic force is greatest when current and field are perpendicular.
• Magnetic force is zero when current and field are parallel.
• Increasing B increases the force.
• Increasing I increases the force.
• Increasing the length of conductor in the field increases the force.
• Fleming's Left-Hand Rule determines the force direction.
• First finger represents magnetic field.
• Second finger represents conventional current.
• Thumb represents force/motion.
• Reversing current reverses the force.
• Reversing the magnetic field reverses the force.
• Reversing both current and field leaves the force direction unchanged.
• Forces on opposite sides of a current-carrying coil can produce torque.
• Electric motors use this effect to convert electrical energy into mechanical energy.
• Loudspeakers use changing magnetic forces to convert electrical signals into sound.

Check Your Understanding

1. Explain why a current-carrying wire can experience a force in a magnetic field.
2. State the equation for the magnetic force on a straight current-carrying conductor.
3. What does each symbol in F = BIL sin θ represent?
4. Under what condition can the equation be simplified to F = BIL?
5. When is the magnetic force maximum?
6. When is the magnetic force zero?
7. Describe Fleming's Left-Hand Rule.
8. Which finger represents the magnetic field?
9. Which finger represents conventional current?
10. Which finger represents force or motion?
11. What happens to the force direction if the current is reversed?
12. What happens if the magnetic-field direction is reversed?
13. A 0.30 m wire carries 4.0 A perpendicular to a 0.50 T field. Calculate the force.
14. A 20 cm wire carries 6.0 A perpendicular to a 300 mT field. Calculate the force.
15. A conductor experiences a force of 0.80 N while carrying 5.0 A through a 0.40 T field. If it is perpendicular to the field, calculate the length of conductor in the field.
16. Explain how increasing current affects the force on a conductor.
17. Explain how opposite forces on a current-carrying coil can produce rotation.
18. Describe the role of the split-ring commutator in a simple DC motor.
19. Explain how the motor effect is used in a loudspeaker.
20. Suggest three changes that could increase the turning effect produced by a current-carrying coil in an electric motor.

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.

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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.

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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.

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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

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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

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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.

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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.

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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.

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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.

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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

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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

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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.

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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

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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.

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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.

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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

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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

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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

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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.

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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

1. Explain why a charged particle moving perpendicular to a magnetic field can follow a circular path.
2. What force provides the centripetal force?
3. State the magnetic-force equation for a particle moving perpendicular to a magnetic field.
4. State the equation for centripetal force.
5. Derive r = mv/(|q|B).
6. How does increasing particle mass affect the radius?
7. How does increasing particle speed affect the radius?
8. How does increasing magnetic flux density affect the radius?
9. How does increasing charge magnitude affect the radius?
10. What determines whether a particle curves clockwise or counterclockwise?
11. Explain why a magnetic field does not change the speed of a particle when it is the only force acting.
12. 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.
13. A particle's speed doubles while all other quantities remain constant. What happens to its radius?
14. The magnetic flux density doubles while all other quantities remain constant. What happens to the radius?
15. Describe the motion of a charged particle travelling exactly parallel to a uniform magnetic field.
16. Explain why a particle entering a magnetic field at an angle can follow a helical path.
17. 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.
18. Explain how circular motion in a magnetic field can be used to separate ions in a mass spectrometer.
19. Explain why magnetic fields are useful for confining extremely hot plasmas.
20. 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.

Motors and Loudspeakers

Electricity, Magnetism, and Motion

Electric motors and loudspeakers are examples of electromechanical devices. They use interactions between electric currents and magnetic fields to produce mechanical motion.

The key principle is the motor effect:

A current-carrying conductor placed in a magnetic field can experience a force.

For a straight conductor:

F = BIL sin θ

where:

• F = magnetic force (N)
• B = magnetic flux density (T)
• I = current (A)
• L = length of conductor in the magnetic field (m)
• θ = angle between the current and magnetic field

When the conductor is perpendicular to the field:

F = BIL

This simple principle allows electrical energy to produce movement.

---

The Motor Effect

Electric current consists of moving charges.

When these moving charges pass through a magnetic field, they experience magnetic forces.

In a wire, these forces are transferred to the conductor.

Therefore:

electric current + magnetic field → force on conductor

The direction of this force can be determined using Fleming's Left-Hand Rule.

---

Fleming's Left-Hand Rule

Hold the thumb, first finger, and second finger of your left hand mutually perpendicular.

First finger

Represents the magnetic:

Field

Second finger

Represents conventional:

Current

Thumb

Represents:

Force or Motion

A useful memory aid is:

First = Field

seCond = Current

thuMb = Motion

---

What Is an Electric Motor?

An electric motor is a device that converts:

electrical energy → mechanical energy

Motors are found in:

• electric vehicles
• fans
• pumps
• washing machines
• refrigerators
• elevators
• drones
• power tools
• computer cooling systems
• industrial machinery
• robots

Although motors can have very different designs, their operation ultimately involves magnetic forces.

---

The Simple DC Motor

A simple DC motor provides a useful model for understanding how electric motors work.

Important components include:

• permanent magnets or electromagnets
• rotating coil
• axle
• split-ring commutator
• brushes
• DC power supply

---

Magnetic Field

A simple motor contains a magnetic field produced by:

• permanent magnets

or:

• electromagnets

The coil is positioned within this field.

A strong magnetic field helps produce a large force because:

F ∝ B

---

The Current-Carrying Coil

Electric current flows through a coil of wire located inside the magnetic field.

Opposite sides of the coil carry current in opposite directions.

Because force direction depends on current direction, the two sides experience forces in opposite directions.

One side may be pushed upward while the other is pushed downward.

These forces create a:

turning effect

---

Torque

The turning effect of a force is called:

torque

The forces on opposite sides of the coil form a couple.

Instead of simply pushing the entire coil in one direction, the forces cause it to rotate.

The motor effect therefore converts magnetic forces into rotational motion.

---

How the Coil Begins to Rotate

Suppose one side of the coil carries current into the page while the opposite side carries current out of the page.

Using Fleming's Left-Hand Rule:

• one side experiences an upward force
• the opposite side experiences a downward force

These forces create torque around the axle.

The coil begins to rotate.

---

The Problem After Half a Turn

If the current direction remained unchanged, the forces would eventually produce torque in the opposite rotational direction after the coil had turned through approximately half a rotation.

The coil would not continue rotating smoothly in the same direction.

A simple DC motor solves this problem using the:

split-ring commutator

---

Split-Ring Commutator

The split-ring commutator reverses the current through the rotating coil every half-turn.

The sequence is:

coil rotates

↓

half-turn completed

↓

commutator changes electrical contact

↓

current through coil reverses

↓

magnetic forces reverse

↓

torque continues rotation in the same overall direction

This allows continuous rotation.

---

Brushes

The brushes provide electrical contact between the stationary power supply and the rotating commutator.

Traditionally, brushes may be made from carbon-based materials.

Their purpose is to allow current to flow into and out of the rotating coil while the motor is operating.

In a simple model:

power supply → brushes → commutator → coil

---

Axle

The rotating coil is attached to an:

axle

The torque on the coil rotates the axle.

The axle can then transfer mechanical energy to another device.

For example, the axle might turn:

• a fan blade
• a wheel
• a pump
• a gear system

---

Energy Transfer in a Motor

A motor converts electrical energy into mechanical energy.

The useful energy pathway can be summarized as:

electrical energy → magnetic interaction → rotational kinetic energy

However, real motors are not 100% efficient.

Some energy is transferred to the surroundings through:

• heating in wires
• friction
• sound
• magnetic losses

Therefore:

electrical input energy = useful mechanical energy + energy dissipated

---

Increasing the Force in a Motor

For a conductor perpendicular to the field:

F = BIL

Therefore, the force can be increased by increasing:

Magnetic flux density

Larger B → larger force

Current

Larger I → larger force

Length of conductor in the magnetic field

Larger L → larger force

A motor can therefore be made more powerful by carefully controlling these factors.

---

Increasing Motor Torque

Motor torque can also be increased by:

• increasing the current
• increasing magnetic field strength
• increasing the number of turns in the coil
• increasing coil area
• optimizing the orientation of the coil and magnetic field

For an ideal current-carrying coil:

τ = NBIA sin θ

where:

• τ = torque (N·m)
• N = number of turns
• B = magnetic flux density (T)
• I = current (A)
• A = coil area (m²)
• θ = angle between the coil's magnetic moment and the field

---

Why More Turns Help

Suppose a coil contains many loops of wire.

Each loop experiences magnetic forces.

The turning effects from the loops combine.

Therefore:

more turns → greater torque

provided other conditions remain the same.

This is why practical motor windings contain many turns rather than a single loop.

---

Reversing Motor Direction

The direction of motor rotation can be reversed by reversing either:

• current direction

or:

• magnetic-field direction

If both are reversed simultaneously, the direction of force remains unchanged.

This follows directly from the motor effect.

---

Permanent-Magnet Motors

Some motors use permanent magnets to provide the magnetic field.

Modern high-strength permanent magnets can produce powerful magnetic fields in compact devices.

Permanent-magnet motors are common in:

• electric vehicles
• drones
• computer equipment
• robotics
• power tools

---

Electromagnetic Motors

Other motors use electromagnets to create some or all of their magnetic fields.

This allows engineers to control magnetic-field strength using electric current.

Modern motors may use sophisticated electronic control systems instead of the simple mechanical commutator found in introductory DC motor models.

The underlying principle remains:

magnetic fields interacting with electric currents produce forces and torque.

---

Motor Components

Not every motor contains all these components. For example, many modern motors do not use brushes or a mechanical commutator.

---

Rotor and Stator

Two important motor terms are:

Rotor

The part that rotates.

Stator

The stationary part surrounding or interacting with the rotor.

Depending on the motor design, permanent magnets or electromagnets may be located on either the rotor or stator.

---

From Motors to Loudspeakers

A loudspeaker uses the same fundamental motor effect.

However, instead of producing continuous rotation, a loudspeaker produces:

back-and-forth motion

This movement creates sound waves.

Therefore, motors and loudspeakers are closely related electromechanical devices.

---

What Is a Loudspeaker?

A loudspeaker converts:

electrical energy → mechanical vibration → sound energy

Important components include:

• permanent magnet
• voice coil
• cone or diaphragm
• flexible suspension
• frame
• electrical connections

---

The Permanent Magnet

A loudspeaker usually contains a strong permanent magnet.

The magnet creates a magnetic field in the region surrounding the:

voice coil

The magnetic structure is designed so that a strong field passes through the coil.

---

The Voice Coil

The voice coil is a lightweight coil of wire connected to the speaker cone.

The audio signal passes through the coil as an electric current.

Because the coil is located in a magnetic field, it experiences a magnetic force.

The force is related to:

F = BIL

for sections of conductor appropriately oriented to the field.

---

Why the Voice Coil Moves

When current flows through the voice coil:

current + magnetic field → magnetic force

The force moves the coil.

Because the coil is attached to the cone:

coil moves → cone moves

When the current reverses:

force reverses

Therefore:

coil moves in the opposite direction

---

Alternating Audio Signals

An audio signal changes continuously.

The current through the voice coil therefore changes:

• magnitude
• direction

As the current changes direction, the magnetic force also changes direction.

Therefore, the voice coil moves:

back and forth

The cone follows this motion.

---

Producing Sound Waves

The moving speaker cone pushes and pulls on the surrounding air.

When the cone moves outward, it compresses nearby air.

When it moves inward, it produces a region of lower pressure.

This produces alternating:

• compressions
• rarefactions

These pressure disturbances travel through the air as:

sound waves

Therefore:

electrical signal → changing current → changing magnetic force → cone vibration → sound wave

---

Frequency and Pitch

If the current changes direction rapidly, the speaker cone vibrates rapidly.

The frequency of the cone's vibration determines the frequency of the sound wave.

Higher frequency generally produces:

higher pitch

Lower frequency generally produces:

lower pitch

For example:

200 Hz electrical signal

↓

speaker vibrates approximately 200 times per second

↓

sound wave frequency ≈ 200 Hz

---

Amplitude and Loudness

A larger current can produce a larger magnetic force.

From:

F ∝ I

a stronger current can produce greater cone displacement, within the operating limits of the speaker.

Greater vibration amplitude generally produces a sound wave with greater amplitude.

This is perceived as:

greater loudness

Therefore:

signal frequency → sound frequency/pitch

signal amplitude → vibration amplitude/loudness

---

Example: Speaker Force

Suppose the effective conductor length in a voice coil is:

L = 0.50 m

The current is:

I = 0.40 A

The magnetic flux density is:

B = 0.60 T

Assume the conductor is perpendicular to the field.

Use:

F = BIL

F = (0.60)(0.40)(0.50)

F = 0.12 N

If the current reverses, the force becomes:

0.12 N in the opposite direction

This reversal allows the cone to vibrate.

---

Why the Cone Is Lightweight

The speaker cone and voice coil should respond rapidly to changes in the electrical signal.

A lower mass allows them to accelerate more easily.

From:

F = ma

a given force produces greater acceleration when mass is smaller.

This helps the speaker reproduce rapidly changing sounds accurately.

---

Speaker Suspension

The speaker cone cannot move freely without control.

A flexible suspension:

• supports the cone
• allows back-and-forth movement
• helps return the cone toward equilibrium
• keeps the voice coil correctly positioned

The speaker therefore behaves partly like a mechanical oscillating system.

---

Motors vs Loudspeakers

Both devices use the same basic principle:

current + magnetic field → force

But they use the force differently.

---

Headphones and Earbuds

Many headphones use the same basic principle as loudspeakers.

An electrical audio signal passes through a small coil.

The coil interacts with a magnetic field.

The resulting force moves a small diaphragm.

The diaphragm produces sound waves.

Because headphones operate close to the ear, the required diaphragm motion can be much smaller than in a large loudspeaker.

---

Microphones: The Reverse Idea

Some microphones use electromagnetic principles in approximately the reverse direction.

In a dynamic microphone:

sound waves

↓

diaphragm vibrates

↓

coil moves in magnetic field

↓

voltage is induced

↓

electrical audio signal

A loudspeaker converts electrical signals into sound.

A dynamic microphone converts sound into electrical signals.

This illustrates the close relationship between electricity, magnetism and mechanical motion.

---

Other Electromechanical Devices

The motor effect is used in many technologies.

Examples include:

• electric motors
• loudspeakers
• headphones
• moving-coil meters
• actuators
• some vibration systems
• robotic mechanisms
• electromagnetic control systems

These devices convert electrical signals into mechanical movement.

---

Moving-Coil Meters

A moving-coil meter contains a coil in a magnetic field.

When current flows, the coil experiences torque.

A larger current produces a larger torque.

The coil rotates against a restoring mechanism and moves a pointer.

The amount of deflection can therefore indicate the current.

This principle has been used in analogue:

• ammeters
• voltmeters
• galvanometers

---

Practical Example: Electric Vehicle Motor

An electric vehicle uses electrical energy stored in its battery.

The motor converts this energy into rotational mechanical energy.

A simplified energy pathway is:

battery

↓

electrical current

↓

controlled magnetic fields

↓

magnetic forces and torque

↓

rotating motor shaft

↓

wheels

Modern electric-vehicle motors are far more sophisticated than the simple classroom DC motor, but magnetic force remains central to their operation.

---

Practical Example: Cooling Fan

A computer cooling fan contains a small electric motor.

Electrical energy produces magnetic interactions within the motor.

These produce torque.

The rotor turns.

The attached fan blades move air through the computer.

Therefore:

electrical energy → rotational kinetic energy → moving air

---

Practical Example: Loudspeaker

For a loudspeaker:

audio electrical signal

↓

changing current in voice coil

↓

changing magnetic force

↓

voice coil moves

↓

cone vibrates

↓

air vibrates

↓

sound reaches listener

The speaker therefore reproduces information contained in the electrical signal as variations in air pressure.

---

Magnetic Force as an Energy-Conversion Mechanism

Magnetic force provides an important connection between electrical and mechanical systems.

A current can produce:

• linear motion
• rotation
• vibration

depending on how the conductor and magnetic field are arranged.

This allows engineers to design many different electromechanical devices using the same fundamental physics.

---

Common Mistakes

Mistake: Saying a motor works because magnets simply attract the coil.
The key effect is the force on a current-carrying conductor in a magnetic field.

Mistake: Saying both sides of a motor coil experience forces in the same direction.
Opposite current directions produce opposite forces, creating torque.

Mistake: Confusing force with torque.
Force acts on the conductors; the arrangement of forces produces torque.

Mistake: Saying the split-ring commutator increases the current.
Its main role in a simple DC motor is to reverse current every half-turn.

Mistake: Saying the brushes produce the magnetic field.
Brushes provide electrical contact with the rotating commutator.

Mistake: Saying a loudspeaker's permanent magnet moves back and forth.
In a typical speaker, the voice coil and attached cone move.

Mistake: Saying sound comes directly from the electric current.
The current produces magnetic force, which moves the cone, which then produces sound waves in the air.

Mistake: Saying frequency mainly determines loudness.
Frequency is closely associated with pitch; amplitude is associated with loudness.

Mistake: Saying a loudspeaker and motor use unrelated physics.
Both rely fundamentally on magnetic forces acting on current-carrying conductors.

---

Key Terms

Electric motor — device that converts electrical energy into mechanical energy.

Motor effect — force experienced by a current-carrying conductor in a magnetic field.

Magnetic force — force produced by the interaction between a current and magnetic field.

Torque — turning effect of a force.

Couple — pair of forces that produces a turning effect.

Rotor — rotating part of a motor.

Stator — stationary part of a motor.

Coil — loops of conducting wire carrying electric current.

Split-ring commutator — component that reverses current through the coil of a simple DC motor every half-turn.

Brushes — stationary electrical contacts that supply current to the rotating commutator.

Axle — shaft that carries rotational motion from a motor.

Loudspeaker — device that converts an electrical audio signal into sound.

Voice coil — current-carrying coil in a loudspeaker that moves within a magnetic field.

Diaphragm — flexible vibrating surface that produces sound waves.

Electromechanical device — device that converts between electrical and mechanical forms of energy or signals.

---

Key Takeaways

• Motors and loudspeakers rely on the motor effect.
• A current-carrying conductor experiences a force in a magnetic field.
• For a straight conductor, F = BIL sin θ.
• Fleming's Left-Hand Rule determines the force direction.
• A motor converts electrical energy into mechanical energy.
• Opposite sides of a current-carrying motor coil experience forces in opposite directions.
• These forces create torque.
• The rotor is the rotating part of a motor.
• The stator is the stationary part.
• In a simple DC motor, the split-ring commutator reverses current every half-turn.
• The brushes maintain electrical contact with the rotating commutator.
• Increasing current or magnetic flux density can increase magnetic force.
• Increasing the number of coil turns can increase motor torque.
• A loudspeaker converts an electrical audio signal into mechanical vibration and then sound.
• Current through the voice coil interacts with a permanent magnetic field.
• Reversing the current reverses the force on the voice coil.
• The voice coil therefore moves back and forth.
• The attached cone or diaphragm produces pressure variations in the air.
• Signal frequency determines the frequency of the produced sound.
• Motors, speakers, headphones and many actuators apply the same fundamental magnetic-force principles.

Check Your Understanding

1. What is the motor effect?
2. State the equation for the force on a current-carrying conductor in a magnetic field.
3. Explain why opposite sides of a motor coil experience forces in opposite directions.
4. What is torque?
5. Describe how magnetic forces produce rotation in a simple DC motor.
6. What is the purpose of the split-ring commutator?
7. Why must the current in a simple DC motor reverse every half-turn?
8. What is the function of the brushes?
9. Distinguish between the rotor and stator.
10. State three ways of increasing the turning effect of a motor.
11. Explain how reversing the current affects motor rotation.
12. Describe the energy conversion occurring in an electric motor.
13. Identify the major components of a simple loudspeaker.
14. What is the function of the voice coil?
15. Explain why the voice coil moves when current passes through it.
16. Why does an alternating audio signal cause the speaker cone to vibrate?
17. Explain how cone vibration produces sound waves.
18. How does the frequency of the electrical signal affect the sound produced?
19. A speaker voice coil has an effective conductor length of 0.80 m in a 0.50 T magnetic field and carries 0.30 A. Assuming perpendicular orientation, calculate the magnetic force.
20. Compare an electric motor and a loudspeaker, explaining how both use the same magnetic-force principle to produce different forms of mechanical motion.

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

---

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.

---

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 →

---

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

---

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

---

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.

---

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.

---

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.

---

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

---

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.

---

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

---

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

---

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

---

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.

---

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

---

Electric vs Magnetic Fields

This distinction is fundamental to charged-particle physics.

---

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.

---

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.

---

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.

---

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

---

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

---

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

---

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.

---

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.

---

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.

---

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

---

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.

---

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

---

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

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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

---

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

---

Choosing the Correct Equation

---

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.

---

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.

---

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.