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.

---

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

---

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.

---

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.

---

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.

---

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

---

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

---

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

---

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.

---

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.

---

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

---

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.

---

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.

---

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

---

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

---

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

---

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

---

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

---

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

---

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

---

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

---

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.

---

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.

---

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.

---

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)

---

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.

---

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

---

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

---

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

---

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.

---

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.

---

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

---

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.

---

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

---

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

---

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

---

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.

---

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

---

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.

---

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.

---

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

---

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

---

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

---

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.

---

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

---

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.

---

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.

---

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.