Magnetic Forces and Fields in Systems
2. Force on current-carrying wires
Learning Outcomes
- I can explain why a current-carrying wire experiences a magnetic force.
- I can determine force direction using Fleming's Left-Hand Rule.
- I can calculate magnetic force on a conductor.
- I can describe factors that affect magnetic force.
- I can apply these principles to practical devices.
- Force formula:
- Right-hand rule for current and force
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:
- calculate the force
- 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
- Explain why a current-carrying wire can experience a force in a magnetic field.
- State the equation for the magnetic force on a straight current-carrying conductor.
- What does each symbol in F = BIL sin θ represent?
- Under what condition can the equation be simplified to F = BIL?
- When is the magnetic force maximum?
- When is the magnetic force zero?
- Describe Fleming's Left-Hand Rule.
- Which finger represents the magnetic field?
- Which finger represents conventional current?
- Which finger represents force or motion?
- What happens to the force direction if the current is reversed?
- What happens if the magnetic-field direction is reversed?
- A 0.30 m wire carries 4.0 A perpendicular to a 0.50 T field. Calculate the force.
- A 20 cm wire carries 6.0 A perpendicular to a 300 mT field. Calculate the force.
- 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.
- Explain how increasing current affects the force on a conductor.
- Explain how opposite forces on a current-carrying coil can produce rotation.
- Describe the role of the split-ring commutator in a simple DC motor.
- Explain how the motor effect is used in a loudspeaker.
- Suggest three changes that could increase the turning effect produced by a current-carrying coil in an electric motor.