Magnetic Force

Overview

A magnetic field can exert a force on:

• a current-carrying conductor
• a moving charged particle
• another current-carrying conductor through magnetic interaction

This topic studies how magnetic fields produce motion, turning effects, and particle deflection.

It builds directly from Magnetic Fields.

Core Ideas

Magnetic-force questions revolve around a few main ideas:

• magnetic force acts only when charge is moving
• the force is perpendicular to the magnetic field and to the current or velocity direction
• magnetic force can change direction of motion without changing speed
• conductor force and moving-charge force follow parallel formula structures
• current-carrying wires interact through the magnetic fields they create

Exam Relevance

Students are expected to:

• calculate force magnitude with the correct angle factor
• determine directions using Fleming’s left-hand rule and charge sign
• explain circular or helical motion in uniform magnetic fields
• distinguish magnetic force from electric force
• decide whether parallel currents attract or repel

Core Physical Idea

Magnetic force acts when charge is moving.

Hence:

• a stationary charge experiences no magnetic force
• a moving charge may experience force
• a conductor carrying current may experience force

Magnetic force is always perpendicular to both:

• magnetic field direction
• motion or current direction

Therefore magnetic force often changes direction of motion rather than speed.

Key Representations

Force on a Current-Carrying Conductor

A conductor of length $L$, carrying current $I$, in magnetic field $\vec{B}$ experiences force:

$$F=BIL\sin \theta$$

where:

• $B$ = magnetic flux density
• $I$ = current
• $L$ = length in field
• $\theta$ = angle between current direction and field

Special Cases

Maximum Force

When:

$$\theta =90^{\circ }$$

Then:

$$F=BIL$$

Zero Force

When the conductor is parallel or anti-parallel to the field:

$$\theta =0^{\circ },180^{\circ }$$

Then:

$$F=0$$

Direction of Force: Fleming’s Left-Hand Rule

Use for conventional current.

• first finger = magnetic field
• second finger = current
• thumb = force

Figure: Fleming’s left-hand rule. The thumb, first finger, and second finger are mutually perpendicular, representing force, field, and conventional current respectively.

Applications of Conductor Force

• electric motors
• loudspeakers
• moving-coil meters
• galvanometers

Definition of Tesla

From:

$$F=BIL$$

when the conductor is perpendicular to the field,

$$B=\frac{F}{IL}$$

Hence 1 tesla is the magnetic flux density that produces a force of 1 N on a 1 m wire carrying 1 A at right angles to the field.

Force on a Moving Charge Overview

A charged particle moving with speed $v$ in a magnetic field experiences force:

$$F=B|q|v\sin \theta$$

where:

• $q$ = charge magnitude
• $v$ = speed
• $\theta$ = angle between velocity and field

Detailed page: Charged Particles in Magnetic Fields

Direction of Particle Force

For a positive charge:

Use Fleming’s left-hand rule with current direction taken as velocity direction.

For a negative charge:

Force direction is opposite.

Circular Motion in a Magnetic Field Overview

If a particle enters perpendicular to a uniform magnetic field:

• magnetic force is always perpendicular to velocity
• it acts as centripetal force
• path is circular

Condition:

$$B|q|v=\frac{m{v}^2}{r}$$

Hence:

$$r=\frac{mv}{B|q|}$$

Related topic: Circular Motion

Period of Circular Motion

$$T=\frac{2\pi r}{v}=\frac{2\pi m}{B|q|}$$

So period is independent of speed.

Fast particles move in larger circles.

Helical Motion Overview

If velocity has:

• a component parallel to the field
• a component perpendicular to the field

Then:

• the perpendicular component causes circular motion
• the parallel component remains unchanged

Result: helical path.

Crossed Fields / Velocity Selector Overview

When electric and magnetic forces act in opposite directions on a moving charge:

For undeflected motion:

$$qE=qvB$$

So:

$$v=\frac{E}{B}$$

Only particles with this speed pass straight through.

Linked topic: Electric Fields

Detailed treatment: Charged Particles in Magnetic Fields

Force Between Parallel Currents Overview

Two parallel current-carrying wires exert forces on each other.

• same direction currents attract
• opposite direction currents repel

Magnitude:

$$F=\frac{{\mu }_0{I}_1{I}_{2}L}{2\pi r}$$

Detailed page: Force Between Parallel Currents

Short Worked Examples

Example 1: Force on Wire

A wire of length $0.40\text{m}$, current $3.0\text{A}$, and field $0.50\text{T}$ is perpendicular to the field.

$$F=BIL=0.50\times 3.0\times 0.40=0.60\text{N}$$

Example 2: Circular Radius

If $v$ doubles while $B$, $q$, and $m$ are unchanged:

$$r=\frac{mv}{Bq}$$

Radius doubles.

Example 3: Parallel Wires

Two wires carrying current in the same direction attract each other.

Common Exam Traps Overview

Detailed page: Magnetic Force Common Exam Traps

Frequent mistakes:

• forgetting $\sin \theta$
• using the wrong hand rule
• getting electron direction wrong
• assuming magnetic force changes speed
• confusing attraction and repulsion of wires
• forgetting force is perpendicular

Summary Sheet

Force on Conductor

$$F=BIL\sin \theta$$

Force on Moving Charge

$$F=Bqv\sin \theta$$

Circular Path

$$Bqv=\frac{m{v}^2}{r}$$ $$r=\frac{mv}{Bq}$$

Period

$$T=\frac{2\pi m}{Bq}$$

Velocity Selector

$$v=\frac{E}{B}$$

Parallel Currents

$$F=\frac{{\mu }_0{I}_1{I}_{2}L}{2\pi r}$$

Key Concepts to Remember

• magnetic force is perpendicular to the field
• magnetic force is perpendicular to motion or current
• magnetic force can bend a path without changing speed
• a stationary charge feels no magnetic force

Links

• Magnetic Fields
• Charged Particles in Magnetic Fields
• Force Between Parallel Currents
• Magnetic Force Common Exam Traps
• Circular Motion
• Electric Fields
• Electromagnetic Induction
• Transformers
• Vectors