Charged Particles in Magnetic Fields

Overview

A moving charged particle entering a magnetic field may experience a magnetic force. This force can deflect the particle, causing:

• circular motion
• helical motion
• straight-line motion in special cases

This topic combines ideas from:

• Magnetic Force
• Magnetic Fields
• Circular Motion

Definition

For a moving charge in a magnetic field, the magnetic-force magnitude is:

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

where $\theta$ is the angle between the velocity and the magnetic field.

Why It Matters

This topic explains how magnetic fields:

• bend beams of charged particles
• separate particles by radius or direction
• allow speed selection in crossed electric and magnetic fields
• change direction of motion without changing kinetic energy

Core Physical Idea

A magnetic field acts only on a moving charge.

• stationary charge: no magnetic force
• moving charge: may experience force

The force is always perpendicular to:

• particle velocity
• magnetic field direction

Hence magnetic force usually changes direction of motion, not speed.

Key Representations

Force on a Moving Charge

Magnitude:

$$F=Bqv\sin \theta$$

where:

• $B$ = magnetic flux density
• $q$ = charge magnitude
• $v$ = particle speed
• $\theta$ = angle between velocity and magnetic field

Special Cases

Maximum Force

When:

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

Then:

$$F=B|q|v$$

Zero Force

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

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

Then:

$$F=0$$

Direction of Force

Positive Charge

Use Fleming’s left-hand rule, treating velocity direction as current direction.

• first finger = field
• second finger = velocity
• thumb = force

Figure: Fleming’s left-hand rule for magnetic force direction.

Negative Charge

Force direction is opposite to that of a positive charge moving the same way.

Quick Direction Logic

If a proton curves upward, an electron entering with the same velocity would curve downward.

Always check the sign of charge.

Figure: Opposite magnetic deflection of positive and negative charges.

Circular Motion in a Uniform Magnetic Field

If velocity is perpendicular to the field:

• force always perpendicular to velocity
• force acts as centripetal force
• speed is constant
• path is circular

Derivation of Radius

Magnetic force provides centripetal force:

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

Hence:

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

using magnitudes.

Meaning of Radius Formula

Larger radius if:

• larger mass $m$
• larger speed $v$

Smaller radius if:

• larger field strength $B$
• larger charge magnitude $q$

Period of Revolution

Using:

$$T=\frac{2\pi r}{v}$$

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

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

Important Consequence

The period does not depend on speed.

So:

• fast particles move in larger circles
• slow particles move in smaller circles
• both can have the same period if $m$, $q$, and $B$ are unchanged

Why Speed Stays Constant

Magnetic force is perpendicular to velocity.

Therefore:

• no component of force acts along the motion
• no work is done
• kinetic energy is unchanged

So speed stays constant while direction changes.

Energy Interpretation

Since work done is zero:

$$W=0$$

Hence:

$$\Delta KE=0$$

This is why magnetic fields bend beams without speeding them up.

Helical Motion

If velocity has two components:

• $v_{\perp }$: perpendicular to the field
• $v_{\parallel }$: parallel to the field

Then:

• $v_{\perp }$ gives circular motion
• $v_{\parallel }$ remains unchanged

Result: helical path.

Helix Features

• stronger field gives a tighter spiral
• faster parallel speed gives a larger pitch
• no force acts on the parallel component

Straight-Line Motion

If the particle enters exactly parallel to the magnetic field:

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

Then:

$$F=0$$

The particle continues straight.

Velocity Selector

Uses perpendicular electric and magnetic fields.

For an undeflected particle:

• electric force balances magnetic force

$$qE=qvB$$

So:

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

Only particles with this speed pass straight through.

Related topic: Electric Fields

Why Velocity Selector Works

If Particle Is Too Slow

Magnetic force is too small.

Electric force dominates.

The particle deflects one way.

If Particle Is Too Fast

Magnetic force is too large.

The particle deflects the opposite way.

Only the correct speed remains undeflected.

Exam-Style Particle Path Reasoning

Step 1

Identify field direction:

• into page $\times$
• out of page $\bullet$

Step 2

Identify sign of charge.

Step 3

Use the left-hand rule for a positive charge.

Step 4

Reverse for a negative charge.

Step 5

Recognise motion type:

• perpendicular entry gives a circle
• angled entry gives a helix
• parallel entry gives a straight line

Short Worked Examples

Example 1: Radius Change

If speed doubles:

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

Radius doubles.

Example 2: Stronger Field

If $B$ doubles:

Radius halves.

Example 3: Proton vs Electron

At the same speed and in the same field:

• they curve in opposite directions
• the electron usually has a much smaller radius due to its smaller mass

Common Mistakes

1. Forgetting charge sign
2. Thinking magnetic force changes speed
3. Using the wrong radius proportionality
4. Using the wrong angle
5. Mixing electric and magnetic force directions

Summary

Force

$$F=Bqv\sin \theta$$

Circular Motion

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

Radius

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

Period

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

Velocity Selector

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

Links

• Magnetic Force
• Magnetic Fields
• Force Between Parallel Currents
• Magnetic Force Common Exam Traps
• Circular Motion
• Electric Fields
• Vectors