Year 12 Physics Module 6 ⏱ ~40 min 5 MC · 2 Short Answer Lesson 4 of 21 IQ1: Charged Particles in Fields

Magnetic Fields and the Lorentz Force

Predict the magnitude and direction of the magnetic force on a moving charge, then explain why a pure magnetic field can turn a particle without changing its speed.

Today's hook: A positive charge moves to the right while the magnetic field points into the page. Which way does the force act? Would an electron moving the same way feel the same force, and can either particle gain speed from the magnetic field alone?
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1
Orient yourself, retrieve prior knowledge, and expose your starting model.
Warm up first

Three quick questions from earlier lessons. Pulling old material back to mind before you learn something new makes the new material stick better, so this is not busywork.

Worksheets

Practise this lesson

Use the available foundations worksheet for guided practice on this lesson.

Build Foundations & guided practice
Before you read, predict

A positively charged particle moves to the right through a region where a magnetic field points into the page.

Before reading on, answer:

  1. In which direction do you think the magnetic force acts: up, down, into the page, out of the page, or along the direction of motion?
  2. If the particle were an electron instead, would the force direction change?
  3. If the particle slowed down while still in the field, what would happen to the magnitude of the magnetic force?

Warm-up, which of the following will experience NO magnetic force in a uniform magnetic field?

Learning Intentions
goals

Know, Lorentz Force

  • Magnetic-force magnitude: $F_B = |q|vB\sin\theta$
  • Force is zero when $v$ is parallel to $B$ ($\theta = 0$ or $180°$)
  • Maximum force when $v$ is perpendicular to $B$ ($\theta = 90°$)

Understand, Direction and Work

  • Right-hand rule for positive charges; opposite for electrons
  • Why magnetic force does no work on the particle
  • Why speed remains constant in a pure magnetic field

Can Do, Predict and Calculate

  • Use the right-hand rule to determine force direction
  • Calculate magnetic force magnitude given $q$, $v$, $B$, $\theta$
  • Explain why kinetic energy is unchanged by magnetic fields
Scan these before reading
vocab
Lorentz forceThe total electromagnetic force is $\vec F=q(\vec E+\vec v\times\vec B)$. This lesson isolates its magnetic part.
Magnetic field ($B$)A vector field that exerts force on moving charges. Measured in teslas (T).
Right-hand ruleThumb = velocity, fingers = field, palm = force (for positive charges).
Tesla (T)SI unit of magnetic field strength. $1\,\text{T} = 1\,\text{N/(A·m)}$.
Cross productThe mathematical operation $\vec{v} \times \vec{B}$ that gives the force direction.
Misconceptions to fix
✗ Wrong: "A magnetic field speeds up charged particles, which is how particle accelerators work."
✓ Right: Magnetic fields only change the direction of motion, never the speed. Particle accelerators use electric fields to increase speed and magnetic fields to bend the path.
✗ Wrong: "Stationary charges experience a magnetic force."
✓ Right: $F_B=|q|vB\sin\theta$. If $v=0$, then $F_B=0$. A stationary charge in a magnetic field has no magnetic force, although an electric force could still act if $\vec E\ne0$.
✗ Wrong: "The magnetic force on an electron is in the same direction as for a proton moving the same way."
✓ Right: The force direction depends on the sign of the charge. Electrons (negative) experience the opposite force direction compared to protons (positive) moving with the same velocity.

A magnetic field can accelerate a charged particle and increase its kinetic energy.

A stationary charged particle in a strong magnetic field experiences a large magnetic force.

Cross-lesson links: L03 used electric fields to change kinetic energy. L04 introduces the magnetic part $q(\vec v\times\vec B)$, which changes direction but does no work. Perpendicular entry leads to circular motion in L05.

Core Content

2
Build the magnetic-force law and connect perpendicular force to zero work.
1
The Lorentz Force Law
+5 XP

Magnitude, direction, and why magnetic forces do no work

Hold a compass next to a wire carrying current: the needle deflects. Now send a charged particle through the same region, it too is deflected, but sideways, never speeding up or slowing down. That sideways deflection is the magnetic force on a moving charge, and its magnitude depends on the charge, the speed, the field strength, and the angle between velocity and field.

Magnetic part of the Lorentz force

$\vec F_B=q(\vec v\times\vec B)$   and   $F_B=|q|vB\sin\theta$

$F_B$ = magnetic-force magnitude (N)  ·  $q$ = signed charge (C)  ·  $v$ = speed (m/s)  ·  $B$ = field strength (T)  ·  $\theta$ = angle between $\vec{v}$ and $\vec{B}$

The direction of the force is given by the right-hand rule for positive charges:

  1. Point your thumb in the direction of velocity $\vec{v}$
  2. Point your fingers in the direction of the magnetic field $\vec{B}$
  3. Your palm pushes in the direction of the force $\vec{F}$ on a positive charge

For a negative charge (electron), the force is in the opposite direction.

Use a physical hand rule

Keep $\vec v$ and $\vec B$ fixed while you orient your hand. First obtain the force direction for a positive charge from $\vec v\times\vec B$; only then reverse that direction if the charge is negative. The next figure provides four unambiguous checks using into-page and out-of-page notation.

A crucial difference from electric forces: magnetic forces do no work. Because $\vec{F}$ is always perpendicular to $\vec{v}$, the displacement in the direction of the force is zero at every instant:

$W = \vec{F} \cdot \vec{d} = 0$   (since $\vec{F} \perp \vec{v}$)

This means the kinetic energy and speed of the particle never change in a pure magnetic field. Only the direction of motion changes.

Stop & Check

An electron moves parallel to a uniform magnetic field. What is the magnetic force on it? Explain why this makes sense using the Lorentz force equation.

Full Lorentz force: $\vec F=q(\vec E+\vec v\times\vec B)$. In a pure magnetic field, $F_B=|q|vB\sin\theta$: maximum at $90°$ and zero for $v\parallel B$. A zero force for parallel entry does not mean $B=0$. Reverse the positive-charge cross-product direction for an electron. The magnetic part does zero work, so in a pure $B$ field speed is constant.

Pause, copy the highlighted law and right-hand rule into your book before moving on.

A charged particle moves at 45° to a uniform magnetic field. The force on it is proportional to which trigonometric function of the angle?

3
Read page-direction symbols and apply the signed cross-product direction rule.
2
Magnetic Field Representations
+5 XP

How to read and draw field directions in 2D diagrams

We just saw that $F_B=|q|vB\sin\theta$ gives the magnetic-force magnitude, with direction from $q(\vec v\times\vec B)$. That raises a question: how do you represent a magnetic field pointing into or out of the page in a 2D diagram? This card answers it → × means into the page (arrow tail), · means out of the page (arrow tip).

Because magnetic fields are three-dimensional, we need symbols to represent directions perpendicular to the page. Mastering these symbols is essential for every force-direction problem.

  • Cross (×): Field points into the page, like the tail feathers of an arrow flying away from you
  • Dot (·): Field points out of the page, like the tip of an arrow flying toward you
Into page Like arrow tail flying away Out of page Like arrow tip flying toward you Force Direction Grid v direction B direction F (+charge) Right × Up Left · Up Up × Left Down · Left

Figure 1, Magnetic field symbols and sample force directions for a positive charge. Reverse each force direction for a negative charge.

Stop & Check

A student draws a magnetic field pointing out of the page and claims a positive charge moving to the right will experience a force upward. Use the right-hand rule to check this. Is the student correct?

Field notation: × = into page (arrow tail); · = out of page (arrow tip). Apply right-hand rule, thumb = $\vec{v}$, fingers = $\vec{B}$, palm = $\vec{F}$ (positive charges); reverse palm direction for electrons.

Add the highlighted notation and rule to your notes before the check below.

An electron moves to the right through a magnetic field pointing out of the page. The force on the electron is directed...

4
Test how charge sign, speed, field strength, and angle affect magnetic force.
Interactive Tool, Magnetic-force direction Open fullscreen ↗

Use in this lesson: vary charge sign, speed, field strength and entry angle to test the force law and direction rule. Any circular-path or mass-analysis controls are previews for Lessons 5 and 18.

A charged particle moving parallel to a magnetic field (angle = 0°) experiences zero magnetic force.

The speed of a charged particle moving in a pure magnetic field remains constant.

A magnetic field can do positive work on a charged particle if the field is strong enough.

5
Calculate magnetic force and distinguish core principles from later applications.
WE
Worked Example, Electron in a Magnetic Field
+5 XP

Calculate force magnitude and direction; explain why speed is constant

Problem

An electron travels at $3.0 \times 10^6$ m/s perpendicular to a uniform magnetic field of strength $0.050$ T. The field points into the page; the electron moves to the right.

  • (a) Calculate the magnitude of the magnetic force on the electron.
  • (b) Determine the direction of the force.
  • (c) Explain why the electron's speed remains constant even though a force acts on it.
Step 1, Part (a): Force magnitude
  1. Given. $q = 1.60 \times 10^{-19}$ C, $v = 3.0 \times 10^6$ m/s, $B = 0.050$ T, $\theta = 90°$.
  2. Find. $F$.
  3. Method. Use $F_B=|q|vB\sin\theta$ for magnitude, then determine direction from the signed cross product.
  4. Solve. $F = (1.60 \times 10^{-19})(3.0 \times 10^6)(0.050)(1) = \mathbf{2.4 \times 10^{-14}}$ N.
Step 2, Part (b): Force direction
  1. Right-hand rule for a positive charge: thumb right ($\vec{v}$), fingers into page ($\vec{B}$), palm pushes upward.
  2. But this is an electron (negative charge), so the force reverses: downward.
Step 3, Part (c): Why speed is constant

The magnetic force is always perpendicular to the velocity. Work done is $W = Fd\cos\phi$ where $\phi = 90°$ between force and displacement. Since $\cos 90° = 0$, the magnetic force does zero work. By the work-energy theorem, $\Delta K = 0$, so speed remains constant. Only the direction of motion changes.

A proton moves at $2.0 \times 10^6$ m/s perpendicular to a $0.050$ T field. The magnetic force on the proton (in units of $10^{-14}$ N, to 2 significant figures) is _____.

3
Real-World Applications
+5 XP

Where the Lorentz force shapes everyday technology and natural phenomena

We just saw the magnetic Lorentz term and 2D direction rule. That raises a question: where does $F_B=|q|vB\sin\theta$ matter? This card answers it → charged-particle steering, auroral motion and selector/analyser devices use the same magnetic deflection principle.

Magnetic deflection appears in charged-particle beam steering and in the motion of charged particles entering Earth's magnetic environment. These examples preview applications; the assessed L04 core is the force vector, its angle dependence and its zero-work consequence.

Earth's magnetic field and the aurora

High-energy charged particles from the Sun (the solar wind) enter the Earth's magnetic field. The Lorentz force deflects them, not slowing them, but curving their path. Near the poles, the field lines converge, funnelling particles into the atmosphere where they collide with gas molecules, producing the aurora. No work is done; the particles' speed is unchanged by the magnetic deflection.

Particle accelerators, electric vs magnetic

In a cyclotron or synchrotron, electric fields do the speeding up because they can do work. Magnetic fields steer the beam without changing its speed. Lesson 5 derives the resulting paths.

Preview: crossed fields

Electric and magnetic forces can be arranged to oppose each other. Lesson 6 develops the conditions for a velocity selector; here, notice only that the magnetic contribution remains perpendicular to the particle's velocity.

A pure magnetic field can steer a moving charged particle because $\vec F_B$ is perpendicular to $\vec v$. The magnetic force does no work, so it changes the velocity direction but not the particle's speed or kinetic energy.

Add the highlighted application principle to your notes before the check below.

Three of these statements about charged particles in magnetic fields are correct. Pick the odd one out.

6
Apply the direction rule, calculate a force, and consolidate the lesson boundary.
Activity 1, Right-Hand Rule Drills
ApplyBand 3

Practise predicting force directions for various charge/field/velocity combinations

  1. A positive charge moves upward through a field pointing to the right. Determine the force direction.
  2. An electron moves to the left through a field pointing out of the page. Determine the force direction.
  3. A proton moves into the page through a field pointing upward. Determine the force direction.
  4. For each case above, state whether the particle would curve left, right, up, down, into or out of the page.

Which scenario has the correct force direction for a positive charge?

Activity 2, Calculation and Reasoning
ApplyBand 4

Apply the Lorentz force equation and explain why kinetic energy is conserved

An electron travels at $2.0 \times 10^6$ m/s perpendicular to a uniform magnetic field of $0.020$ T. The field points into the page; the electron initially moves to the right.

  1. Calculate the magnitude of the magnetic force on the electron. (1 mark)
  2. Determine the direction of the force. Show your reasoning. (1 mark)
  3. Explain why the electron's kinetic energy remains constant as it moves through the field. (1 mark)
  4. Predict the shape of the electron's path and justify your answer. (1 mark)
Wrap-up, Key Formulae and Summary
Essential formulae, Lorentz force

Magnetic-force magnitude: $F_B=|q|vB\sin\theta$

Maximum force ($\theta = 90°$): $F_B=|q|vB$

Zero force: when $v \parallel B$ (i.e. $\theta = 0°$ or $180°$)

Pure magnetic field: $W = Fd\cos90° = 0$, so the magnetic force does not change speed

Misconceptions, final check

✗ "Doubling the angle always doubles the force."
✓ Force depends on $\sin\theta$. Going from 30° to 60° changes $\sin$ from 0.5 to 0.866, not doubled. Only doubling $q$, $v$, or $B$ doubles the force.
✗ "The electron's path in a magnetic field is a straight line because force and velocity are perpendicular."
✓ With perpendicular entry into a uniform magnetic field, the force continually turns the velocity and produces a circular path. Lesson 5 derives this result; parallel and oblique entry require separate treatment.

Copy into your books

Key Formula

  • $F_B=|q|vB\sin\theta$
  • Max when $\theta = 90°$
  • Zero when $\theta = 0°$

Right-Hand Rule

  • Thumb = $\vec{v}$
  • Fingers = $\vec{B}$
  • Palm = $\vec{F}$ (positive charges)

Field Symbols

  • × = into page
  • · = out of page

Key Principles

  • No work → speed constant
  • Magnetic force changes direction, not speed
  • Electrons: reverse direction

A proton moves in a circular path in a uniform magnetic field. Which statement is correct?

7
Retrieve the force law, direction convention, and zero-work reasoning.
Quick recall, Lorentz force & magnetic fields
+5 XP

A fresh five-question set drawn from this lesson's bank, feedback shown immediately. +5 XP per correct · +25 XP all correct

Pick your answer, then rate your confidence, that tells the system what to drill next.

Short Answer, 6 marks
+5 XP

ApplyBand 4(3 marks) 1. A positive ion of charge $+2e$ moves at $4.0 \times 10^5$ m/s at an angle of $30°$ to a uniform magnetic field of $0.080$ T.

1 mark: correct substitution into $F_B=|q|vB\sin\theta$ · 1 mark: correct force magnitude · 1 mark: correct statement about whether the force changes the ion's speed

EvaluateBand 5(3 marks) 2. A student claims: "Because a magnetic field exerts a force on a moving charge, it must do work on the charge and change its kinetic energy." Evaluate this claim fully.

1 mark: identifies the force is always perpendicular to velocity · 1 mark: uses $W = Fd\cos90° = 0$ or equivalent reasoning · 1 mark: correctly concludes speed/KE is unchanged and provides a real-world example

Show all answers

Multiple choice

MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set drawn from the lesson bank.

Short Answer, Model Answers

Q1 (3 marks): $|q|=2(1.60\times10^{-19})=3.20\times10^{-19}$ C (1 mark). $F_B=|q|vB\sin\theta=(3.20\times10^{-19})(4.0\times10^5)(0.080)\sin30°=5.12\times10^{-15}$ N (1 mark). The magnetic force is perpendicular to velocity, so it does no work and changes direction rather than speed (1 mark).

Q2 (3 marks): The claim is incorrect. The magnetic force on a moving charge is perpendicular to the velocity (from $\vec{F}_B = q\vec{v} \times \vec{B}$) (1 mark). Work is $W = Fd\cos\phi$, where $\phi$ is the angle between force and displacement. Since $\phi = 90°$, $W = 0$ (1 mark). Therefore the particle's kinetic energy and speed remain constant even though its velocity can change direction, so it is accelerated without being sped up. For example, a magnetic field can deflect an electron beam, while an electric field is required to increase its kinetic energy (1 mark).

8
Check retention and explain how force direction and energy fit together.
Check what actually stuck
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quiz

A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.

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How did your thinking change?

At the start, a positive charge moved right while $\vec B$ pointed into the page. For a positive charge, $\vec v\times\vec B$ points upward. For an electron, the negative charge reverses that direction, so the force points downward.

In both cases the force is perpendicular to the instantaneous velocity. The magnetic field can therefore change the velocity vector without changing the particle's speed or kinetic energy.

Explain the evidence: State the direction rule, explain the charge-sign reversal, and use the work-energy theorem to justify why the speed remains constant.

← Previous: Work, Energy and Potential Next: Circular Motion in Magnetic Fields →

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