Magnetic Fields and the Lorentz Force
In 1918, Arthur Dempster at the University of Chicago built the first practical magnetic deflection mass spectrometer. He separated uranium isotopes by firing ionised atoms through a magnetic field: lighter ions (U-235) curved into a smaller radius (r = mv/qB) than heavier ions (U-238). This same principle was later scaled up into the Manhattan Project's calutrons, each separating roughly 1 gram of enriched uranium per day.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
A positively charged particle moves to the right through a region where a magnetic field points into the page.
Before reading on, answer:
- 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?
- If the particle were an electron instead, would the force direction change?
- 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?
Know — Lorentz Force
- Magnetic force on a moving charge: $F = qvB\sin\theta$
- 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
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.
Core Content
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.
$F = qvB\sin\theta$
$F$ = magnetic force (N) · $q$ = 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:
- Point your thumb in the direction of velocity $\vec{v}$
- Point your fingers in the direction of the magnetic field $\vec{B}$
- 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.
Figure 1 — Right-hand rule: thumb = v, fingers = B, palm = F (for positive charges)
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.
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.
Lorentz force: $F = qvB\sin\theta$ (N). Maximum when $\theta = 90°$; zero when $v \parallel B$. Right-hand rule: thumb = $\vec{v}$, fingers = $\vec{B}$, palm = $\vec{F}$ (positive charges; reverse for electrons). Magnetic force does zero work ($W = 0$) → speed is always 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?
How to read and draw field directions in 2D diagrams
We just saw that $F = qvB\sin\theta$ gives the Lorentz force magnitude, with direction from the right-hand rule. 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
Figure 2 — Magnetic field symbols and sample force directions for a positive charge
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...
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.
Calculate force magnitude and direction; explain why speed is constant
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.
- Given. $q = 1.60 \times 10^{-19}$ C, $v = 3.0 \times 10^6$ m/s, $B = 0.050$ T, $\theta = 90°$.
- Find. $F$.
- Method. $F = qvB\sin\theta$.
- Solve. $F = (1.60 \times 10^{-19})(3.0 \times 10^6)(0.050)(1) = \mathbf{2.4 \times 10^{-14}}$ N.
- Right-hand rule for a positive charge: thumb right ($\vec{v}$), fingers into page ($\vec{B}$), palm pushes downward.
- But this is an electron (negative charge), so the force reverses: upward.
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 _____.
Where the Lorentz force shapes everyday technology and natural phenomena
We just saw the Lorentz force law and how to apply the right-hand rule in 2D diagrams. That raises a question: where does $F = qvB\sin\theta$ appear in real-world technology and natural phenomena? This card answers it → aurora borealis, particle accelerators, and the velocity selector all depend on the same magnetic deflection principle.
The Lorentz force underpins technologies from hospital MRI machines to the aurora borealis. Understanding how a sideways force curves a charged particle is the key to all of them.
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.
In a cyclotron or synchrotron, electric fields do the accelerating (they do work, increasing KE). Magnetic fields do the steering — they bend the beam into a circular or spiral path without changing its speed. The Large Hadron Collider uses superconducting magnets producing ~8 T to keep 7 TeV protons in a 27 km ring.
A velocity selector uses perpendicular electric ($\vec{E}$) and magnetic ($\vec{B}$) fields. The electric force ($qE$) and magnetic force ($qvB$) act in opposite directions. Particles with speed $v = E/B$ experience zero net force and pass straight through — all others are deflected. This selects a precise speed from a beam.
In particle accelerators, electric fields do the accelerating (increase KE) while magnetic fields steer (circular path, no work). Velocity selector: $qE = qvB$ → only particles with $v = E/B$ pass straight through. Aurora: solar wind particles curved by Earth's B field with no change in speed.
Add the highlighted applications and velocity selector condition to your notes before the check below.
Three of these statements about charged particles in magnetic fields are correct. Pick the odd one out.
Practise predicting force directions for various charge/field/velocity combinations
- A positive charge moves upward through a field pointing to the right. Determine the force direction.
- An electron moves to the left through a field pointing out of the page. Determine the force direction.
- A proton moves into the page through a field pointing upward. Determine the force direction.
- For each case above, state whether the particle would curve left, right, up, down, into or out of the page.
Match each scenario to the correct force direction for a positive charge.
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.
- Calculate the magnitude of the magnetic force on the electron. (1 mark)
- Determine the direction of the force. Show your reasoning. (1 mark)
- Explain why the electron's kinetic energy remains constant as it moves through the field. (1 mark)
- Predict the shape of the electron's path and justify your answer. (1 mark)
Lorentz force magnitude: $F = qvB\sin\theta$
Maximum force ($\theta = 90°$): $F = qvB$
Zero force: when $v \parallel B$ (i.e. $\theta = 0°$ or $180°$)
No work done: $W = Fd\cos90° = 0$ — speed is always constant
Misconceptions — final check
Copy into your books
Key Formula
- $F = qvB\sin\theta$
- 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
- $F \perp v$ → circular motion
- Electrons: reverse direction
A proton moves in a circular path in a uniform magnetic field. Which statement is correct?
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.
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 $F = qvB\sin\theta$ · 1 mark: correct calculation of 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): $q = 2 \times 1.60 \times 10^{-19} = 3.20 \times 10^{-19}$ C (1 mark for identifying $q = 2e$). $F = qvB\sin\theta = (3.20 \times 10^{-19})(4.0 \times 10^5)(0.080)\sin30° = (3.20 \times 10^{-19})(4.0 \times 10^5)(0.080)(0.5) = 5.12 \times 10^{-15}$ N (1 mark for correct calculation). The magnetic force is always perpendicular to the ion's velocity, so it does no work and the ion's speed remains constant — only its direction changes (1 mark).
Q2 (3 marks): The claim is incorrect. The magnetic force on a moving charge is always perpendicular to the velocity (by the right-hand rule and $\vec{F} = q\vec{v} \times \vec{B}$) (1 mark). Work is defined as $W = Fd\cos\phi$ where $\phi$ is the angle between force and displacement. Since the force is perpendicular to velocity ($\phi = 90°$), $\cos90° = 0$ and $W = 0$ — the magnetic force does zero work (1 mark). Therefore the kinetic energy and speed of the charged particle remain constant. For example, electrons in a CRT screen are deflected in direction by magnetic fields but not accelerated — a separate electric field is needed to change their speed (1 mark).
Five timed questions on the Lorentz force. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
Enter the arenaAt the start you were asked about Dempster's 1918 mass spectrometer at the University of Chicago — which of two uranium ions curves more tightly, and whether reversing charge flips the curve direction.
The answer: U-235 (lighter, 3.90 × 10⁻²⁵ kg) curves more tightly because $r = mv/(qB)$ — smaller mass gives smaller radius. A negative ion curves in the opposite direction because $F = qvB$ reverses sign when charge is negative.
The Dempster spectrometer's power: by measuring where each ion hit the detector, scientists could separate U-235 from U-238 and measure their masses — the first proof that stable isotopes of the same element exist.
Extend your thinking: As an ion curves in the Dempster spectrometer, its velocity direction changes. What happens to the direction of the magnetic force as the ion curves? Does it stay in one direction, or does it rotate too? Explain why this leads to circular motion.