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5.3.3 Charge in Magnetic Fields

By the end of this lesson you can:

  • Calculate the force on a moving charge in a magnetic field
  • Explain why the path is circular
  • Calculate the radius of the path
Find your course: NSW Year 11NSW Year 12: Module 6, IQ1QLD Unit 3: ElectromagnetismVIC Unit 3: Fields and motionAP Physics 2: Unit 12IB: D.3 Motion in electromagnetic fields
🎥 Watch📝 Practise🕹️ Explore🎯 Exam ready🍎 Teachers
🎥Watch
Learn the idea: a short explanation, then a check.

Hand Rules Explained

Before we continue it's an important to learn the hand rules that are used in studying electromagnetism.
Hand rules are 'tool's used to establish the correct relationship between the vectors  of electrical current, force, EMF and magnetic field.
There are two in predominant use the first is Fleming's hand rules the second is the Palm rule. This video discusses both.
Both are equally valid, however it is best to consistently use one or the other.

Charge behaviour in an Magnetic field

As demonstrated in the simulation, a moving charged particle deflected by a magnetic field follows a distinct circular arc.

The Lorentz Force & Circular Motion

When a charge q moves with velocity v through an external magnetic field B, it experiences a magnetic force (FB) described by the magnetic component of the Lorentz Force Law:

FB = qvB sin(θ)

Where θ is the angle between the particle's velocity vector and the magnetic field vector. According to the Right-Hand Palm Rule (or Fleming's Left-Hand Rule), this magnetic force acts strictly perpendicular to both the direction of motion and the magnetic field lines.

Because FB is always perpendicular to velocity, it acts as a centripetal force (Fc = mv2⁄r). It changes the direction of the particle's motion without altering its speed or kinetic energy. Equating centripetal force to the magnetic force when moving perpendicularly (θ = 90°) allows us to derive the radius (r) of the path:

qvB = mv2⁄r  & implies;  r = mv⁄qB

Watch the video below as we elaborate further on these mathematical relationships and explore how varying mass, charge, velocity, and field strength influence particle trajectories.

📝Practise
Worked examples first, then problems to try, then worksheets.

Check Your Understanding: Charges in Magnetic Fields

1. Which diagram correctly shows the deflection of a cathode ray by a bar magnet?
2. A charged particle, q, enters a uniform magnetic field B at velocity v. The particle follows a circular path of radius r as shown. If the magnitude of the magnetic field were doubled and the other variables were kept constant, what would the new radius be?
A charged particle following a circular path in a uniform magnetic field
3. An electron and a proton enter the same uniform magnetic field with equal velocities perpendicular to the field lines. How do the radii (r) and directions of their circular paths compare?

We are now ready to try a sample problem
Below is a sample problem with a video that explain how to solve it. It is suggested you try the problem beforehand, as this actually aids understanding, even if you are unsure if you are correct.

Picture

Problems to try

Practice Problems: Charged Particles in a Magnetic Field

In each problem the particle moves at right angles to the field, so the magnetic force provides the centripetal force: qvB = mv2 / r, which gives r = mv / (qB).

1. An unknown particle having a mass of 2.2 × 10-27 kg and a charge of 3.3 × 10-19 C passes through a magnetic field of 5.4 × 10-1 T. The velocity of the particle is 6.4 × 103 m/s. What is the radius of its path?

Given: m = 2.2 × 10-27 kg, q = 3.3 × 10-19 C, B = 5.4 × 10-1 T, v = 6.4 × 103 m/s

Formula: r = mv / (qB)

Calculation: r = (2.2 × 10-27 × 6.4 × 103) / (3.3 × 10-19 × 5.4 × 10-1) = (1.408 × 10-23) / (1.782 × 10-19)

Answer: r = 7.9 × 10-5 m
2. A particle with a mass of 3.8 × 10-27 kg and a charge of 6.2 × 10-19 C crosses a magnetic field that measures 2.7 × 10-2 T. The particle assumes a circular path with a radius of 1.5 × 10-1 m. At what speed is the particle moving?

Given: m = 3.8 × 10-27 kg, q = 6.2 × 10-19 C, B = 2.7 × 10-2 T, r = 1.5 × 10-1 m

Formula: r = mv / (qB) → v = qBr / m

Calculation: v = (6.2 × 10-19 × 2.7 × 10-2 × 1.5 × 10-1) / (3.8 × 10-27) = (2.511 × 10-21) / (3.8 × 10-27)

Answer: v = 6.6 × 105 m/s (about 660,790 m/s)
3. A particle passing through a magnetic field has a mass of 6.3 × 10-27 kg and is moving at 3.9 × 104 m/s. The charge on the particle is 2.4 × 10-18 C and the radius of its circular path through the field is 4.4 × 10-2 m. What is the strength of the magnetic field?

Given: m = 6.3 × 10-27 kg, v = 3.9 × 104 m/s, q = 2.4 × 10-18 C, r = 4.4 × 10-2 m

Formula: r = mv / (qB) → B = mv / (qr)

Calculation: B = (6.3 × 10-27 × 3.9 × 104) / (2.4 × 10-18 × 4.4 × 10-2) = (2.457 × 10-22) / (1.056 × 10-19)

Answer: B = 2.33 × 10-3 T (0.00233 T)
4. A particle with a mass of 3.34 × 10-27 kg and a charge of 1.2 × 10-19 C passes through a magnetic field of 3.4 × 10-3 T. This causes the nucleus to assume a circular path with a radius of 0.065 m. What is its velocity?

Given: m = 3.34 × 10-27 kg, q = 1.2 × 10-19 C, B = 3.4 × 10-3 T, r = 0.065 m

Formula: v = qBr / m

Calculation: v = (1.2 × 10-19 × 3.4 × 10-3 × 0.065) / (3.34 × 10-27) = (2.652 × 10-23) / (3.34 × 10-27)

Answer: v = 7.94 × 103 m/s (about 7,940 m/s)
🕹️Explore
Test your predictions in a simulation.
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The instructions below allow you to consolidate and test your understanding.

  1. Set the variables to some value
  2. Fire a trace
  3. Change ONE variable, keeping all others constant and PREDICT what will happen
  4. Fire the trace to test your prediction.
    Can you explain the trace?
  5. Repeat for another variable

🔬 Interactive Investigation: Charged Particle Magnetic Deflection

Simulation Guide

Use the Lorentz Force Engine simulator above to explore how a moving charge behaves when entering a uniform magnetic field. Work through the inquiry challenges below to build your physical intuition before analysing the equations.

⚡ Quick Start Protocol:
  1. Set initial parameters: Charge (q = +1), Velocity (v = 2), Mass (m = 2), and Field Strength (B = 2).
  2. Click Fire to launch the particle into the field zone.
  3. Hit Clear Paths between experiments, or stack multiple runs to compare trajectories directly.

🎯 Guided Inquiry Questions

Observation 1
1. Trajectory Shape
What path geometric shape does the particle follow upon entering the magnetic field region, and why doesn't its speed change during deflection?
Explanation: The particle follows a circular arc. Because the magnetic force (FB = qvB) acts perpendicular to the velocity vector at every instant, it acts strictly as a centripetal force. It changes the particle's direction without performing work or altering kinetic energy (speed remains constant).
Predict & Test
2. Velocity & Radius
If you double the initial velocity (v) while keeping mass, charge, and field strength constant, what happens to the radius (r) of the path?
Explanation: The radius doubles (r is directly proportional to v). Faster particles have greater momentum (mv), requiring a larger circular turn radius for a given magnetic field force to bend them.
Predict & Test
3. Mass vs. Field Strength
What happens to the radius if you double the mass (m) versus if you double the magnetic field strength (B)?
Explanation:
  • Doubling Mass (m): Doubles the radius (r ∝ m) because greater inertia resists bending.
  • Doubling Field Strength (B): Halves the radius (r ∝ 1/B) because a stronger magnetic force exerts a sharper deflection.
Directional Analysis
4. Field Orientation & Charge Sign
Switch the field direction from Into Page (×) to Out of Page (•). How does the path change? What would happen if the charge was negative (e.g., an electron)?
Explanation: Reversing the magnetic field direction flips the force direction 180° (e.g., from downward deflection to upward deflection). Likewise, switching to a negative charge reverses the sign of q, causing opposite deflection according to the Right-Hand Palm Rule.
Synthesis Challenge
5. Zero Deflection Challenge
Can you configure the variables so that two particles with different masses (m1 and m2 = 2m1) follow the exact same circular trajectory?
Explanation: Yes! Since r = mv / qB, you can keep r constant by doubling the charge (q2 = 2q1) or doubling the field strength (B), or by halving the second particle's velocity (v2 = ½v1).
🎯Exam ready
Check you can do it under exam conditions.

How well do you know charge behaviour in fields?

Having now covered charge behaviour in both electric and magnetic fields, test your understanding of charges in fields.
Do the Quiz and try to get full marks

Then check your understanding if necessary with the video