Difficulty: Intermediate | Prerequisites: Electric fields and Gauss's Law (Part 1), vector cross products, basic calculus.
Magnetism enters the course in two stages. First, you learn how magnetic fields exert forces on moving charges and current-carrying wires, including the behaviour of magnetic dipoles in external fields. Second, you learn how currents create magnetic fields (Ampere's Law, solenoids) and how changing magnetic fields create electric fields and EMFs (Faraday's Law, Lenz's Law). These ideas lead directly to inductance and are the foundation for AC circuits, transformers, and electromagnetic waves. If Gauss's Law was the workhorse of electrostatics, Ampere's Law and Faraday's Law are the workhorses of magnetism.
A moving charge in a magnetic field experiences a force perpendicular to both its velocity and the field (F = qv × B), which means the magnetic force does no work. Ampere's Law lets you calculate the field from symmetric current distributions, and Faraday's Law says a changing magnetic flux induces an EMF. Lenz's Law tells you the direction: the induced current opposes the change.
Magnetic force on a moving charge
F = q(v × B). The force is perpendicular to both velocity and field. In simple terms, a magnetic field can deflect a moving charge but cannot speed it up or slow it down.
Lorentz force
The full electromagnetic force on a charge: F = q(E + v × B). Combines the electric and magnetic forces into one expression.
Ampere's Law
∮ B · dl = μ₀I_enc. The line integral of the magnetic field around a closed loop equals μ₀ times the current threading through the loop. Think of it as the magnetic analogue of Gauss's Law.
Solenoid
A tightly wound coil of wire. Inside an ideal (infinite) solenoid, the magnetic field is uniform: B = μ₀nI, where n = N/l is the number of turns per unit length.
Magnetic dipole moment (μ)
For a current loop: μ = NIA, where N is the number of turns, I is the current, and A is the area vector (perpendicular to the loop, direction from the right-hand rule). In simple terms, it measures how strongly a current loop responds to an external magnetic field.
Faraday's Law of Induction
EMF = −dΦ_B/dt. A changing magnetic flux through a loop induces an electromotive force. The minus sign encodes Lenz's Law.
Lenz's Law
The direction of the induced current is such that it opposes the change in magnetic flux that produced it. In simple terms, nature resists changes in flux.
Self-inductance (L)
A measure of how much flux a coil generates through itself per ampere of current: L = NΦ_B / I. For a solenoid, L = μ₀N²A / l. Measured in henrys (H).
F = q(v × B), magnitude F = qvB sin θ, where θ is the angle between v and B.
If v is parallel to B, then sin θ = 0 and F = 0. The charge passes through undeflected.
If v is perpendicular to B, the force is maximum (F = qvB) and the charge moves in a circle.
Circular motion radius: r = mv / (qB). This comes from setting qvB = mv²/r (magnetic force = centripetal force).
Given: m = 1.67 × 10⁻²⁷ kg, q = 1.6 × 10⁻¹⁹ C, B = 0.5 T, v = 10⁶ m/s, perpendicular to B.
r = mv/(qB) = (1.67 × 10⁻²⁷ × 10⁶) / (1.6 × 10⁻¹⁹ × 0.5)
r = 1.67 × 10⁻²¹ / 8.0 × 10⁻²⁰ = 0.0209 m ≈ 2.09 cm.
A current loop with dipole moment μ = NIA placed in a uniform external field B experiences:
A torque: τ = μ × B, magnitude τ = μB sin θ. This torque tends to align μ with B.
A potential energy: U = −μ · B = −μB cos θ. Energy is minimised when μ is parallel to B.
In a uniform field, there is no net force on the dipole, only torque. A net force appears only in a non-uniform field.
For an infinitely long straight wire carrying current I, choose a circular Amperian loop of radius r.
∮ B · dl = B(2πr) = μ₀I, so B = μ₀I / (2πr).
The field circles the wire (right-hand rule: thumb along current, fingers curl in the direction of B).
Use a rectangular Amperian loop with one side inside the solenoid (length l) and one side outside (where B ≈ 0).
∮ B · dl = Bl = μ₀nIl, so B = μ₀nI.
The field is uniform inside and essentially zero outside (for an ideal infinite solenoid).
n = N/l is the turns per unit length. Do not confuse N (total turns) with n.
EMF = −dΦ_B/dt, where Φ_B = ∫ B · dA is the magnetic flux through the loop.
Flux changes when B changes, the area changes, or the angle between B and A changes.
Lenz's Law in practice: If flux into the page is decreasing, the induced current flows clockwise (to maintain the inward flux by the right-hand rule). If flux into the page is increasing, the current flows counter-clockwise.
The flux through one turn: Φ_B = BA = μ₀(N/l)IA.
Total flux linkage: NΦ_B = μ₀(N²/l)IA.
Inductance: L = μ₀N²A / l.
The same formula is sometimes written L = μ₀n²lA (since n = N/l and N² / l = n²l).
These two forms are equivalent, but the exam may present both. Recognise that option A (μ₀n²lA) and option D (μ₀N²A/l) are the same thing written differently. The correct form given N, l, A is L = μ₀N²A/l.
Quantity | Expression |
|---|---|
Magnetic force | F = q(v × B), F = qvB sin θ |
Circular orbit radius | r = mv / (qB) |
Cyclotron frequency | f = qB / (2πm) |
Magnetic dipole moment | μ = NIA |
Torque on dipole | τ = μ × B |
Potential energy of dipole | U = −μ · B |
Ampere's Law | ∮ B · dl = μ₀I_enc |
Field from long wire | B = μ₀I / (2πr) |
Field inside solenoid | B = μ₀nI |
Faraday's Law | EMF = −dΦ_B/dt |
Self-inductance (solenoid) | L = μ₀N²A / l |
μ₀ (permeability of free space) | 4π × 10⁻⁷ T·m/A |
Cyclotrons and mass spectrometers exploit r = mv/(qB) to separate particles by mass. Electric motors and generators are direct applications of the torque on a current loop and Faraday's Law. The inductors in your phone's power supply are tiny solenoids whose self-inductance smooths out voltage spikes.
Students often think a magnetic field can do work on a charge. It cannot: the force is always perpendicular to velocity, so the magnetic force does zero work. Any change in kinetic energy comes from an electric field, not the magnetic field.
Confusing N (total turns) and n (turns per unit length) in solenoid problems is a common source of wrong answers. The field is B = μ₀nI, not μ₀NI.
When applying Lenz's Law, students sometimes confuse "the flux is into the page" with "the flux is increasing into the page." Lenz's Law cares about the change in flux, not the flux itself.
Students sometimes apply the point-charge force formula (F = kqQ/r²) to magnetic problems. Electric and magnetic forces are distinct: use F = qvB sin θ for magnetic force.
⚠️ If velocity is parallel to B, the magnetic force is zero. This is a very common multiple-choice question.
⚠️ The cyclotron radius calculation (r = mv/qB) is a standard 5-mark short-answer problem. Include units.
⚠️ Know both forms of solenoid inductance: L = μ₀N²A/l and L = μ₀n²lA. The exam may present both and ask which is correct.
⚠️ Lenz's Law direction problems (clockwise vs. counter-clockwise) appear nearly every exam. Draw the flux, determine whether it is increasing or decreasing, then find the current direction that opposes the change.
⚠️ The magnetic dipole moment question (define μ, explain torque, mention U = −μ·B) is a common 5-mark short-answer.
True or false: A charged particle moving parallel to a magnetic field experiences no magnetic force.
Fill in the blank: The radius of a charged particle's circular orbit in a magnetic field is r = _______.
True or false: Inside an ideal solenoid, the magnetic field depends on the radius of the solenoid.
Fill in the blank: When flux into the page through a loop is decreasing, Lenz's Law predicts the induced current flows _______ (clockwise / counter-clockwise).
True or false: The self-inductance of a solenoid is proportional to the square of the number of turns.
Answers: 1. True. 2. mv/(qB). 3. False (it is uniform: B = μ₀nI, independent of radius). 4. Clockwise (to maintain inward flux). 5. True (L = μ₀N²A/l).
Q: A charged particle moves with velocity parallel to a uniform magnetic field. What is the magnetic force on the particle?
A: Zero. F = qvB sin θ, and θ = 0 when v is parallel to B, so F = 0.
Q: What is the magnetic field at distance r from an infinitely long wire carrying current I?
A: B = μ₀I / (2πr), from Ampere's Law applied to a circular loop of radius r.
Q: A circular loop sits in a magnetic field pointing into the page. The field magnitude is decreasing. What is the direction of the induced current?
A: Clockwise. By Lenz's Law, the induced current must create a field into the page to oppose the decreasing inward flux. By the right-hand rule, this requires a clockwise current.
Q: What is the self-inductance of a solenoid with N turns, length l, and cross-sectional area A?
A: L = μ₀N²A / l. Derived from L = NΦ_B/I with B = μ₀(N/l)I inside the solenoid.
Q: Define the magnetic dipole moment for a current loop and state the torque it experiences in an external field.
A: The dipole moment is μ = NIA, where A is the area vector perpendicular to the loop. The torque is τ = μ × B, which acts to align μ with B. The potential energy is U = −μ · B.
The cyclotron radius connects to mass spectrometry and particle physics. Faraday's Law and inductance lead directly into RL circuits and the time constant τ = L/R (covered in the circuits notes). The interplay between changing electric and magnetic fields eventually produces electromagnetic waves (covered in the EM waves notes). Ampere's Law is extended by Maxwell's displacement current to complete the set of Maxwell's equations.
magnetic force, Lorentz force, cross product, cyclotron radius, cyclotron frequency, mass spectrometer, Ampere's Law, Biot-Savart Law, solenoid, magnetic field inside solenoid, turns per unit length, Faraday's Law, Lenz's Law, electromagnetic induction, induced EMF, magnetic flux, self-inductance, inductor, henry, magnetic dipole moment, torque on current loop, right-hand rule, PHYS 212, University Physics electricity and magnetism