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Magnetism & Induction
Magnetic fields, forces, Faraday induction, inductors and motors. 62 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Magnetism is closely tied to electricity. This area covers magnetic fields and field lines, the force on a moving charge and a current-carrying wire, the motion of charged particles in magnetic fields, and electromagnetic induction described by Faraday's and Lenz's laws. It explains how motors, generators, transformers and inductors work.
How the topics connect
Magnetism starts with magnetic poles and fields, then shows that moving charges feel magnetic forces. Faraday's law explains how changing magnetic flux makes a voltage, which is the basis of motors, generators and transformers. Maxwell's equations join electricity and magnetism into a single theory that predicts light.
Real-world uses
Explain how a current in a magnetic field produces torque in a motor, and how moving a coil in a field produces electricity in a generator.
Understand how a transformer steps voltage up for long-distance transmission and down for homes, using induction.
Use magnetic fields to steer and focus charged particles in accelerators, and to produce strong fields in medical scanners.
Relate the Earth's magnetic field to how a compass needle aligns, and why the field varies across the planet.
Engineers use the force on current-carrying wires to design motors, and the induced voltage from moving coils to design generators.
MRI scanners use strong magnetic fields, and Hall sensors in cars measure the position and speed of moving parts.
Magnets and poles
Like poles repel and opposite poles attract.
Every magnet has a north and south pole; like poles repel and unlike attract. Cutting a magnet gives two smaller magnets, never isolated poles.
A compass needle's north end points to Earth's geographic north, which is a magnetic south pole.
Magnetic poles always come in pairs.
Magnetic field lines
Lines run from north to south outside the magnet.
Field lines show the direction a north pole would be pushed; they leave north poles, enter south poles, and form closed loops. Denser lines mean a stronger field.
Iron filings reveal the pattern around a bar magnet.
Field lines never cross.
Earth's magnetic field
Earth acts as a giant bar magnet.
Earth's field is roughly a dipole of 25–65 μT from circulating liquid iron in the outer core (the geodynamo). It deflects solar wind and guides compasses.
The field has reversed many times over geological history.
The magnetic poles drift and are offset from the geographic poles.
Magnetic force on a moving charge
Force perpendicular to velocity and field.
F = qv×BF = qv×B: perpendicular to both velocity and B, with magnitude qvB sinθ. It changes the direction but not the speed, so does no work.
An electron at 10⁶ m/s in 0.1 T: 1.6×10⁻¹⁴ N.
No force on a charge at rest or moving along B.
Circular motion in B field
A charge moves in a circle with radius r = mv/qB.
r = mv/qBWith v ⊥ B the force provides centripetal acceleration: qvB = mv²/r, so r = mv/(qB). Faster or heavier particles have larger radii.
A proton at 10⁶ m/s in 1 T: r = 1.04 cm.
The period is independent of speed.
Cyclotron frequency
Frequency is independent of speed.
f = qB/2πmf = qB/(2πm), independent of speed and radius, which is why cyclotrons work with a fixed-frequency electric field.
Proton in 1 T: 15.2 MHz.
Relativity breaks this at high energies.
Helical motion
Velocity component along B gives a helix.
If velocity has a component along B, the particle follows a helix: circular motion in the perpendicular plane plus uniform drift along B, pitch = v_∥·T.
Charged particles spiral along field lines in the aurora.
Parallel velocity is unaffected.
Velocity selector
Crossed E and B fields select speed v = E/B.
v = E/BCrossed E and B fields allow only particles with v = E/B to pass undeflected (qE = qvB). It's used before mass spectrometers.
E = 10⁵ V/m, B = 0.1 T: v = 10⁶ m/s.
Independent of charge and mass.
Mass spectrometer
Separates ions by mass using a B field.
Ions of velocity v enter a magnetic field and travel on semicircles of radius r = mv/(qB), landing at positions that identify their mass-to-charge ratios. Used to detect isotopes and molecules.
Carbon-12 vs carbon-14 are separated by about 17%.
Ions must be accelerated first.
Cyclotron
Accelerates particles using alternating electric field and magnetic field.
Particles circulate between two D-shaped electrodes; an alternating electric field accelerates them every half-turn, and the magnetic field keeps them in a spiral of growing radius.
Used for producing medical isotopes.
Final energy K = q²B²R²/2m.
Force on a current-carrying wire
Force on a wire in a magnetic field.
F = BIL sinθF = I L × B, magnitude BIL sinθ. It is the sum of forces on moving charges. The right-hand rule gives direction.
A 0.5 m wire carrying 10 A in 0.2 T: 1 N.
Zero when the wire is parallel to B.
Force between parallel wires
Parallel currents attract.
F/L = μ₀I₁I₂/2πdParallel currents attract and antiparallel currents repel with F/L = μ₀I₁I₂/(2πd).
Two 10 A wires 1 cm apart: 2 mN/m.
Opposite of electrostatic like charges.
Definition of the ampere
Based on the force between wires.
Historically 1 A was the current giving 2×10⁻⁷ N/m between wires 1 m apart. Since 2019 it's defined by fixing e = 1.602176634×10⁻¹⁹ C.
The definition fixed μ₀ = 4π×10⁻⁷ H/m.
Now μ₀ is measured, not exact.
Torque on a current loop
A loop rotates in a field.
τ = NBIA sinθA loop in a uniform field feels torque τ = NIAB sinθ, trying to align its moment with B. It's the principle of the motor and galvanometer.
N = 100, I = 1 A, A = 0.01 m², B = 0.5 T: 0.5 N·m max.
No net force in a uniform field.
Moving coil galvanometer
Deflection is proportional to current.
The coil in a radial field feels constant torque NIAB, balanced by a spring κθ, so deflection ∝ I. Current sensitivity = NAB/κ.
Radial field keeps the scale linear.
Damping avoids overshoot.
Electric motor
Converts electrical energy to rotation.
A current loop in a magnetic field with a commutator that reverses current every half turn gives continuous rotation. Torque is largest when the loop is parallel to the field.
Brushless motors use electronics instead.
Spinning generates back EMF.
Biot–Savart law
Field from a current element.
dB = μ₀Idl×r̂/4πr²The field from a current element: dB = (μ₀/4π) I dl × r̂/r². Integrating gives fields of wires, loops and coils.
Field of a loop at the centre: μ₀I/2R.
Direction: right-hand rule.
Field of a long straight wire
Circles around the wire.
B = μ₀I/2πrB = μ₀I/(2πr), with circular field lines around the wire in the direction of the right-hand grip rule.
10 A at 5 cm: 40 μT.
Falls as 1/r.
Field at centre of a loop
Field at the centre of a circular coil.
B = μ₀I/2RB = μ₀NI/(2R) for N turns of radius R, pointing along the axis.
N = 50, I = 2 A, R = 10 cm: 0.63 mT.
Right-hand rule gives the direction.
Field on axis of a coil
Field falls off along the axis.
B = μ₀IR²/(2(R² + x²)^{3/2}) on the axis, maximal at the centre and falling as 1/x³ far away.
At x = R it's 35% of the centre value.
Used for Helmholtz coils.
Field of a solenoid
Uniform inside a long coil.
B = μ₀nIInside a long solenoid B = μ₀nI (n turns per metre), uniform, with nearly zero field outside. Adding an iron core multiplies B by relative permeability.
1000 turns/m at 5 A: 6.3 mT.
Uniform only in the middle of a long coil.
Toroid
Field within a doughnut-shaped coil.
B = μ₀NI/2πrA doughnut-shaped coil confines the field entirely inside: B = μ₀NI/(2πr), depending on radius.
N = 500, I = 2 A, r = 10 cm: 2 mT.
No field outside the toroid.
Ampere's law
Line integral of B equals μ₀ times enclosed current.
∮B·dl = μ₀I∮B·dl = μ₀I_enc: like Gauss's law for magnetism, it gives fields with symmetry easily. Maxwell added displacement current.
Straight wire: B·2πr = μ₀I.
Choose an amperian loop that exploits symmetry.
Helmholtz coils
Two coils giving near uniform field.
Two identical coils separated by their radius produce a nearly uniform field in the centre: B = (4/5)^{3/2} μ₀NI/R.
Used to cancel Earth's field and in experiments.
Uniform over a small volume only.
Magnetic dipole moment
Current loop acts as a tiny magnet.
m = NIAA current loop has magnetic moment m = NIA (direction by right-hand rule). In a field it feels torque m × B and has energy −m·B.
N = 10, I = 2 A, A = 0.01 m²: 0.2 A·m².
Atoms and electrons also have magnetic moments.
Magnetic materials
Diamagnetic, paramagnetic and ferromagnetic behaviour.
Diamagnetic materials are weakly repelled, paramagnetic weakly attracted, ferromagnetic strongly attracted and keep magnetisation. Classified by magnetic susceptibility.
Water diamagnetic, aluminium paramagnetic, iron ferromagnetic.
Strong magnets also levitate frogs via diamagnetism.
Ferromagnetism and domains
Aligned domains give strong magnetisation.
Ferromagnets contain microscopic domains with aligned atomic moments; an external field grows aligned domains and rotates others, giving a very large magnetisation.
Unmagnetised iron has random domains.
Strong exchange interaction needed.
Hysteresis
Magnetisation lags the field, tracing a loop.
Magnetisation lags behind the field, forming a loop in B–H; the area is energy lost per cycle. Hard magnets have wide loops; transformer cores have narrow loops.
Permanent magnets need a wide loop; electromagnets a narrow one.
Remanence and coercivity describe the loop.
Curie temperature
Ferromagnetism disappears above this temperature.
Above T_C thermal agitation destroys ferromagnetic order and the material becomes paramagnetic. Iron 770 °C, nickel 358 °C.
Heating a magnet above T_C demagnetises it.
Cooling in a field restores magnetisation.
Electromagnets
Soft iron cores concentrate field from a coil.
A coil with a soft iron core creates a strong, switchable magnet. Field ∝ NI. Used in cranes, relays, loudspeakers and MRI.
Scrapyard cranes lift cars.
Soft iron loses magnetism when current stops.
Permanent magnets
Hard magnetic materials retain magnetisation.
Made of hard ferromagnetic materials (steel, alnico, neodymium) that retain magnetisation. Neodymium magnets reach 1.4 T.
Fridge magnets, hard drives, motors.
Can be demagnetised by heat or knocks.
Magnetic flux
Field passing through a surface.
Φ = BA cosθΦ = BA cosθ counts field lines through a loop; unit weber (T·m²). It changes if B, A or the angle changes.
0.5 T through 0.02 m² face-on: 10 mWb.
Flux is zero if B is parallel to the surface.
Faraday's law
Induced emf equals the rate of change of flux.
ε = −dΦ/dtA changing flux through a circuit induces an EMF: ε = −N dΦ/dt. The minus sign is Lenz's law. Faster changes give more EMF.
A coil of 100 turns with flux dropping by 1 mWb in 0.1 s: 1 V.
EMF arises from change of flux, not flux itself.
Lenz's law
Induced current opposes the change causing it.
The induced current's magnetic effect opposes the change that created it. It's conservation of energy: otherwise you'd get free energy.
Dropping a magnet through a copper tube slows it.
It opposes the change, not the flux itself.
Motional emf
A rod moving in a field generates emf.
ε = BLvA rod of length L moving with speed v perpendicular to B has ε = BLv, from the magnetic force on charges in the rod.
L = 0.5 m, v = 4 m/s, B = 0.2 T: 0.4 V.
Only the component of velocity perpendicular to B and L counts.
Eddy currents
Circulating currents induced in bulk conductors.
Changing flux induces circulating currents in bulk conductors, heating them and opposing motion. Laminated cores reduce them in transformers.
Induction cookers heat pans with eddy currents.
Useful or harmful depending on the application.
Eddy current braking
Magnets slow motion in conductors.
A moving conductor in a magnetic field has eddy currents that oppose its motion, giving smooth, contactless braking used on trains and roller coasters.
Magnetic brakes in coasters work at any speed with no wear.
Force decreases as speed falls, so it can't hold at rest.
Self-inductance
Coil opposes changing current.
ε = −L dI/dtA coil resists changes in its own current: ε = −L dI/dt, where L = NΦ/I in henrys. It acts like inertia for current.
L = 0.5 H, current changing 10 A/s: 5 V.
Switching off an inductive load can cause a spark.
Inductance of a solenoid
Depends on turns and geometry.
L = μ₀N²A/lL = μ₀N²A/l: grows as the square of turns and with an iron core by μ_r.
N = 1000, A = 10 cm², l = 0.2 m: 6.3 mH.
Doubling turns quadruples L.
Energy in an inductor
Energy stored in magnetic field.
U = ½LI²U = ½LI², stored in the magnetic field. Energy density u = B²/2μ₀.
L = 2 H, I = 3 A: 9 J.
Releases suddenly when the current is cut.
Mutual inductance
Changing current in one coil induces emf in another.
M = Φ₂/I₁A changing current in one coil induces an EMF in a nearby coil: ε₂ = −M dI₁/dt. M depends on geometry and coupling.
Wireless charging uses mutual inductance.
M₁₂ = M₂₁.
Transformer
Changes AC voltage using mutual induction.
V₂/V₁ = N₂/N₁Two coils on a shared core: V₂/V₁ = N₂/N₁. Step-up raises voltage and lowers current, keeping power nearly constant: V₁I₁ ≈ V₂I₂.
230 V to 12 V with 1000 turns primary: 52 turns secondary.
Only works with AC.
Transformer losses
Copper, iron and flux leakage losses.
Copper loss (I²R), eddy-current and hysteresis loss in the core, and flux leakage. Large power transformers reach 99% efficiency.
Laminated cores and oil cooling.
Losses become heat.
AC generator
A rotating coil produces sinusoidal emf.
ε = NBAω sinωtA coil rotating at ω in a field produces ε = NBAω sin ωt. Power stations rotate coils with turbines.
N = 100, A = 0.05 m², B = 0.5 T, 50 Hz: peak 785 V.
Peak EMF scales with speed.
Dynamo and alternator
Generating electricity using rotation.
A dynamo outputs DC using a commutator; an alternator gives AC using slip rings and is more efficient. Car alternators charge the battery.
Bicycle dynamo lamps.
Output rises with speed.
Back emf in a motor
A spinning motor generates opposing emf.
A spinning motor coil generates an EMF opposing the supply, so I = (V − ε_b)/R. At start-up there's no back EMF, hence large starting current.
12 V motor, R = 1 Ω, back EMF 10 V: 2 A.
If stalled the current is dangerous.
LR circuit growth and decay
Current rises exponentially with time constant L/R.
τ = L/RCurrent in an LR circuit rises as I = (V/R)(1 − e^{−Rt/L}) and decays as I₀e^{−Rt/L}, time constant τ = L/R.
L = 0.1 H, R = 10 Ω: τ = 10 ms.
Analogous to RC.
Maxwell's displacement current
Changing electric fields also produce magnetic fields.
I_d = ε₀dΦ_E/dtA changing electric field acts like a current: I_d = ε₀ dΦ_E/dt. With it, Ampère's law becomes ∮B·dl = μ₀(I + I_d), predicting EM waves.
It makes the current continuous through a charging capacitor.
Needed for charge conservation.
Maxwell's equations
Four equations unifying electricity, magnetism and light.
Four equations: Gauss's law for E, Gauss's law for B (no monopoles), Faraday's law and Ampère–Maxwell law. With the Lorentz force they describe all classical electromagnetism and predict light at c = 1/√(μ₀ε₀).
A massive unification in 1865.
Foundation of relativity.
Magnetic monopoles
Hypothetical isolated poles, never observed.
Isolated magnetic charges would make ∇·B ≠ 0 and symmetric Maxwell's equations. Dirac showed they'd explain charge quantisation. None have been found.
Searches in cosmic rays and colliders.
Not seen in nature so far.
Vector potential
A field A with B = ∇×A.
B = ∇×A defines the vector potential, simplifying electrodynamics; E = −∇V − ∂A/∂t. The Aharonov–Bohm effect shows A has physical effects.
Used in quantum theory.
A isn't uniquely defined (gauge freedom).
Magnetic levitation
Trains levitated by magnetic force.
Maglev trains use magnets to levitate and propel along a track without contact. Superconductors can levitate above magnets via flux pinning.
Shanghai maglev reaches 430 km/h.
Stable levitation requires special arrangements (Earnshaw).
Magnetic resonance imaging
Uses nuclear spin in strong fields.
Strong static field (1.5–3 T) aligns hydrogen nuclei; radio pulses tip them, and the emitted signal gives images with contrast from tissue differences. Field gradients encode position.
No ionising radiation.
Metal implants can be hazardous.
Aurorae
Charged solar particles spiralling in Earth's field excite atmosphere.
Charged solar wind particles are guided by Earth's field to the poles, where they excite atmospheric oxygen (green, red) and nitrogen (blue/purple).
Seen near latitudes 65–70°.
Stronger during solar storms.
Van Allen belts
Charged particles trapped in Earth's magnetosphere.
Charged particles are trapped in Earth's magnetosphere, bouncing between poles in two belts at roughly 1000–6000 km and 13,000–60,000 km altitude. They are hazards to spacecraft.
Discovered in 1958 by Explorer 1.
Intensity varies with solar activity.
Superconducting magnets
Zero-resistance coils make huge fields.
Coils of niobium–titanium or similar cooled with liquid helium carry large currents without losses, producing fields of 10+ T. Used in MRI, LHC and fusion devices.
LHC dipoles are 8.3 T at 1.9 K.
A quench can destroy the coil.
Magnetic field of moving charge
Field from a charge in motion.
B = μ₀qv×r̂/4πr²A point charge with velocity v produces B = (μ₀/4π) q v × r̂/r² (for v ≪ c). The field forms circles around the line of motion.
An electron beam creates a field.
Only valid at low speeds.
Hall effect and sensors
Hall probes measure magnetic field.
Hall probes use the transverse voltage to measure B; they are used in position sensors, wheel-speed sensors and phones.
Output voltage ∝ B.
Temperature drift needs compensation.
Magnetic energy density
Energy stored per volume.
u = B²/2μ₀u = B²/(2μ₀); for B = 1 T that's 400 kJ/m³, much more than the electric energy density at practical fields.
MRI magnet stores megajoules.
Compare ½ε₀E².
Magnetic pressure
Fields exert pressure on plasma.
Magnetic fields exert pressure P = B²/2μ₀ and tension along the field lines, important in plasmas and stars. At 1 T the pressure is 4 atm.
Tokamaks confine plasma with magnetic pressure.
Sunspots are regions of strong fields.
Lorentz force law
Combined electric and magnetic force.
F = q(E + v×B)F = q(E + v×B) gives the complete electromagnetic force on a charge. It governs particle accelerators, mass spectrometers and plasmas.
An electron in crossed fields drifts at E/B.
Magnetic part does no work.
Magnetic shielding
High-permeability material diverts field lines.
High-permeability materials such as mu-metal divert field lines around a region, shielding it. Superconductors expel fields. Simple conductors only shield changing fields.
Shielding CRTs and sensitive instruments.
Static fields can't be shielded by copper.
Frequently asked questions
Why does a magnet not attract all metals?
Only ferromagnetic materials such as iron, nickel and cobalt, and a few others, respond strongly. In most materials the magnetic effect is too weak to notice.
What does Lenz's law say?
The induced current in a circuit opposes the change in magnetic flux that causes it. This is why a magnet falling through a copper tube slows down, and why it is a consequence of energy conservation.
Why does a magnetic field not change the speed of a charged particle?
The magnetic force is always perpendicular to the velocity, so it changes the direction of motion but does no work. The kinetic energy and speed stay constant.
Why does a transformer only work with AC?
A transformer needs a changing magnetic flux to induce a voltage in the second coil. Steady DC gives constant flux, so no voltage is induced.
Are electricity and magnetism really the same thing?
They are two aspects of one electromagnetic field. A moving charge produces a magnetic field, and changing magnetic fields produce electric fields.
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