Magnetic flux
The magnetic flux through a loop measures how much magnetic field passes through it: for a uniform field B at right angles to a flat loop of area A, Φ = B A, in webers (tesla times square meters). If the loop is tilted by an angle θ, only the part of the field along its axis counts: Φ = B A cos θ. A coil of N turns links N times the flux of one turn.
Faraday's law
In 1831 Michael Faraday found that a current flows in a coil whenever the flux through it changes, and only then. The EMF (the voltage the changing flux drives round the circuit) is
ε = −N dΦ ÷ dt
The flux can change because the magnet moves, the coil moves or turns, the field strength changes, or the coil changes shape. A magnet sitting still inside a coil, however strong, induces nothing. Doubling the speed doubles the EMF; doubling the turns doubles it again.
Lenz's law: the minus sign
The induced current always flows so that its own magnetic field opposes the change in flux. Push a north pole into a coil and the near end of the coil becomes a north pole too, repelling the magnet; pull it out and that end becomes a south pole, pulling it back. You have to push against this force, and the work you do is the electrical energy that appears in the circuit. If the induced current helped the change instead, a magnet would accelerate into a coil on its own and energy would come from nowhere.
A worked example: a magnet and a coil
A neodymium magnet with a dipole moment of 1 A·m² passes at 1 m/s along the axis of a 200-turn coil of radius 2 cm.
- The flux of a small magnet through one loop of radius a at distance d along its axis is Φ = μ₀ m a² ÷ (2 (a² + d²)3/2). At the center (d = 0) this is μ₀ m ÷ 2a = 1.26 × 10−6 ÷ 0.04 = 3.1 × 10−5 Wb.
- The flux changes fastest at d = a ÷ 2 = 1 cm from the coil, at a rate of 0.43 μ₀ m ÷ a² = 1.35 × 10−3 Wb per meter of travel.
- At 1 m/s: ε = 200 × 1.35 × 10−3 × 1 = 0.27 V at the peak, positive going in and negative coming out, and zero as the magnet passes the center, where the flux is greatest but momentarily not changing.
- With 4 Ω of coil wire and a 10 Ω load, the peak current is 0.27 ÷ 14 = 19 mA.
Generators
Spin a coil of N turns and area A at angular speed ω in a uniform field B. The flux through it is N B A cos ωt, so
ε = N B A ω sin ωt, peak ε₀ = N B A ω, RMS = ε₀ ÷ √2
The output is a sine wave that reverses twice each turn: alternating current. Example: 200 turns of 100 cm² in 0.2 T, turning 50 times a second (ω = 314 rad/s), give a peak of 200 × 0.2 × 0.01 × 314 = 126 V, an RMS voltage of 89 V. The EMF is largest when the coil's plane is along the field, where the flux through it is zero but changing fastest. Power-station generators work the same way, usually with the magnet turning inside fixed coils, at 3,000 or 3,600 revolutions a minute for 50 or 60 Hz.
Transformers
A transformer has two coils on a shared iron core. Alternating current in the primary makes a changing flux in the core, which passes through the secondary and induces a voltage there. Every turn of either coil sees the same flux change, so the voltage per turn is the same:
Vs ÷ Vp = Ns ÷ Np, and for an ideal transformer Vp Ip = Vs Is
An old-style mains adapter's transformer might step 230 V down to 12 V with 1,150 turns on the primary and 60 on the secondary. Power lines do the opposite: stepping the voltage up 10 times cuts the current 10 times, and the I²R heating in the lines 100 times, which is why electricity is sent over long distances at hundreds of kilovolts.
Using the simulation
In the Electromagnetic Induction Simulator, things to try:
- Choose By hand and move the position slider slowly, then quickly: the needle swings further the faster you move, and returns to zero when you stop.
- Watch the letters over the coil as the magnet comes in and goes out: the coil's near end always matches the approaching pole and opposes the leaving one.
- Turn the magnet round: every current reverses.
- In the generator, double the frequency: the peak voltage doubles too, because the flux changes twice as fast.
- In the transformer, swap the turns to step down, then tick direct current: the secondary gives nothing.
What the model assumes
- A point-dipole magnet. Its field is that of an ideal dipole at its center, which is accurate far from the coil but rough when the magnet is inside it.
- A short coil. The flux is averaged over nine turns spread along 4 cm of a 2 cm radius coil, and the coil's resistance is 0.02 Ω per turn.
- No self-inductance. The current follows the EMF instantly, and the current's own field does not act back on it or slow the magnet.
- A uniform field between the generator's poles, and a coil turning at a perfectly steady speed.
- An ideal transformer: no resistance in the windings, no losses in the core, all the flux passing through both coils, and no magnetizing current.
- Slow motion. The generator is drawn at no more than one turn a second and the transformer's flux at half a cycle a second; the numbers are for the real frequency.
Edge cases
- Magnet at rest: zero EMF anywhere, even at the center of the coil where the flux is greatest.
- Magnet at the center, moving: the EMF passes through zero there, because the flux is at its maximum and momentarily not changing.
- Equal turns on a transformer: Vs = Vp; it then only isolates one circuit from the other, which is useful for safety.
- Direct current in a transformer: no output, and a primary current limited only by its wire.
Where the model stops being right
- Back-EMF and drag. A large induced current really does push back on the magnet, slowing it; drop a strong magnet down a copper pipe and it falls slowly because of the eddy currents it induces. The simulation moves the magnet at whatever speed you set.
- Inductance. A real coil resists changes in its own current, so with many turns and a small load the current lags and is smaller than ε ÷ R, especially at high frequency.
- Real transformers lose 1 to 5% of the power to winding resistance and to hysteresis and eddy currents in the core, which is why the core is laminated. Their output voltage sags under load, and an iron core saturates if driven too hard or at too low a frequency.
- Changing electric fields and radiation. At high frequencies the changing fields travel as electromagnetic waves, and circuit ideas like a single EMF round a loop no longer hold.
Related tools
Work out currents and power in the load with the Ohm's Law Calculator, see the force a magnetic field puts on a moving charge, the other half of electromagnetism, in the Charged Particle in a Magnetic Field Simulator, and wire up the load in the Series and Parallel Circuit Simulator.