Electrodynamics and induction
Faraday discovered that a changing magnetic field generates voltage. That one sentence contains the whole electrical industry: the generators in power plants, the transformers on the grid, the inductors in filters, and the spark that burns out transistors when a relay is switched off.
01Faraday’s law
In 1831 Michael Faraday ran the decisive experiment: he brought a magnet close to a coil connected to a galvanometer. While the magnet was moving, the needle deflected. With the magnet at rest — even inside the coil — the needle read zero.
A magnetic field does not generate voltage. A magnetic field that changes does. You can achieve this by moving the magnet, moving the coil, changing the current that produces the field, or rotating the loop inside a fixed field. That last one is what an alternator does.
A coil of 500 turns sees its flux change from 0 to 4 mWb in 20 ms:
e = −500 × (0.004 / 0.020) = −100 V
If the same change took place in 1 ms, the voltage would be 2000 V. That dependence on the speed of the change is what explains the destructive spikes when inductive circuits are opened.
02Lenz’s law
It explains the minus sign: the induced current always flows in the direction that opposes the cause that produces it.
If the induced current helped the motion instead of opposing it, the magnet would accelerate on its own and the coil would generate energy that nobody supplies: it would be a perpetual motion machine. Lenz’s law guarantees that the electrical energy generated is supplied by whoever moves the magnet. That is exactly why a generator is harder to turn when a load is connected to it.
A neodymium magnet is dropped inside a copper or aluminum tube (non-magnetic materials). Instead of falling freely, it descends very slowly, as if floating. The changing field induces circular currents in the tube (eddy currents) that, by Lenz’s law, create a field that opposes the motion. This is the principle of the magnetic brake on trucks and trains, and of the damping of the galvanometer seen in analog instruments.
EMF from a moving conductor
03Self-induction
A coil also induces voltage in itself: if the current flowing through it changes, its own flux changes, and that generates an EMF that — by Lenz’s law — opposes the change in current.
| Capacitor | Inductor |
|---|---|
| Opposes changes in voltage | Opposes changes in current |
| Stores energy in the electric field | Stores energy in the magnetic field |
| W = ½·C·V² | W = ½·L·I² |
| In DC it is an open circuit | In DC it is a short circuit (only its ohmic resistance) |
| Its voltage cannot change instantly | Its current cannot change instantly |
| τ = R·C | τ = L/R |
The most important practical consequence of the whole topic. When you open a circuit containing an inductor, the current tries to drop to zero in a very short time. Since e = −L·Δi/Δt, as Δt tends to zero the induced voltage shoots up to hundreds or thousands of volts, no matter that the circuit runs at 12 V.
That overvoltage:
- Destroys the transistor or the IC that did the switching.
- Burns the contacts of a relay or a switch, through the electric arc.
- It is put to use on purpose in the ignition coil of an internal combustion engine, where 12 V becomes 20,000 V for the spark plug.
It is controlled with a freewheeling diode in antiparallel with the inductor (in DC) or with an RC snubber network (in AC). It is not an optional accessory.
The RL circuit
04Mutual induction: the transformer
If two coils share the same core, a change in current in one induces voltage in the other. That is mutual induction, and its application is the transformer.
A 220 V to 12 V transformer with 1100 turns on the primary, supplying a 2 A load.
- Ratio: 220/12 = 18.3 : 1
- Secondary turns: N₂ = 1100 / 18.3 = 60 turns
- Primary current: I₁ = I₂/18.3 = 2/18.3 = 0.109 A
- Power: 12 V × 2 A = 24 W; 220 V × 0.109 A = 24 W ✓
Design consequence: the secondary carries 18 times more current, so its wire must be much thicker than the primary’s. It is the quickest way to tell which winding is which in an unknown transformer: the one with thin wire and many turns is the high-voltage one.
How that coil behaves with alternating current — and why its reactance grows with frequency — can be seen live in the vector study of AC.
Without a change in flux there is no induction: with DC, the secondary delivers zero volts and the primary behaves like a plain low-resistance winding, and burns out. The only exception is switching power supplies, which chop the DC into pulses of tens of kHz — and precisely because the frequency is high, the transformer can be tiny.
05Hysteresis and iron losses
When a ferromagnetic material is magnetized and demagnetized repeatedly, the B-H curve does not retrace its path: it describes a closed loop called the hysteresis loop.
Narrow loop: low remanence and low coercive field. They magnetize and demagnetize easily, so they lose little energy per cycle.
Soft iron, silicon-iron sheet, soft ferrite, permalloy. Used in transformer, motor and relay cores.
Wide loop: high remanence and high coercive field. Once magnetized, they are hard to demagnetize.
Alnico, hard ferrite, neodymium-iron-boron, samarium-cobalt. Used to make permanent magnets.
The two iron losses
| Loss | Cause | How it is reduced |
|---|---|---|
| Hysteresis loss | Energy spent re-orienting the domains in each cycle. Proportional to the loop area and to the frequency. | Using magnetically soft material (silicon-iron, ferrite). |
| Eddy-current loss | The core is a conductor: the changing flux induces circular currents in its own mass, which heat it. | Laminating the core into thin sheets insulated from one another, perpendicular to those currents. At high frequency ferrite is used, which is an insulator. |
These two, added to the copper losses (I²·R in the windings), are what keep a real transformer from being 100% efficient. A small transformer is around 80%; a distribution transformer, more than 98%.
This is the classic exam question. If the core were a solid piece, eddy currents would flow freely through its whole cross-section and heat it red-hot. By cutting it into 0.35 mm sheets insulated with varnish, those paths are interrupted: the currents are confined to each sheet, where the cross-section is tiny and the loss negligible. It is exactly the opposite effect to the magnetic braking of the magnet in the copper tube, where those currents are sought on purpose.
06In the lab
A coil of several hundred turns connected to a center-zero galvanometer. Bring a magnet close: the needle deflects to one side. Move it away: it deflects to the other. Leave it still inside: zero. Repeat moving it faster and check that the deflection increases. Flip the magnet and verify that all the directions reverse.
Simultaneously drop a neodymium magnet and a screw of equal weight into two identical copper or aluminum tubes. The screw falls immediately; the magnet takes several seconds. Repeat with a PVC tube: both fall the same. It is the most convincing demonstration of Lenz’s law and of eddy currents.
The coil of a 12 V relay powered through a switch, with the oscilloscope connected in parallel (×10 probe, DC coupling, fast timebase and trigger in SINGLE mode on a falling edge). When you open the switch, a negative spike of several tens or hundreds of volts appears. Then fit the freewheeling diode and repeat: the spike is limited to −0.7 V. This is the lab that justifies once and for all the diode in every circuit with a relay.
On a 220/12 V transformer with a center tap: measure with the voltmeter the voltages of each winding at no load and verify the ratio. Then connect a load (a 12 V lamp) and measure the two currents: check that they are in inverse ratio to the voltages and that the powers are similar. The difference between them is the efficiency of the transformer.
Work at 220 V: only with the teacher present, with the primary insulated with heat-shrink tubing and the equipment unplugged for every connection change.
07Common mistakes
| Symptom or confusion | Clarification |
|---|---|
| “A magnet at rest inside the coil generates voltage” | No. A change in flux is needed. With the magnet at rest, ΔΦ = 0 and e = 0. |
| The transistor burns out when a relay is switched off | Breaking extra-current. The freewheeling diode is missing. |
| The transformer hums and heats up with no load | Excessive primary voltage (saturated core), loose laminations, or it was connected to a voltage lower than the design value at the wrong frequency. |
| A transformer was connected to DC and burned out | Without a change in flux there is no induced EMF to oppose the current: the primary is left with only its ohmic resistance, which is very low. |
| Faraday’s law and Lenz’s law get mixed up | Faraday gives the magnitude of the induced voltage; Lenz gives its direction (always opposing the cause). |
| The solid core gets extremely hot | Eddy currents. The core must be made of thin sheets insulated from one another, or of ferrite at high frequency. |
| L is calculated and gives an absurd value when the turns are doubled | L depends on N², not on N: doubling the turns quadruples the inductance. |
08Self-assessment
What condition is essential for an EMF to be induced?
That the magnetic flux linked by the loop varies in time. It can vary because the magnet moves, the coil moves, the loop rotates, or the current that creates the field changes.
A coil of 200 turns sees its flux change from 3 mWb to 1 mWb in 10 ms. What EMF is induced?
e = −200 × (−0.002 / 0.010) = +40 V. The magnitude is 40 V; the sign indicates that the induced current will try to sustain the flux that is disappearing.
State Lenz’s law and say what principle it guarantees.
The induced current flows in the direction such that it opposes the cause that produces it. It guarantees the conservation of energy: if it were the other way around, the system would accelerate on its own and generate energy from nothing.
Why does a magnet fall slowly inside a copper tube if copper is not magnetic?
Because the changing field induces eddy currents in the wall of the tube, and those currents create a field that, by Lenz’s law, opposes the motion. The material does not need to be magnetic: being a good conductor is enough.
Why is it dangerous to open a circuit containing an unprotected inductor?
Because e = −L·Δi/Δt: when the current is cut in a very short time, the induced voltage shoots up to hundreds or thousands of volts. That punctures semiconductors and produces arcing at the contacts. It is avoided with a freewheeling diode in DC or an RC snubber in AC.
A transformer with N₁ = 800, N₂ = 40, supplied at 220 V with 3 A in the secondary. What are V₂ and I₁?
Ratio 800/40 = 20. V₂ = 220/20 = 11 V. I₁ = 3/20 = 0.15 A. Power check: 11 × 3 = 33 W and 220 × 0.15 = 33 W ✓.
What is remanence and what is it good for?
It is the field B that remains in the material when the magnetizing current goes back to zero. In hard materials it is high and makes it possible to build permanent magnets; in soft materials it is kept low, so that the core of a transformer or relay does not stay magnetized.
Why is a transformer core made of insulated sheets and not of a solid piece?
To interrupt the paths of the eddy currents, which in a solid core would flow freely and heat it enormously. Thin sheets insulated with varnish confine those currents to tiny cross-sections, where the loss is negligible.
A 100 mH inductor in series with 50 Ω. What is the time constant?
τ = L/R = 0.1 / 50 = 2 ms. The current reaches its final value after 5τ, that is 10 ms. Note that increasing R speeds up the RL circuit, the opposite of the RC.