Electromagnetic Induction

Currents make magnetic fields. Faraday asked the reverse question, and found that a changing magnetic field makes a current. This discovery powers every generator on Earth.

Faraday’s experiments#

Put a bar magnet on the axis of a wire loop connected to a galvanometer. Held still, nothing happens. Push the magnet toward the loop and the needle swings; pull it away and it swings the other way. The current lasts only while the magnet moves.

Faraday found the rule behind this and many similar experiments:

Φ is the magnetic flux, measured in webers (1 Wb = 1 T m²). It can change because B changes, because the loop’s area changes, or because the angle between them changes. The emf is the induced emf; in a loop of resistance R it drives an induced current:

Which way does the current go?#

Using the sign in Faraday’s law:

1. Put an arrow on the loop to choose a positive sense for current (any choice).
2. Curl your right-hand fingers along the arrow: the thumb gives the positive normal.
3. Find Φ with that normal. If Φ is increasing, ℰ is negative: the current flows against your arrow. If Φ is decreasing, it flows along it.

Lenz’s law#

A quicker way. The induced current makes its own magnetic field, which can strengthen or weaken the original flux. Nature always chooses:

• An N pole pushed toward a loop raises the flux; the induced current makes a field pointing back at the magnet, so the near face of the loop acts as an N pole and repels it.
• A sliding wire that shrinks a circuit in a field into the page lowers the flux; the induced current makes field into the page, i.e. it runs clockwise.
• A loop beside a straight wire whose current is increasing: the flux through the loop grows, so the induced current circulates to make field opposite to the wire’s.

Where does an induced emf actually come from? When a conductor moves through a field, the answer is simply the magnetic force on its moving electrons.

Two ways to change the flux#

Flux can change because (a) the field is steady but the loop (or part of it) moves, (b) the loop is still but the field changes, or both. The mechanism is different in the two cases. This lesson covers (a), motional emf.

A rod moving through a field#

A rod PQ of length l moves at velocity v⃗ through a field B⃗, with rod, velocity and field all mutually perpendicular. Its free electrons share the velocity v⃗, so each feels qv⃗ × B⃗ along the rod. Electrons pile up at one end (P) and leave positive charge at the other (Q). The resulting electric field E inside the rod pushes back. Charge stops building up when the two balance: qE = qvB, i.e. E = vB.

Join P and Q by a resistor and current flows. The magnetic force keeps driving electrons back up the rod, maintaining the potential difference. With external resistance R and rod resistance r: i = vBl/(R + r).

Faraday’s law agrees: if x is the length of the circuit in the field, Φ = Blx and |dΦ/dt| = Bl dx/dt = vBl.

Forces and energy#

Once current i flows in the rod, the field pushes on it with ilB, against its motion (Lenz again). To keep it moving at constant v something must push with force F = ilB = B²l²v/R. That force delivers power Fv = B²l²v²/R, and the circuit produces heat i²R = (vBl/R)²R = B²l²v²/R. Exactly equal: your mechanical work becomes the heat in the circuit.

If the loop isn’t moving, the magnetic force can’t start its electrons going. Something else must: an electric field produced by the changing magnetic field itself.

The induced electric field#

Hold a loop still in a steady field, then let the field change. A current starts. But the electrons were at rest, and a magnetic field cannot push charges at rest, so it must be an electric field that sets them moving. It is not made by charges (Coulomb’s law has nothing to say about it); it is made by the changing magnetic field.

• It is non-electrostatic and non-conservative: no potential can be defined for it.
• Its field lines are closed loops, with no beginning or end.
• It exists whenever B changes, loop or no loop. A conducting loop merely lets its free electrons respond, giving a current.

The work per unit charge round a loop is the emf, so Faraday’s law becomes a statement about the field:

Eddy currents#

A solid metal plate moving into a field has no single wire to guide the current, but every closed path drawn on it has a changing flux. Currents swirl round many such paths at once: eddy currents.

They heat the plate, and that energy comes out of its kinetic energy: the plate slows down. This is electromagnetic damping. Cutting slots in the plate breaks up the current paths and greatly reduces it.

A coil carrying a current threads its own magnetic flux. Change the current and the coil fights back with an emf of its own: self-induction.

Self-inductance#

The field of a current, and so the flux through the loop carrying it, is proportional to the current:

The unit is the weber per ampere, equivalently volt-second per ampere, named the henry (H). For a coil of N turns, add the emf of every turn: ℰ = −N d(∫B⃗·dS⃗)/dt. By Lenz’s law the self-induced emf opposes the change: it pushes against a rising current and tries to keep a falling one going.

Inductance of a long solenoid#

A solenoid of radius r with n turns per length carries i: B = μ₀ni inside, so each turn links μ₀niπr². A length l has nl turns, giving a total emf −(nl)(μ₀nπr²) di/dt. Comparing with −L di/dt:

A coil of thick wire has negligible resistance but appreciable inductance: an ideal inductor, drawn as a row of loops.

Inductance resists sudden change#

An inductor gives a circuit inertia. Current can’t jump on or off; it rises and falls exponentially, with a time constant L/R, and the inductor stores energy in its magnetic field.

Growth of current#

An inductor L, a resistor R and a cell ℰ are joined in series and the switch is closed at t = 0. As the current rises, the inductor’s emf −L di/dt opposes it. The loop law gives

L/R has the dimensions of time: the time constant. After one time constant the current is i₀(1 − 1/e) = 63 % of its final value. In principle the final value is reached only as t → ∞; in practice a few time constants is enough. A small τ means a steep rise.

Decay of current#

With a steady current i₀ flowing, the cell is replaced by a plain wire (a special switch closes the new path before opening the old one, so the circuit is never broken). Now the inductor’s emf is the only one: −L di/dt = Ri, so

Energy stored in an inductor#

Multiply the growth equation by i dt and integrate from 0 to t:

The work done by the cell equals the Joule heat in R plus a remaining term: the energy stored in the inductor’s magnetic field.

Energy density of a magnetic field#

For a long solenoid (radius r, length l, n turns per length, no magnetic material), L = μ₀n²πr²l and B = μ₀ni, so U = ½Li² = (B²/2μ₀)πr²l. Dividing by the volume πr²l:

A changing current in one circuit can induce an emf in a completely separate circuit nearby. That coupling is the heart of transformers, induction coils and wireless chargers.

Mutual inductance#

Current i in one circuit produces a field whose flux through a second circuit is proportional to i:

For circuits with several turns, add the flux through every turn before writing Φ = Mi.

The induction coil#

Ruhmkorff’s induction coil makes a very high emf from a low-voltage cell. A primary coil T is wound on a laminated soft-iron core; a secondary coil S of many turns is wound over it and ends in two rods G₁ and G₂ with an adjustable gap. The primary circuit has a cell and an automatic interrupter: a springy metal strip A carrying a soft-iron hammer H, touching a screw D. A capacitor C is connected across the interrupter.

How it works

1. Make and break. Current through the primary magnetizes the core, which pulls the hammer and breaks the contact. The current stops, the core lets go, the strip springs back and the contact closes again, over and over.
2. Slow rise, fast fall. On make, the primary’s self-inductance makes the current grow slowly. On break, it collapses very quickly.
3. Secondary emf. The secondary emf −M di/dt is therefore small on make and very large on break (opposite in sign), enough to throw sparks across the gap: about 50 000 V from a 12 V cell, enough to run a discharge tube.
4. The capacitor. On break, the primary’s own induced emf would spark across the interrupter, keeping current flowing, slowing its fall and pitting the contacts. The capacitor absorbs that surge instead, so the current drops faster; it then discharges back through the primary, swinging the current almost to −i. The change in flux nearly doubles and so does the secondary emf.