Electromagnetic Induction: a Magnet and a Coil

Move a magnet through a coil and an EMF and a current appear in it. See how field lines “thread” the turns, where the current reverses, why it brakes the magnet and where the energy goes.
The simulator shows Faraday’s law ε = −N·dΦ₁/dt, Lenz’s rule, the current I = ε/R, the Lenz force and the heat — in real units and to real scale.

where the magnet is now (scrub)

N / S — magnet poles — field lines that pass through the coil (their number = the flux) ↓ v — velocity of the magnet ↓ F — Lenz force on the magnet ● — charges in the turns (they run with the current) ↓ B′ — field of the coil’s induced current G — galvanometer needle (the scale adjusts itself)

The chain: from the magnet’s motion to the force on it

The EMF appears because a changing magnetic field generates a vortex electric field: more in the “Vortex electric field” simulator.

The scene is to scale: the ruler on the left is in centimetres. The magnet is treated as a point dipole (accurate when it is clearly smaller than the coil), the coil is thin — all turns in one plane — and self-inductance is neglected.

The idea in 30 seconds

Motion is needed

A magnet that just sits next to the coil produces no current, however close it is. An EMF appears only when the number of field lines passing through the coil CHANGES. Move the magnet faster — the lines change faster — the EMF is bigger.

EMF = how fast the flux changes

The flux Φ is “how many field lines pass through the coil” (they are light blue on the scene). The EMF equals the rate of change of the flux: ε = −N·dΦ₁/dt. In the centre of the coil the flux is largest, but at that instant it is not changing — so ε = 0 and the current reverses sign: you get a curve with two humps.

The current resists its cause

Lenz’s rule: the induced current flows so as to oppose the change of the flux. That is why the magnet is always braked: the Lenz force points against the velocity. This is energy conservation — to get a current you must do work, and the magnet’s energy ends up as heat.

Try it yourself

Switch on “By hand” (or just grab the magnet with the mouse) and move it back and forth: faster — bigger EMF; stop — EMF = 0, even if the magnet is inside the coil.
Press “Magnet at the centre”: the flux is maximal (all the Φ cells are filled) yet the current is zero. Why? Hint: look at the ε(t) graph.
Flip the magnet: all the currents reverse, but the Lenz force still brakes the magnet. Try to predict it before you start.
Open the circuit: there is an EMF but no current — and the magnet falls like a free body (the ghost coincides with the magnet).
“Free fall” with the “Floating magnet” experiment: in a closed circuit with a small R and large N the magnet falls almost uniformly — the Lenz force balances gravity and all the energy goes into heat.

Where it works in real life

Power-station generators: coils rotate in a magnetic field — the flux changes and an EMF is induced in them.
Induction hob: an alternating field induces eddy currents in the bottom of the pan and the cookware heats itself.
Wireless charging: the charger coil creates an alternating field and an EMF is induced in the coil of the phone.
Guitar pickup and dynamic microphone: the vibration of a string or a diaphragm changes the flux and produces an electrical signal.
Magnetic brakes of trains and rides: a copper plate is braked in the field of a magnet with no friction — that is the Lenz force.

Graphs of one pass

The flux Φ, the EMF ε and the Lenz force F during one pass. The shaded area under the ε(t) curve equals the loss of flux: ∫ε dt = −ΔΦ. Note that the current changes direction but the Lenz force always has one sign — it brakes both on the way in and on the way out.

Slow motion

Ready-made experiments

How the magnet moves

v magnet speed

Direction of motion

Which pole is on top

N number of turns

a coil radius

m magnetic moment (and mass)

Closed circuit

R circuit resistance

Show

magnet field lines (light blue — those passing through the coil)

charges in the turns (beads)

“ghost”: where the magnet would be without the coil

Model: the magnet is a point dipole on the coil axis, the flux through one turn is Φ₁ = μ₀·m·a² / (2·(a²+z²)^{3/2}), the EMF is ε = −N·dΦ₁/dt and the current direction follows Lenz’s rule. The motion is computed exactly, so the numbers do not depend on the frame rate or the playback speed: playback can be slowed down while the numbers and graphs stay in real time. The magnet’s mass is 7.5 g for every A·m² (neodymium).

The key point: the EMF is produced not by the flux itself but by the rate at which it changes. Hence the ε(t) curve with two humps of opposite sign and a zero at the centre of the coil — while the Lenz force brakes the magnet all the time.

KEY FORMULAS

Flux of the magnet through one turn

$$\Phi_1(z) = \frac{\mu_0\, m\, a^2}{2\,(a^2+z^2)^{3/2}}, \qquad \Phi = N\,\Phi_1$$

A point dipole m on the axis of a turn of radius a, z is the distance from the magnet to the plane of the turn; the coil flux is the sum over N turns

▶ Detailed Explanation

The flux can be pictured as “the number of field lines that pass through the turn”. At the centre of the coil (z = 0) it is largest: Φ₁ = μ₀m/(2a); the radius sits in the denominator, so a wide coil collects less flux than a narrow one. Far away (|z| ≫ a) the flux falls off as 1/|z|³ — the magnet has to pass close to the turns.

The sign of the flux is set by the orientation of the magnet: with the north pole up and the normal up the flux is positive, for a flipped magnet it is negative. The total flux of the coil Φ = N·Φ₁ grows with the number of turns, so the EMF is proportional to N as well.

Faraday’s law

$$\varepsilon = -N\,\frac{d\Phi_1}{dt} = -N\,\frac{d\Phi_1}{dz}\,v, \qquad \frac{d\Phi_1}{dz} = -\frac{3\,\mu_0\, m\, a^2\, z}{2\,(a^2+z^2)^{5/2}}$$

The EMF is the rate of change of the flux: you need both a flux that varies with position and a speed v; at the centre dΦ₁/dz = 0

▶ Detailed Explanation

Substituting Φ₁(z) into Faraday’s law gives ε = 3·N·μ₀·m·a²·z·v / (2·(a²+z²)^{5/2}). A magnet at rest (v = 0) gives no EMF however close it is. In the centre of the coil (z = 0) the flux is maximal, its derivative with respect to z is zero, so ε = 0 — which is why the ε(t) graph passes through zero between the two humps.

The minus sign is Lenz’s rule: the current opposes the change of the flux. In the simulator a positive EMF means a counter-clockwise current seen from above.

Peak EMF

$$\varepsilon_{\max} = \frac{24\sqrt5}{125}\,\frac{\mu_0\, N\, m\, v}{a^2} \approx 0.43\,\frac{\mu_0\, N\, m\, v}{a^2} \quad (z = \pm a/2)$$

At constant speed the EMF peaks when the magnet is at a distance a/2 from the plane of the turns; ε_max ∝ N·m·v/a²

▶ Detailed Explanation

The flux changes fastest where |dΦ₁/dz| is largest — at z = ±a/2. Substituting z = a/2 gives ε_max = (24√5/125)·μ₀·N·m·v/a². The formula tells how to raise the EMF: more turns N, a stronger magnet m, faster motion v and a narrower coil (the factor 1/a²). For N = 100, m = 1 A·m², a = 4 cm, v = 1 m/s this gives 33.7 mV — the number the simulator shows.

It is handy for checking: in the “Mystery coil” lab ε_max is proportional to N, so N = 100·ε_max/ε₁₀₀.

Current, the Lenz force and heat

$$I = \frac{\varepsilon}{R}, \qquad F = -\frac{\left(N\,\dfrac{d\Phi_1}{dz}\right)^{2}}{R}\,v, \qquad P = \frac{\varepsilon^2}{R} = -F\,v$$

The Lenz force always points against the velocity; its power goes entirely into the heat ε²/R

▶ Detailed Explanation

The current in a closed circuit is I = ε/R. The coil’s current creates its own magnetic field, which acts on the magnet with a force F. Energy conservation gives F·v = −ε·I = −ε²/R, hence F = −(N·dΦ₁/dz)²·v/R. The force is always opposite to the velocity: the magnet is braked both on entering and on leaving, even though the current changes direction.

The work against this force turns into heat: Q = ∫ε²/R dt. In an open circuit I = 0, so F = 0 and Q = 0: there is an EMF, but it brakes nothing. A smaller R means a larger current and stronger braking.

Area under the ε(t) curve and the charge

$$\int \varepsilon\,dt = -N\,\Delta\Phi_1, \qquad q = \int I\,dt = -\frac{N\,\Delta\Phi_1}{R}$$

The area under the EMF graph equals the loss of flux; the charge that has passed through the circuit depends only on the change of flux and on R

▶ Detailed Explanation

Integrating Faraday’s law gives ∫ε dt = −N·ΔΦ₁: on the ε(t) graph the shaded area up to the current moment equals the loss of flux Φ(0) − Φ(t). The speed of motion does not affect this area — faster motion gives taller, narrower humps of the same area.

If the magnet starts and ends far from the coil, where Φ ≈ 0, the total area is zero: the two humps are opposite. The charge q = −N·ΔΦ₁/R also returns to zero — the golden “beads” in the turns on the scene first run forward and then back.

A magnet falling through a coil

$$M\ddot z = -Mg + F, \qquad v_t = \frac{M\,g\,R}{\left(N\,d\Phi_1/dz\right)^{2}}, \qquad Mg\,\Delta h = \tfrac12 M v^2 + Q$$

Gravity and the Lenz force; when they are equal the speed is constant; lost potential energy = kinetic + heat

▶ Detailed Explanation

Without a current the magnet falls freely: it covers 40 cm in 0.286 s. In a closed circuit a Lenz force that grows with speed is added to gravity. If N and m are large and R is small, in the coil zone it balances M·g, and the magnet falls at the constant speed v_t = M·g·R/(N·dΦ₁/dz)² — it “floats”, as in a copper tube.

The magnet’s mass in the simulator is 7.5 g for every A·m² (neodymium). Energy is conserved: M·g·Δh = ½Mv² + Q, and while it floats almost all the lost potential energy immediately becomes heat in the circuit.

The field of a small permanent magnet is well described by a point dipole with moment m. The flux of this field through one turn of radius a lying on the axis at a distance z from the magnet is Φ₁(z) = μ₀·m·a² / (2·(a²+z²)^{3/2}). At the centre of the coil (z = 0) it is largest and equals μ₀m/(2a), and far away it falls off as 1/|z|³. It is handy to picture the flux as “the number of field lines passing through the coil”: on the scene exactly those lines that thread the turns are highlighted in light blue, and their number is proportional to Φ. For a coil of N identical turns the total flux is Φ = N·Φ₁.

Faraday’s law: the EMF equals minus the rate of change of the flux, ε = −N·dΦ₁/dt = −N·(dΦ₁/dz)·v. So a magnet at rest gives no EMF however large the flux is, and the faster the magnet moves, the larger the EMF. In the centre of the coil the flux is largest but its derivative is zero — there ε = 0 and the current reverses. The EMF peaks at z = ±a/2, where the flux changes fastest: ε_max = (24√5/125)·μ₀·N·m·v/a² ≈ 0.43·μ₀·N·m·v/a². At constant speed the two humps of the ε(t) curve have equal height and opposite sign.

The area under the ε(t) curve over any interval equals the change of the flux: ∫ε dt = −N·ΔΦ₁. The magnet starts and ends far from the coil, where the flux is almost zero, so the areas of the two humps are equal in size and opposite in sign, and the total charge that has passed through the circuit, q = −N·ΔΦ₁/R, is zero: the charges in the turns first run one way and then come back.

The direction of the current is set by Lenz’s rule: the induced current flows so that its own field opposes the change of the flux. As the magnet approaches, the coil turns a like pole towards it and repels it; as the magnet recedes, the coil turns an unlike pole towards it and attracts it. The Lenz force F = −(N·dΦ₁/dz)²·v/R always points against the velocity, so it can only brake: the current changes direction, the force does not. The power this force “takes away”, P = −F·v = ε²/R, turns entirely into heat in the circuit — this is conservation of energy.

In “Free fall” mode the magnet moves under gravity and the Lenz force: M·z̈ = −M·g + F. With an open circuit there is no current, F = 0, and the magnet flies from +20 to −20 cm in 0.286 s like a free body. In a closed circuit with a small resistance and many turns the Lenz force can balance gravity: the magnet falls at the constant speed v = M·g·R/(N·dΦ₁/dz)², and all its potential energy goes into heat: M·g·Δh = ½Mv² + Q (the “Floating magnet” experiment). The physical cause of the EMF is the vortex electric field generated by the changing magnetic field: the EMF appears in the loop itself and does not depend on the material of the coil, while a current flows only in a closed circuit.