EM-20 · Time-varying field

Faraday's Law

When the magnetic flux through a loop changes, an emf appears in the loop. In a rotating loop, see for yourself that the induced emf is largest at the instant the flux changes fastest, and that it is 0 at the instant the flux is at its maximum. ε = -dΦ/dt.

Rotate the loop to see the induced emf

The sky-blue arrows are a uniform magnetic field, and the slider sets the angle of the loop sitting in it. When the loop faces the field head-on (maximum flux), the gold current arrows vanish; when it stands edge-on (flux 0), they are at their longest. Why the emf is smallest exactly when the flux is largest is the question of this lesson.

Loop angle θ (rotation phase)θ = 45°

Drag to orbit the view. The slider sets the loop angle θ.

The loop's flux and induced emf

ε = -dΦ/dt · the faster the flux changes, the bigger the emf

Φ = ∫B·dA = BA cosθ · the flux through the loop

Flux 71% · emf 71%

Slanted

A changing flux drives a current

In electrostatics and magnetostatics, electricity and magnetism were worked out as two separate fields. Faraday then found that a current flows in a coil only while a magnet is moving toward it or away from it; hold the magnet still and nothing happens. What drove the current was not the magnetic field itself but its change. Generators, transformers, electric-guitar pickups, and the transit card that runs with no battery, powered by the reader you tap it on, all come from this discovery.

Flux and its change

The magnetic flux through a loop is Φ = ∫B·dA, the field piercing the loop's surface, added up over the whole surface (EM-05). Faraday's law says the induced emf equals the rate of change of this flux with a minus sign: ε = -dΦ/dt. There are three ways to change the flux: change the field B (move a magnet or ramp a current), change the loop's area, or turn the loop so its surface points a different way. In every case only how fast the flux changes sets the emf; whether the flux is large or small does not matter. Wind the coil N turns and the same change adds up N times, so the emf is N times larger.

Observeε = −dΦ/dt

The emf is set by how fast the magnetic flux changes.

ChooseΦ = ?

Magnetic flux is B integrated over the loop's surface.

The minus sign and Lenz's law

The minus sign in front sets the direction of the induced current. The induced current always flows so as to oppose the flux change that produced it; this is Lenz's law. Push a magnet's north pole into a loop and the loop's current makes a field that pushes the pole back; pull the magnet out and the current makes a field that holds it in. If the induced current helped the change instead, a tiny wobble would grow on its own and energy would keep appearing from nothing, so Lenz's law is energy conservation written as a rule about current direction. It is also why a generator gets harder to turn once you draw current from it: the electrical energy taken from the loop comes from the work done by whatever turns it.

The generator and the differential form

Turn the loop at a steady angular speed ω and the flux oscillates as Φ = BA cosωt, so the emf is ε = -dΦ/dt = BAω sinωt. This is the sinusoidal AC a generator puts out. Just as on the first screen, at the instant the loop faces the field and the flux is at its maximum, the rate of change is 0, so the emf is 0; at the instant it stands edge-on and the flux is 0, the change is fastest, so the emf is at its maximum. Written at a single point, Faraday's law becomes the differential form ∇ × E = -∂B/∂t: a changing magnetic field comes with a swirling electric field (the curl of EM-16). In a case like this picture, though, the field stays the same and the wire moves, so there is no swirling electric field; the emf comes from the Lorentz force on the charges carried along by the moving wire. That one flux rule gives the right value in both cases is what makes Faraday's law so useful.

Fill inε = BAω ?

A loop turning steadily gives an emf that rises and falls as a sine.

On your own∇ × E = ?

A changing magnetic field comes with a swirling electric field.

Back to the first screen

On the first screen, when the loop faced the field head-on (maximum flux) the induced emf was 0, and when it stood edge-on (flux 0) the emf was largest. That is because the emf follows not the size of the flux but how fast it changes. The moment of maximum flux is the top of the curve, where rising turns into falling and the change pauses for an instant, so the emf is 0; the moment the flux passes through 0 is the steepest part of the curve, so the emf peaks. That is why flux (cos) and emf (sin) are a quarter cycle out of step. ε = -dΦ/dt: a changing flux makes electricity.

Faraday's law says that when the magnetic flux through a loop changes, an emf appears. ε = -dΦ/dt. The faster the flux Φ = ∫B·dA changes, the larger the induced emf, and the minus sign means the induced current flows so as to oppose that change (Lenz's law = energy conservation). A loop turning steadily gives the AC emf ε = BAω sin(ωt) (a generator). The differential form is ∇ × E = -∂B/∂t.

What comes next

Faraday's law says a changing magnetic field makes an electric field. Maxwell asked whether the reverse also holds, and the answer is yes: a changing electric field makes a magnetic field too, and that term is the displacement current (EM-21). Add it to Ampère's law and the equations of electricity and magnetism come in matching pairs, which makes electromagnetic waves possible, the two fields producing each other as they race through empty space.