EM-20 · Time-varying field

Faraday's Law

A changing magnetic flux makes an electric field (an emf). In a rotating loop, see that the induced emf is largest at the instant the flux changes fastest, and is zero when the flux is at its maximum. ε = -dΦ/dt.

Rotate the loop to see the induced emf

Stop a rotating loop at one moment at a time in a uniform field. When the loop faces the field head-on (flux maximum) versus when it stands edge-on (flux zero), what is the induced emf in each case? Why are the flux and the emf out of step?

Loop angle θ (rotation phase)θ = 45°

Drag to orbit. The slider sets the loop's 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 field makes electricity

Until now electricity and magnetism stood apart; electrostatics and magnetostatics ignored each other. Faraday broke that wall. Bring a magnet near a coil or pull it away and a current flows in the coil; hold the magnet still and nothing happens. A changing magnetic field has made an electric field — an emf. For the first time, magnetism gives birth to electricity. Generators and transformers all come from here.

Flux and its change

The flux through a loop is Φ = ∫B·dA, the amount of field passing perpendicularly through it (EM-05). Faraday's law says the induced emf is proportional to the rate of change of this flux: ε = -dΦ/dt. There are three ways to change the flux: change the field B (move a magnet), change the loop's area, or rotate the loop to change its orientation. Whichever it is, the faster Φ changes, the bigger the emf.

Observeε = −dΦ/dt

The emf is the rate of change of the magnetic flux.

ChooseΦ = ?

The flux is the surface integral of B through the loop.

Lenz, opposing the change

The minus sign is the conscience of Faraday's law. The induced current always flows so as to oppose the very change in flux that created it — this is Lenz's law. Push a magnet into a loop and the loop makes a field that pushes it back; pull it out and the loop makes a field that holds it. If instead it aided the change, the current would grow of itself and energy would pour out endlessly. So Lenz's law is really another face of energy conservation.

The generator and the differential form

Rotate the loop at a steady rate and the flux oscillates as Φ = BA cosωt, so the emf is ε = -dΦ/dt = BAω sinωt — a sinusoidal alternating current. This is the principle of the generator. As the first screen showed, at the instant the loop faces the field with flux at its maximum, the rate of change is zero and so is the emf; at the instant it stands edge-on with zero flux, the change is fastest and the emf is greatest. Written at a point, Faraday's law becomes the differential form ∇ × E = -∂B/∂t: a changing magnetic field makes a swirling electric field (the curl of EM-16).

Fill inε = BAω ?

A rotating loop's emf oscillates as a sine.

On your own∇ × E = ?

A changing magnetic field makes a swirling electric field.

Back to the first screen

On the first screen, the induced emf was zero with the loop face-on (flux at its maximum) and largest with the loop edge-on (zero flux). That is because the emf follows not the flux but the rate of change of the flux. At the moment of maximum flux the change momentarily stops at the peak, so the emf is zero; at the moment the flux crosses zero it is changing most steeply, so the emf is greatest. That is why flux (cos) and emf (sin) are a beat out of step. ε = -dΦ/dt. A changing magnetic field gives birth to electricity.

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

What comes next

Faraday said "a changing magnetic field makes an electric field." Maxwell asks the mirror question: shouldn’t a changing electric field make a magnetic field too? The answer is the displacement current (EM-21). Adding this term to Ampère’s law makes electricity and magnetism perfectly symmetric, and only then can an electromagnetic wave — each field begetting the other — run through space.