Faraday's Law: How a Changing Magnetic Field Creates a Current

A changing magnetic field creates an electric field

In 1831, Michael Faraday found that moving a magnet near a coil of wire induces a current in the wire — with no battery, no physical contact, nothing but motion and a magnetic field. It's one of the handful of experiments that directly built the modern electrified world, and its mathematical statement is deceptively short: the induced electromotive force (EMF) around a loop equals the negative rate of change of magnetic flux through it.

Magnetic flux Φ measures how much magnetic field passes through a surface bounded by the loop — it grows with field strength, with the loop's area, and with how directly the field points through the loop rather than along it:

Φ = ∫∫ B · dA = B·A·cos(θ)          (for a uniform field)

Faraday's law:   EMF = −N · dΦ/dt

Φ = magnetic flux (webers)       B = magnetic field strength
N = number of turns in the coil  θ = angle between B and the loop's normal
dΦ/dt = rate of change of flux

The formula says nothing about why flux is changing — sliding a magnet in, pulling it out, rotating the coil, or ramping a nearby electromagnet's current up and down all produce the same effect, because all four change Φ over time. This is exactly why a generator, a transformer and a wireless charging pad, despite looking nothing alike, all run on the same equation.

Lenz's law: the minus sign has a purpose

The negative sign in Faraday's law is not a bookkeeping convention — it's a physical law in its own right, called Lenz's law: the induced current always flows in the direction that opposes the change producing it. Push a magnet's north pole toward a coil and the induced current creates its own magnetic field that pushes back against the incoming magnet. This is really conservation of energy in disguise: if the induced current instead helped the magnet along, you could push the magnet once and get free, ever-accelerating energy out of the system — a violation no experiment has ever observed.

Why more turns and faster motion both help

Every one of the N turns in a coil independently intercepts the same changing flux and contributes its own EMF; because the turns are wired in series, their EMFs add, so a 100-turn coil produces one hundred times the EMF of a single loop experiencing the identical flux change. Speed matters for the same underlying reason as the derivative in the formula: moving the magnet twice as fast doubles dΦ/dt, and doubles the induced EMF — this is why hand-crank generators and bicycle dynamos feel stiffer to turn the faster you spin them, since Lenz's law opposition scales with the very same rate of change.

From a lab demonstration to the power grid

Every power station generator, whatever the source of its rotational energy — falling water, steam from burning coal or gas, nuclear fission, or a spinning wind turbine — is fundamentally a coil rotating inside a magnetic field (or a magnet rotating inside a coil), continuously changing the flux and producing an alternating EMF. A transformer uses the same law differently: an alternating current in one coil produces a changing flux in a shared iron core, and that changing flux induces an EMF in a second coil with a different number of turns, stepping voltage up or down by the turns ratio. Faraday's original hand-wound coil and magnet, unglamorous as it looks, is the direct ancestor of both machines.

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