We know that moving magnets can create currents. But to predict how much voltage is generated, we need to quantify the magnetic field passing through a circuit. This quantity is called magnetic flux.

Imagine holding a loop of wire. The magnetic flux ($\\Phi_B$) is a measure of how many magnetic field lines pass through the area enclosed by that loop. It depends on three factors: the strength of the magnetic field ($B$), the area of the loop ($A$), and the angle ($\\theta$) between the field lines and a line perpendicular to the loop's surface (the 'normal').

To induce a voltage, or an electromotive force (EMF), the magnetic flux through the loop must change. You can do this in three ways:

1. Change the magnetic field strength: Move a magnet closer or further away.
2. Change the area of the loop: Stretch or shrink the loop.
3. Change the orientation: Rotate the loop within the magnetic field.

In the 1830s, Michael Faraday discovered the precise relationship between a changing magnetic flux and the voltage it creates. He found that the induced EMF in any closed circuit is directly proportional to the rate of change of the magnetic flux through that circuit. In simple terms, the faster the flux changes, the greater the induced voltage.

This relationship is summarized in Faraday's Law of Induction. The equation includes a term, $N$, for the number of turns in a coil, since the EMF is amplified with each additional loop.

Notice the negative sign. It's not just a mathematical detail—it represents a fundamental law of physics about the direction of the induced current.

That minus sign is the essence of Lenz’s Law which is a consequence of the conservation of energy. It states that the direction of the induced current will be such that its own magnetic field opposes the change in magnetic flux that produced it. Nature pushes back against the change.

Without this opposition, you could create a runaway feedback loop. Pushing a magnet in would create a current that pulled it in faster, creating more current, and so on—generating free energy from nothing, which is impossible. Lenz's Law ensures energy is conserved.

A straightforward application of Faraday's Law is motional EMF. This occurs when a conductor moves through a constant magnetic field. Consider a conducting rod of length $L$ moving at a constant velocity $v$ perpendicular to a uniform magnetic field $B$.

As the rod moves, the area of the loop formed by the rod and rails increases. This change in area causes a change in magnetic flux, which induces an EMF. The magnitude of this motional EMF is given by a simple formula.

Beyond pushing charges in a wire, a changing magnetic field creates an electric field in the space around it. This induced electric field is what drives the current. Unlike the static electric fields created by charges, this induced E-field forms closed loops and is non-conservative, meaning the work done moving a charge around a closed path is not zero. This is the very principle that allows transformers to work.