Wave a magnet in and out of a coil of wire hooked to a tiny bulb, and the bulb flickers — with no battery anywhere in sight. Hold the magnet perfectly still, right next to the coil, and the light dies. That is the whole surprise of induction: it is not the presence of the magnet that makes electricity, it is the changing of the magnet's reach through the coil. Every power plant on Earth — wind, hydro, coal, nuclear — is really just an elaborate way of sweeping magnets and coils past each other.
To turn that into numbers we track the magnetic flux $\Phi_B$, basically how much magnetic field pokes through the loop: $\Phi_B = BA$ (field strength times the loop area it threads). Faraday's law says the induced voltage, or EMF, equals how fast that flux changes, times the number of turns $N$. In its simplest form:
Try a real number. A coil of $N = 100$ turns wrapping an area $A = 0.01$ m² sits in a field that climbs from $0$ to $0.5$ T in $0.1$ s. The flux change is $\Delta\Phi_B = 0.5\times0.01 = 0.005$ Wb, so $\mathcal{E} = 100\times0.005/0.1 = 5$ V — plenty to light an LED, produced purely because the field changed.
The exact statement uses a derivative and a minus sign: $\mathcal{E} = -N\,\dfrac{d\Phi_B}{dt}$, with $\Phi_B = \int \vec{B}\cdot d\vec{A}$. That minus sign is Lenz's law: the induced current always flows in the direction that fights the change that created it. Push a magnet in and the coil pushes back; pull it out and the coil tries to hang on. This is energy conservation in disguise — you must do work to move the magnet, and that work is exactly what becomes electrical energy. A faster change, more turns, a stronger field, or a larger area each raises the EMF. In the sim, the B field, Loop area, and No. turns N sliders set how much flux you have to play with, while Resistance R and Inductance L shape how the resulting current builds up, through the RL time constant $\tau = L/R$.
Try this in the sim above: (1) In Faraday mode, raise No. turns N and watch the EMF curve grow in exact proportion to $N$. (2) Switch to Lenz mode and notice the induced current is always oriented to oppose the flux change. (3) Open RL Circuit mode and increase Inductance L (or lower R): the current now takes longer to reach its final value, because $\tau = L/R$ grows — the inductor resists any sudden change in current.
| Symbol | Quantity | SI Unit |
|---|---|---|
| $\mathcal{E}$ | EMF (electromotive force) | V |
| $\Phi_B=BA\cos\theta$ | Magnetic flux | Wb = T m² |
| $L$ | Self-inductance | H = V s A⁻¹ |
| $\tau=L/R$ | RL time constant | s |