The AC generator
Take the third lever — changing the angle — and turn it into a machine. Spin a coil of N turns and area A at a steady angular frequency \omega inside a field B. The angle grows as \theta = \omega t, so the flux is \Phi_B = BA\cos(\omega t). Feed that into Faraday's law and its rate of change is a sine: the coil delivers a smoothly alternating EMF that swings positive and negative many times a second. This is how essentially all grid electricity is generated.
The generator EMF and its peak value ε₀. Faster spin (larger ω), more turns, stronger field or bigger area all raise the output.
Motors and back-EMF
Run the machine in reverse and you have a motor. Feed current into a coil in a field and it feels a torque and spins: that is an electric motor. But here is the twist — as the motor's coil spins, it is also a generator, and by Faraday's law it produces its own EMF opposing the supply. This is the back-EMF, and it grows with speed.
Back-EMF is what limits a motor's current. At the instant of start-up the coil is not yet turning, there is no back-EMF, and a large inrush current floods in — which is why the lights dim for a moment when a fridge compressor or a power tool kicks on. As the motor reaches full speed, the back-EMF rises until it nearly cancels the supply and the current settles to just what the mechanical load demands.
Self-inductance: electrical inertia
A coil can even induce an EMF in itself. When the current through a coil changes, it changes the coil's own magnetic flux, which by Faraday's law induces an EMF that opposes the change. This property is self-inductance L, measured in henries (H). A coil built to have a useful L is an inductor. It is defined by linking flux to current, N\Phi_B = LI, which makes its self-induced EMF proportional to how fast the current changes.
Self-induced EMF opposes the change in current; the inductance L links flux to current. Bigger L means a stiffer resistance to any change of current.
An inductor resists changes in current the way mass resists changes in velocity — it is a kind of electrical inertia. It has no quarrel with a large steady current; it fights only sudden change, snapping back against any attempt to speed the current up or slow it down.
LR circuits and stored energy
Put an inductor, a resistor and a battery in series (an RL circuit). Because the inductor forbids a jump in current, the current does not snap to its final value \varepsilon/R — it climbs there exponentially, with a characteristic time constant \tau = L/R. After one \tau the current has reached about 63% of its final value.
The current in an RL circuit rises exponentially toward ε/R with time constant τ = L/R.
While the current builds, the inductor stores energy in its magnetic field — energy the source had to supply against the back-EMF. The stored amount parallels the \tfrac12 C V^2 of a capacitor and the \tfrac12 m v^2 of a moving mass, reinforcing the picture of L as inertia.
Energy stored in an inductor's magnetic field — the magnetic cousin of ½CV² and ½mv².