Inductance
Change the current quickly and see the back EMF
The gold helix is the coil, and the cyan lines are the magnetic flux passing through it. Raising the rate of change of the current with the slider lengthens the red arrows on the coil, the back EMF that opposes the change. The arrows differ greatly between changing the current slowly and quickly, and when the current is steady (rate 0), the back EMF disappears.
Magnetic inertia
Running current through a coil creates a magnetic field around and inside it (EM-13·14), and magnetic flux builds up through the coil. This flux is proportional to the current. Change the current and the flux changes with it, and a changing flux inside the coil produces an EMF in the coil that opposes the change. Just as a heavy object resists changes in velocity, a coil resists changes in current, and the size of this magnetic inertia is the inductance.
Self-inductance L = NΦ/I
If the coil has N turns, the flux links the current N times over. This bundle of flux, the flux linkage, divided by the current is the inductance. L = NΦ/I. If the current doubles, the flux doubles too, so their ratio, the inductance, does not change. What sets inductance is not the current or the flux but the shape of the coil, that is, the number of turns, the cross-sectional area, and the length, along with the material filling it. This is exactly the same structure as capacitance being set by shape and material.
The back-emf that opposes change
When the current changes, the flux changes, and a changing flux produces an EMF according to Faraday's law (EM-20). For a coil this EMF is ε = -L dI/dt. The key is the minus sign. The EMF always appears in the direction that opposes the change in current; this is Lenz's law. It resists an increase and holds back a decrease, so the current through a coil cannot change suddenly and changes smoothly instead. The faster the change (large dI/dt), the stronger the back EMF.
The coil's energy
Raising the current in a coil takes work against the back EMF, and that work is stored as energy. The stored energy is U = ½LI², closely resembling a capacitor's ½CV², and this energy is held in the magnetic field the coil creates (EM-19). The inductance of a solenoid is L = μ₀N²A/l, proportional to the square of the number of turns. Filling it with a magnetic material such as an iron core greatly increases the inductance (EM-18), but if the current grows so large that the core saturates, the inductance drops again. Inductors are thus used as components that smooth currents and briefly store energy.
Back to the first screen
On the first screen, when you changed the current slowly the coil's back EMF was small, and the faster you changed it, the more steeply it grew, because the back EMF is proportional to the rate of change. ε = -L dI/dt. With the current steady (rate 0), the back EMF was 0 too. That constant of proportionality is the coil's magnetic inertia, the inductance. Inductance is set by the coil's shape, not by the current (L = NΦ/I), and because of this inertia the current in a coil cannot change suddenly.
Filling a coil with a magnetic material such as an iron core makes the magnetic field μr times stronger through magnetization, and the inductance grows by the same factor. It is the magnetic partner of a dielectric raising capacitance. The next unit (EM-18 magnetic materials and magnetization) treats magnetization M and relative permeability μr, and the ½LI² stored in a coil is revisited as field energy in EM-19 magnetic energy.