EM-17 · Magnetostatics field

Inductance

Inductance is a coil's magnetic inertia — its resistance to change in its own current. See that the faster you change the current, the bigger the back-emf the coil raises to oppose the change. ε = -L dI/dt.

Change the current fast to see the back-emf

Use the slider to raise the rate at which the coil's current changes. How does the back-emf the coil makes differ when you change the current slowly versus fast? And when the current is steady (zero rate), what is the back-emf?

Rate of current change dI/dtdI/dt = 4.0

Drag to orbit. The slider sets the rate of current change dI/dt.

The back-emf the coil makes

ε = -L dI/dt · the back-emf opposing the current change

L = NΦ/I · the flux linkage per current, set by geometry

Back-emf (vs maximum) · 40%

Middle

Magnetic inertia

Send a current through a coil and a magnetic field arises around and inside it (EM-13, EM-14), building up a flux Φ through the coil. This flux is proportional to the current. Try to change the current and the flux must change with it — but the coil resists that change. Just as a massive object resists a change in its velocity, a coil resists a change in its current. The size of this magnetic inertia is the inductance L.

Self-inductance L = NΦ/I

If the coil is wound N times, the flux links the current N-fold. This flux linkage divided by the current is the inductance: L = NΦ/I. Double the current and the flux doubles too, so L is unchanged. L is set not by the current or flux but by the coil's geometry — turns, cross-section, length, and the material filling it. It is exactly the structure by which capacitance C was set by geometry.

ObserveL = NΦI

Inductance is the flux linkage per current.

The back-emf that opposes change

When the current changes, the flux changes, and a changing flux makes an emf by Faraday's law (EM-20). For a coil this emf is ε = -L dI/dt. The minus sign is the point: the emf always arises so as to oppose the change in current (Lenz's law). Try to increase the current and it pushes back; try to decrease it and it holds on. So the current in a coil cannot jump suddenly but changes smoothly. The faster the change (larger dI/dt), the stronger the back-emf.

Chooseε = ?

The back-emf is inductance times the rate of current change.

The coil's energy

To build up the current in a coil you must work against the back-emf, and that work is stored as energy: U = ½LI². It closely mirrors the capacitor's ½CV². This energy resides in the magnetic field the coil makes (EM-19). A solenoid's inductance is L = μ₀N²A/l, proportional to the square of the turns. Filling it with a magnetic material makes L larger still (EM-18). That is why inductors serve in electronics to smooth current and store energy briefly.

Fill inL = μ₀ ?

A solenoid's inductance scales with the square of the turns.

On your ownU = ?

The energy stored in a coil is ½LI².

Back to the first screen

On the first screen, changing the current slowly gave a small back-emf, 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. When the current was steady (zero rate), the back-emf was zero too. That constant of proportionality L is the coil's magnetic inertia — its inductance. L is set not by the current but by the coil's geometry (L = NΦ/I), and with this inertia the coil guards its current against sudden change.

Inductance L is a coil's magnetic inertia, its resistance to change in its own current. The flux linkage made by the current is proportional to the current, and that constant is L (L = NΦ/I). When the current changes the coil raises a back-emf ε = -L dI/dt to oppose the change (Lenz). The faster you change it, the stronger the emf. A solenoid has L = μ₀N²A/l, and the stored energy is ½LI².

What comes next

Fill a coil with a magnetic material (an iron core) and magnetization boosts the field by μ_r, raising the inductance by the same factor — the magnetic counterpart of how a dielectric raised capacitance. The next unit (EM-18, magnetic materials) treats the magnetization M and the relative permeability μ_r. The ½LI² stored in a coil is re-seen as field energy in EM-19.