13.4 Inductance

A current in a coil makes a magnetic field, and that field passes through the coil's own turns. If the current changes, the flux through the coil changes, and Faraday's law says the coil induces an emf in itself. That's self-induction.

The flux through a coil is proportional to its current, so the induced emf is proportional to how fast the current changes: $\mathcal{E} = -L\dfrac{dI}{dt}$. The constant L is the coil's inductance, measured in henries: 1 H = 1 V·s/A.

Inductance is defined by $L = \dfrac{N\Phi_B}{I}$, the total flux through all N turns per unit current. Like capacitance, it depends only on the shape, size and core of the device, not on the current.

Every conductor has some inductance, but a straight wire's is so small that it's treated as zero. An inductor is a part built to have a large inductance, usually a coil such as a solenoid.

By Lenz's law, the self-induced emf, often called a back emf, opposes the change in current. If the current is rising, the inductor pushes against it. If the current is falling, the inductor pushes to keep it going.

So an inductor acts like inertia for current. Mass resists changes in velocity; inductance resists changes in current. A steady current, even a large one, makes no emf across an ideal inductor. Because of this, the current through an inductor can't change instantly.

The potential difference across an ideal inductor is $\lvert\Delta V_L\rvert = L\left\lvert\dfrac{dI}{dt}\right\rvert$. It depends on the slope of the current-versus-time graph, not on the current's value.

For a long solenoid with N turns, length $\ell$ and cross-sectional area A, the field inside is $B = \mu_0\dfrac{N}{\ell}I$ (12.4). The flux through each turn is BA, so $L = \dfrac{N(BA)}{I} = \dfrac{\mu_0N^2A}{\ell}$.

More turns, a fatter coil or a shorter coil (with the same N) all raise L. N appears squared because more turns make both a stronger field and more turns for that field to link. Filling the core with a ferromagnetic material replaces $\mu_0$ with a much larger permeability $\mu$, raising L a lot.

To build up current in an inductor, the source must push against the back emf, at a rate $P = LI\dfrac{dI}{dt}$. Integrating from 0 to I gives the stored energy: $U_L = \tfrac{1}{2}LI^2$.

That energy lives in the magnetic field the current makes. When the current falls, the inductor gives the energy back: it can be turned into thermal energy in a resistor (13.5) or used to charge a capacitor (13.6), and the total energy is conserved either way. Compare $U_L = \tfrac{1}{2}LI^2$ with a capacitor's $U_C = \tfrac{1}{2}C(\Delta V)^2$ and kinetic energy $\tfrac{1}{2}mv^2$.