Self-Inductance

Consider an isolated circuit (coil) carrying a current that is CHANGING with time. The changing current itself continuously alters the magnetic flux that this same current sets up and links through its own circuit -- and by Faraday's law, this self-produced changing flux must itself induce an emf back in the very same circuit. This phenomenon -- a circuit inducing an emf in itself purely because ITS OWN current is changing -- is called self-inductance.

If $\Phi$ is the flux linked with the circuit (due to its own current i) at some instant, then since $\vec{B}$ (and hence $\Phi$) is always directly proportional to the current producing it, $\Phi \propto i$, i.e. $\Phi = Li$, where the constant of proportionality L is called the self-inductance (or coefficient of self-induction) of the circuit; L depends only on the geometry of the circuit and any magnetic material present, never on the current itself. For a closely-wound coil of N turns (where the same flux links every turn), the total flux LINKAGE is $N\Phi_B = Li$.

Differentiating $\Phi=Li$ with respect to time (with L constant) and using Faraday's law gives the induced (self-induced) emf directly in terms of L: $e = -\frac{d\Phi}{dt} = -L\frac{di}{dt}$. This is often taken as the DEFINING relation for L: the self-inductance of a circuit is the induced emf produced per unit rate of change of current in it, so that $L = -\frac{e}{di/dt}$ (or, from the flux definition, $L=\Phi/i$ -- the flux linked per unit current). Comparing units: since $|e|$ is in volts and $\left|\frac{di}{dt}\right|$ is in amperes per second, L is measured in volt-seconds-per-ampere, a unit given the name henry (H); 1 henry = 1 ohm-second -- corresponding to an induced emf of 1 V for a rate of change of current of 1 A/s.

For a long solenoid of N turns, length l, cross-sectional area A and n = N/l turns per unit length, the interior field is $B=\mu_0 ni$, so the flux linkage over the interior (of volume Al) is $N\Phi_B = (nl)(BA) = \mu_0 n^2 i A l$, giving inductance $L = \mu_0 n^2 A l$, i.e. inductance PER UNIT LENGTH near the middle of a long solenoid is $\frac{L}{l}=\mu_0 n^2 A = \mu_0 n^2\frac{\pi d^2}{4}$ (d the solenoid's diameter). Since L must have the dimensions of $\mu_0\times[\text{length}]$ (as n is a number per unit length), this also shows that $\mu_0$ itself can be expressed in henry/metre (H/m).

For inductors combined in a circuit, series and parallel combination rules mirror those for resistors: $L_{Total}=L_1+L_2+L_3+\ldots$ in series, and $\frac{1}{L_{Total}}=\frac{1}{L_1}+\frac{1}{L_2}+\frac{1}{L_3}+\ldots$ in parallel -- so a parallel combination's inductance is always LESS than that of the smallest individual inductor, exactly analogous to resistors in parallel. …