Inductance and Magnetic Energy

A current \(I\) in a circuit sets up a magnetic flux \(\Phi\) that threads the circuit itself. If the loop is wound \(N\) times, each turn catches that flux, and the total flux linkage is \(\Lambda = N\Phi\). The ratio of linkage to the current that produced it is the self-inductance:

Because \(B\) is proportional to \(I\), the current cancels: \(L\) depends only on the geometry of the circuit and the permeability of the medium — never on how much current flows (for linear materials). It is the magnetic counterpart of capacitance from Chapter 8, where \(C=Q/V\) was likewise pure geometry. The unit is the henry (Wb/A = V·s/A).

Now place a second circuit nearby. Some of the flux produced by current \(I_1\) in circuit 1 links the \(N_2\) turns of circuit 2. The mutual inductance measures that shared linkage:

The equality \(M_{12}=M_{21}\) is not obvious but always holds; it falls straight out of the symmetric Neumann formula \(M = \dfrac{\mu}{4\pi}\oint\!\oint \dfrac{d\vec{l}_1\cdot d\vec{l}_2}{R}\). How tightly the two coils share their flux is captured by the dimensionless coupling coefficient \(k\):

With \(k=1\) every flux line links both coils (a perfectly coupled, tightly wound transformer); with \(k\to 0\) the coils ignore each other. Mutual inductance is the entire basis of the transformer and of every coupled-resonator and wireless-power link.

The recipe is always the same: assume a current \(I\), find \(\vec{B}\) (Chapters 11–12), integrate for the flux \(\Phi\), multiply by \(N\), and divide by \(I\). Applied to the standard geometries it gives a short table worth memorising:

Notice the family resemblance to the capacitance results of Chapter 8: the coax has \(C' = 2\pi\varepsilon/\ln(b/a)\) and \(L' = (\mu/2\pi)\ln(b/a)\), so their product \(L'C' = \mu\varepsilon\) is independent of the conductor sizes — a fact that returns with a vengeance when we reach transmission lines in Part 6.

The flux that produces inductance lives in two places: outside the conductor (external inductance) and inside it, where the current itself threads its own field (internal inductance). For a long solid wire carrying a uniform current, integrating the field energy across the cross-section gives a result that is famously independent of the wire's radius:

The total is \(L = L_{\text{int}} + L_{\text{ext}}\). At low frequency both matter; but as frequency rises the skin effect crowds the current onto the conductor surface, hollowing out the interior. With no current deep inside there is no internal flux, so \(L_{\text{int}}\to 0\) and the inductance approaches its external value alone — one reason high-frequency line inductance is slightly lower than the DC figure.

Building a current from zero to \(I\) takes work, because the rising field induces a back-emf that opposes the change. Integrating that power against the linear \(\Lambda = LI\) relation gives the energy stored — the area of the shaded triangle below:

But the energy does not really sit "in the coil" — it sits in the field, spread through all the space the field occupies. Rewriting the same energy as a volume integral gives the magnetic energy density, the exact twin of the electric \(\tfrac12\varepsilon E^2\) from Chapter 8:

When two coils share flux, the total stored energy is no longer a simple sum — the cross term carries the sign of the coupling:

Connect the same two coils in series and the equivalent inductance depends on whether their fields add or oppose — set by the winding sense (the dot convention):

For each item, decide first whether it is a flux-linkage, a standard-formula, an energy, or a coupling problem, then apply the matching relation. Difficulty rises down the list.