Induction and Inductance

In the previous chapter we found that a current produces a magnetic field — a fact that surprised the scientists who discovered it. More surprising still was the reverse effect: a magnetic field can produce an electric field that drives a current. The link between a changing magnetic field and the electric field it induces is now called Faraday's law of induction.

What began as basic science in Michael Faraday's hands is now nearly everywhere. Induction is the basis of the electric guitars that drive rock and metal, of the electric generators that power cities and railways, and of the induction furnaces that melt metal in foundries. Before reaching such applications, though, we must look at two simple experiments.

First experiment. A conducting loop is wired to a sensitive ammeter, with no battery in the circuit. Move a bar magnet toward the loop and a current suddenly appears; it vanishes the instant the magnet stops. Pull the magnet away and a current appears again, now in the opposite direction. Three lessons emerge: a current flows only while there is relative motion between loop and magnet; faster motion gives a larger current; and reversing either the motion or the pole reverses the current.

Second experiment. Place two loops near each other but not touching. Close a switch to turn on a current in the first loop and the meter on the second loop twitches briefly; open the switch and it twitches the other way. A current is induced in the second loop only while the current in the first loop is changing — never when it is steady, however large. The induced current is called an induced current, the work per unit charge driving it an induced emf, and the whole process induction. In both experiments, something is changing — and Faraday saw what.

Faraday realized that the "something" is the number of magnetic field lines passing through the loop. To make this quantitative we define the magnetic flux, exactly as we defined electric flux in Chapter 23 — the field integrated over the area of the loop.

Faraday's law then says that the induced emf equals the rate at which the flux changes — not the flux itself. The minus sign (justified in the next section by Lenz's law) records that the induced emf opposes the change.

Faraday's law gives the size of the emf; Lenz's law gives its direction:

Two equivalent readings make it concrete. Opposition to pole movement: as a magnet's north pole approaches a loop, the loop responds by becoming a magnetic dipole that presents its own north pole toward the intruder, repelling it; pull the magnet away and the loop instead presents a south pole, trying to hold it back. Opposition to flux change: the induced field \(\vec{B}_{\text{ind}}\) points opposite an increasing applied flux and along a decreasing one — always fighting the change, never the field itself. Note carefully that \(\vec{B}_{\text{ind}}\) opposes the change in \(\Phi_B\), which does not always mean it points opposite \(\vec{B}\).

Consider a rectangular loop of width \(L\) being pulled at constant velocity \(v\) out of a region of uniform field \(\vec{B}\). As the loop leaves, the area in the field shrinks, the flux \(\Phi_B = BLx\) falls, and an emf is induced. Replacing \(dx/dt\) with \(v\) gives the motional emf.

Place a copper ring in a magnetic field whose strength is rising steadily. A current appears in the ring — so an electric field must be present to push its electrons. That field was produced not by charges but by the changing magnetic flux. Remarkably, the field is there even if the ring is removed: a changing magnetic field induces an electric field in empty space. By symmetry the induced \(\vec{E}\) forms closed concentric circles around the region of changing flux.

For the symmetric case of a uniform field confined to a cylinder of radius \(R\) and changing at rate \(dB/dt\), a circular path of radius \(r\) gives \(E(2\pi r) = (\text{enclosed area})\,dB/dt\), and the field has a tidy two-part profile.

Just as a capacitor stores a desired electric field, an inductor stores a desired magnetic field; the long solenoid is our prototype. Drive a current \(i\) through its \(N\) windings and the flux linkage \(N\Phi_B\) measures how much field the device bottles up per unit current.

A coil's own changing current changes its own flux, so it induces an emf in itself — self-induction. Combining the flux linkage \(N\Phi_B = Li\) with Faraday's law gives the self-induced emf directly.

Put an emf \(\varepsilon\), a resistor \(R\), and an inductor \(L\) in series. The loop rule gives \(L\,di/dt + Ri = \varepsilon\) — the same form as the RC circuit with \(L\to R\) and \(R\to 1/C\). The current does not jump to \(\varepsilon/R\) instantly; the inductor opposes the change, so it rises exponentially.

Multiply the RL loop equation by \(i\) and read the terms by energy conservation: \(\varepsilon i\) is the power the battery supplies, \(i^{2}R\) the thermal loss, and the remainder \(Li\,di/dt\) the rate at which energy is stored in the magnetic field. Integrating gives the stored energy.

Return to two nearby coils. A current \(i_1\) in coil 1 links flux through coil 2; change \(i_1\) and an emf appears in coil 2. The coupling is measured by the mutual inductance \(M = N_2\Phi_{21}/i_1\). A non-obvious but exact result is that the coupling is symmetric: \(M_{21} = M_{12} = M\).

Take \(\mu_0 = 4\pi\times10^{-7}\,\mathrm{T\cdot m/A}\). For each problem, first identify which of the three flux knobs is turning — \(B\), \(A\), or \(\theta\) — then apply Faraday's law for the magnitude and Lenz's law for the direction. In RL circuits, lean on the two limits: an inductor is an open wire just after switching and a plain wire long after.