Unit 8
Induction & Inductance
Unit 7 showed that currents make magnetic fields. This unit runs the story backwards: a changing magnetic field makes a voltage. That single fact, Faraday's law, is how essentially all of the world's electricity is generated, how every transformer and wireless charger works, and why an inductor resists sudden changes of current. You will learn to compute the induced emf, to get its direction right with Lenz's law (the step everyone trips on), and to treat the inductor as a circuit element alongside the resistor and capacitor. The lab lets you push a magnet through a coil and watch the flux and emf graphs together.
1 · Magnetic flux
Flux measures “how much field passes through a surface.” Picture field lines as arrows streaming through a wire loop: more field, a bigger loop, or a loop turned more squarely to face the field all mean more lines through it.
\(\phi\) is measured from the normal, not from the plane of the loop. If a problem says “the field makes \(30^\circ\) with the plane of the coil,” then \(\phi = 60^\circ\) and \(\Phi_B = BA\cos60^\circ\). Draw the normal every time.
2 · Faraday's law
Michael Faraday found (1831) that a steady magnetic field through a loop does nothing, but a changing flux makes a current flow — as if a battery had appeared in the wire. The “battery” is an induced emf \(\mathcal{E}\), measured in volts.
Since \(\Phi_B = BA\cos\phi\), there are exactly three ways to change it — and every induction device uses one:
| Change | emf (magnitude) | Device |
|---|---|---|
| \(B\) changes | \(NA\cos\phi\,\left|dB/dt\right|\) | Transformer, wireless charger, guitar pickup |
| \(A\) changes | \(NB\cos\phi\,\left|dA/dt\right|\) | Sliding rod on rails (Section 4) |
| \(\phi\) changes | \(NBA\left|d(\cos\phi)/dt\right|\) | Rotating-coil generator (Section 5) |
If the loop has resistance \(R\), the induced current is \(I = \mathcal{E}/R\). An open loop (a coil with its ends not connected) still has the emf across its terminals — that is how you measure it with a voltmeter — but no current flows.
A dynamic microphone is a coil glued to a diaphragm, sitting in a magnet's field; sound moves the coil and Faraday's law turns motion into an audio voltage. An electric guitar's pickup is a coil around a magnet: the vibrating steel string changes the flux through the coil. A current transformer clamps around a power line and measures AC current through the emf induced by its changing field — the AC version of the clamp meter in Unit 7.
3 · Lenz's law: which way does the current go?
Faraday's law gives the size of the emf. Its direction comes from Lenz's law:
The induced current flows in the direction that makes its own magnetic field oppose the change in flux that produced it.
Nature is conservative about flux: if the flux through a loop is growing, the loop makes a field that pushes back against the growth; if the flux is shrinking, the loop makes a field that tries to replace what is being lost. It has to be this way — if the induced current helped the change, it would increase the flux further, inducing more current, without limit: free energy. Lenz's law is energy conservation.
- Which way does the external \(\mathbf{B}\) point through the loop?
- Is the flux increasing or decreasing? (Stronger field, bigger area, loop turning to face the field → increasing.)
- Induced \(\mathbf{B}\): opposite to the external \(\mathbf{B}\) if the flux is increasing; the same direction if it is decreasing.
- Current: use the loop right-hand rule — thumb along the induced \(\mathbf{B}\), fingers give the current. State the answer with a viewpoint (“clockwise as seen from the magnet”).
| Situation (loop to the right of the magnet) | External B at loop | Flux | Induced B | Current, seen from magnet |
|---|---|---|---|---|
| N pole approaches | right | increasing | left | counterclockwise |
| N pole withdraws | right | decreasing | right | clockwise |
| S pole approaches | left | increasing (leftward) | right | clockwise |
| S pole withdraws | left | decreasing | left | counterclockwise |
| Magnet held still | right | constant | none | no current |
- “The induced field is always opposite to the external field.” No — it opposes the change. When the flux is decreasing, the induced field points the same way as the external one.
- No viewpoint. “Clockwise” means nothing until you say from which side you are looking. Clockwise from the left is counterclockwise from the right.
- Big flux, big emf. A huge steady field induces nothing. Only the rate of change matters.
- Forgetting the force. Lenz's law also tells you the mechanical force: the induced current always resists the motion that caused it (an approaching magnet is repelled; a withdrawing one is attracted back).
When a relay or solenoid actuator is switched off, its collapsing field induces an emf that tries to keep the current flowing (Lenz again). With nowhere to go, that emf can reach hundreds of volts and destroy the transistor that switched it. The fix on every relay driver schematic is a flyback diode across the coil, giving the current a safe path while it dies away (Section 7 has the numbers).
4 · Motional emf: the rod on rails
A conducting rod of length \(L\) slides at speed \(v\) along two parallel rails, joined at one end by a resistor \(R\). A uniform field \(\mathbf{B}\) points into the page. The circuit's area grows at rate \(Lv\), so the flux grows at rate \(BLv\).
Energy check. To keep the rod moving at constant \(v\), an external agent must push with force \(F = B^2L^2v/R\), delivering mechanical power \(Fv = B^2L^2v^2/R\). The resistor dissipates \(I^2R = (BLv/R)^2R = B^2L^2v^2/R\). Identical — the mechanical work you do becomes heat in the resistor. This is a generator in its simplest form.
Run it backwards and you have a motor: push current through the rod from a battery and the force \(ILB\) accelerates it — the principle of linear motors and railguns. As the rod speeds up, its motional emf opposes the battery (“back-emf”), which is why a DC motor draws a huge current at start-up and much less once spinning.
5 · Generators, eddy currents and induced electric fields
The AC generator
Spin a coil of \(N\) turns and area \(A\) at angular speed \(\omega\) in a uniform field \(B\). The angle between the coil's normal and \(\mathbf{B}\) is \(\phi = \omega t\), so the flux through each turn is \(BA\cos\omega t\).
The emf follows the slope of \(\Phi_B(t)\), not its height. At the top of a cosine the slope is zero. Sketch \(\Phi_B\) and \(\mathcal{E}\) one above the other and you will never mix them up again.
Every power station — coal, gas, nuclear, hydro, wind — ends in this equation. Grid generators spin at a speed locked to 60 Hz (50 Hz in much of the world), and the rotating part is usually an electromagnet (the field winding) turning inside fixed coils. Unit 9 analyses the AC this produces.
Eddy currents
A changing flux through a solid piece of metal induces emfs around loops inside the metal itself, so currents swirl in it — eddy currents. By Lenz's law they oppose the change: a metal plate swinging into a magnet's field is braked; a metal core in an AC coil gets hot.
Useful: induction cooktops (a coil driven at tens of kilohertz heats the pan by eddy currents), eddy-current brakes on trains and roller coasters (no contact, no wear), metal detectors, and induction furnaces. Harmful: eddy currents waste power in transformer and motor cores, so cores are made of thin insulated laminations (or of ferrite, a poor conductor) to break the current loops. In MRI, the rapidly switched gradient coils induce eddy currents in the magnet's metal structure that engineers must compensate for.
Induced electric fields
What actually pushes the charges around a stationary loop when \(B\) changes? There is no motion, so it is not \(q\mathbf{v}\times\mathbf{B}\). The answer: a changing magnetic field creates an electric field, whose field lines circle around like water swirling in a basin.
Example: inside a long solenoid of radius \(R\) whose field changes at rate \(dB/dt\), a circle of radius \(r < R\) gives \(E(2\pi r) = \pi r^2\,dB/dt\), so \(E = \tfrac{r}{2}\,dB/dt\). With \(dB/dt = 0.10\) T/s and \(r = 5.0\) cm, \(E = 2.5\times10^{-3}\) V/m — circling the axis even in empty space.
6 · Self-inductance
A coil carrying current \(I\) makes a field, and that field threads the coil itself. Change \(I\) and the coil's own flux changes, so Faraday's law induces an emf in the coil — opposing the change in current. This is self-induction, and the coil is now a circuit element: an inductor.
The long solenoid. \(N\) turns, length \(\ell\), cross-section \(A\). From Unit 7, \(B = \mu_0 (N/\ell) I\), so the flux through each turn is \(\Phi_B = BA = \mu_0 N I A/\ell\). Then:
How to think about an inductor: it is electrical inertia. A mass resists changes in velocity; an inductor resists changes in current. The voltage across it is \(L\,dI/dt\) — zero for a steady current (a DC inductor is just a wire), huge for a fast change.
- The current through an inductor cannot jump. A sudden jump would need \(dI/dt = \infty\) and an infinite voltage. Right after a switch moves, the inductor current equals what it was just before. (Its voltage can jump.) Compare: a capacitor's voltage cannot jump.
- The sign. In a circuit, write the voltage drop across an inductor in the direction of the current as \(+L\,dI/dt\) — exactly like \(+IR\) for a resistor. The minus in \(\mathcal{E}_L = -L\,dI/dt\) is already accounted for by calling it a drop.
A buck converter (the circuit that turns 12 V into 3.3 V for a processor, at 90%+ efficiency) works by repeatedly switching a voltage across an inductor and relying on it to keep the current flowing smoothly between switching events. A car's ignition coil does the opposite: it interrupts the current abruptly so \(L\,dI/dt\) (stepped up by a secondary winding) reaches tens of kilovolts at the spark plug.
7 · RL circuits and magnetic energy
Current growth
A battery \(V\), resistor \(R\) and inductor \(L\) in series; the switch closes at \(t = 0\) with \(I(0) = 0\). Kirchhoff's loop rule, with the inductor drop \(L\,dI/dt\):
Current decay
Energy stored in an inductor
While the current grows, the battery pushes charge through the inductor against its emf, delivering power \(P = I\cdot L\,dI/dt\). Integrating from 0 to \(I\):
For the long solenoid, \(U = \tfrac12\dfrac{\mu_0N^2A}{\ell}I^2\) and \(B = \mu_0NI/\ell\). Eliminating \(I\) gives \(U = \dfrac{B^2}{2\mu_0}(A\ell)\), and \(A\ell\) is the volume inside. So every cubic metre of field holds:
A modest 1.0 T field stores \(u_B = 1/(2\cdot4\pi\times10^{-7})\approx4.0\times10^5\) J/m³. The strongest electric field air tolerates before sparking is about \(3\times10^6\) V/m, storing \(u_E = \tfrac12(8.85\times10^{-12})(3\times10^6)^2\approx40\) J/m³ — ten thousand times less. That is why practical motors, generators and power converters store and move energy magnetically.
8 · Mutual inductance, transformers and LC oscillations
Mutual inductance
Put two coils near each other. A changing current \(I_1\) in coil 1 makes a changing flux through coil 2, inducing an emf there — no wires between them.
A phone charging pad is a flat transmitter coil driven at roughly 100–200 kHz (the Qi standard); the phone holds a receiver coil. Their mutual inductance couples power across the gap. Because the field of a coil falls off quickly (Unit 7), \(M\) drops sharply with distance and misalignment — which is why the phone has to sit right on the pad. NFC payment cards and RFID tags use the same principle at 13.56 MHz.
The ideal transformer
Wind a primary (\(N_p\) turns) and a secondary (\(N_s\) turns) on one iron core, so the same flux \(\Phi_B\) passes through every turn of both. Faraday gives \(V_p = N_p\,d\Phi_B/dt\) and \(V_s = N_s\,d\Phi_B/dt\). An ideal transformer also wastes no power, so \(V_pI_p = V_sI_s\):
A 120 V primary has 500 turns; the secondary has 50 turns and feeds a \(6.0\ \Omega\) load. Then \(V_s = 120\times\tfrac{50}{500} = 12\) V, \(I_s = 12/6.0 = 2.0\) A, \(P = 24\) W, and \(I_p = 24/120 = 0.20\) A. Seen from the primary, the load looks like \(120/0.20 = 600\ \Omega = (N_p/N_s)^2\times6.0\ \Omega\) — transformers also match impedances.
LC oscillations
Charge a capacitor and connect it across an inductor. The capacitor drives current through the inductor; the inductor, having built up current, will not let it stop, and charges the capacitor the other way; and so on. Energy sloshes between the electric field of \(C\) and the magnetic field of \(L\), like a mass on a spring.
The LC circuit is the electrical tuning fork. It sets the frequency of radio oscillators, selects one station from many in a receiver, and forms the resonant tank in wireless chargers. Real circuits have resistance, so the oscillation decays — the RLC circuit of Unit 9.
9 · Worked examples
1. Ramping a field through a coil
A square coil of \(200\) turns, \(10\) cm on a side, total resistance \(5.0\ \Omega\), lies in the page. A uniform field pointing out of the page rises steadily from 0 to \(0.50\) T in \(0.10\) s. Find the emf, the current and its direction, and the total charge that flows.
Direction (Lenz): external \(\mathbf{B}\) out of the page, increasing → induced \(\mathbf{B}\) into the page → thumb into the page → current clockwise as seen from the front (looking at the page).
The second form shows that the charge depends only on the total flux change, not on how fast it happened — the basis of the “flux meter.”
2. Rod on rails: emf, force and power balance
In the rail figure, \(B = 0.40\) T, \(L = 0.50\) m, \(R = 2.0\ \Omega\), and the rod is pushed at a steady \(3.0\) m/s. Find the emf, current, required force, and check energy conservation.
Current runs up the rod (counterclockwise), and the magnetic force on the rod points against \(\mathbf{v}\), so the agent must push with 0.060 N to keep the speed constant.
3. A 60 Hz generator · EE
A generator coil has \(100\) turns of area \(0.020\ \text{m}^2\) and spins at 60 rev/s in a \(0.25\) T field. Find the peak and RMS emf, and the average power into a \(100\ \Omega\) load (ignore the coil's resistance).
The RMS value (Unit 9) is the DC voltage that would deliver the same average power. Note \(NBA = 0.50\) Wb is the peak flux linkage; multiplying by \(\omega\) converts “how much flux” into “how fast it changes.”
4. Inductance of a solenoid
An air-core solenoid has \(500\) turns, length \(10\) cm and radius \(1.0\) cm. Find \(L\); the emf when the current ramps by \(2.0\) A in \(1.0\) ms; and the energy stored at \(2.0\) A.
About 1 mH. The emf opposes the increase in current (it acts against the source). A ferrite core with \(\mu_r = 100\) would make it about 0.1 H in the same space.
5. An RL circuit switching on · EE
A \(12\) V supply, \(10\ \Omega\) resistor and \(50\) mH inductor are in series; the switch closes at \(t = 0\). Find \(\tau\), the final current, the current at \(t = 5.0\) ms, the time to reach half the final current, and the final stored energy.
Switching off: if the switch simply opens, the current must still be 1.2 A an instant later, through an open circuit — so the voltage across the gap shoots up until it arcs. With a flyback diode across the inductor, the current instead decays through the coil's own resistance and the diode, and the 36 mJ is dissipated harmlessly.
6. LC oscillation
A \(1.0\ \mu\text{F}\) capacitor charged to \(10\) V is connected across a \(10\) mH inductor. Find the frequency, the period, the maximum current, and check energy conservation.
A quarter period (0.157 ms) after connection, the capacitor is empty and all 50 μJ is in the inductor's magnetic field; at half a period (0.314 ms) it is back in the capacitor with the opposite polarity.
Checklist
- I can compute \(\Phi_B = BA\cos\phi\) with \(\phi\) measured from the normal.
- I know \(\mathcal{E} = -N\,d\Phi_B/dt\) and the three ways to change the flux.
- I run the four-step Lenz recipe and state current directions with a viewpoint.
- I can derive \(\mathcal{E} = BLv\) two ways and check the power balance.
- I know the generator emf \(NBA\omega\sin\omega t\), and that max emf occurs at zero flux.
- I can explain eddy currents, laminations, and the non-conservative induced \(\mathbf{E}\).
- I know \(L = \mu_0N^2A/\ell\), that inductor current cannot jump, and \(\tau = L/R\).
- I know \(U = \tfrac12LI^2\), \(u_B = B^2/2\mu_0\), the ideal-transformer ratios, and \(\omega = 1/\sqrt{LC}\).