Electromagnetic Induction

On August 29, 1831, Michael Faraday wrote in his lab notebook: "Have had an iron ring made... Will call this the induction ring." He wound a primary coil and a secondary coil on opposite sides of an iron ring, hooked the primary to a battery through a switch, and watched the galvanometer on the secondary.

When he closed the switch, he saw a momentary deflection. When he opened it, another deflection — in the opposite direction. Between flipping the switch, while current in the primary was steady, nothing happened.

Faraday had discovered two things simultaneously: (1) you can induce currents without any physical motion, and (2) induction is entirely about change. Steady state produces nothing; transitions produce everything.

This was the day electricity became practical. Within 40 years, Edison and Tesla would build power grids. Within 60 years, Marconi would send radio signals across the Atlantic. Within 100 years, radar, television, and computers would exist. All from one notebook entry about an iron ring.

Since Φ = BA cos φ, there are exactly three ways to make flux change — and therefore three ways to induce an EMF:

• Change B: turn an electromagnet on/off, move a permanent magnet closer or farther, or vary the current in a nearby coil. This is how RFID tags and wireless chargers work.
• Change A: stretch, shrink, or slide a coil. This is how a slide-wire generator works (Slide 8).
• Change φ: rotate a coil in a fixed B field. This is how every AC generator on Earth works (Slide 9).

Every induction problem reduces to identifying WHICH variable is changing. Once you know, you can compute dΦ/dt by taking the derivative of the changing variable and keeping the others constant. B changes → dΦ/dt = (dB/dt)·A·cos φ. A changes → dΦ/dt = B·(dA/dt)·cos φ. φ changes → dΦ/dt = B·A·(d(cos φ)/dt) = −BA·sin φ·(dφ/dt).

If your coil has N turns, each turn sees the same flux change dΦ/dt. Each turn produces its own tiny EMF equal to dΦ/dt. These N tiny EMFs are connected in series around the coil, so they add up. Total: ε = N × (dΦ/dt).

This is why real generators and transformers have coils with thousands of turns. A single loop in a field that's changing at 1 T/s through an area of 1 m² produces only 1 V. But 1000 turns gives 1000 V — enough to be useful. Coil windings are essentially free voltage multipliers.

The minus sign is Lenz's Law (next slide). It means the induced EMF opposes the change in flux that created it. If you try to increase the flux, the induced current fights back to reduce it. If you try to decrease it, the current fights back to maintain it. Nature resists change.

The "linear change" form |ε| = N|ΔΦ/Δt| is just a finite-difference version of the derivative, valid when the change is approximately constant over the time interval. For curved or sinusoidal changes, you need the true derivative dΦ/dt.

When you write ΔΦ = Δ(BA cos φ), be careful: the Δ (change) operator only applies to the variables that actually vary. If B and A are fixed, ΔΦ = BA·Δ(cos φ) = BA·(cos φ_final − cos φ_initial). The B and A come OUT of the Δ because they don't change.

Common mistake: students compute Δφ = 40° and try to use cos(40°) directly. But that's wrong — you must evaluate cos at each endpoint separately. cos(30°) − cos(70°) ≠ cos(40°).

Another common mistake: forgetting that "angle from the plane" is the complement of "angle from the normal." If a problem says "B is 20° from the plane of the loop," then φ (from the normal) = 70°, and cos φ = cos 70° = 0.342. Get this wrong and your answer is off by a factor.

Here's the deepest insight in all of induction: Lenz's Law is a direct consequence of energy conservation. If the induced current flowed the other way, it would attract the incoming magnet instead of repelling it. The magnet would accelerate toward the coil, the induced current would grow stronger, the attraction would grow stronger, and you'd get free energy from nothing. Impossible.

So the universe enforces Lenz's Law by making induced currents always oppose the change. If you want to push a magnet into a coil, you have to do work against the induced repulsive force. That work is where the electrical energy comes from — it's mechanical energy converted by the magnet's motion.

This is how all generators work: some external agent (a turbine, water wheel, wind turbine) does mechanical work to move magnets through coils. The induced currents create opposing forces (Lenz's Law), and the external agent must push against them. The mechanical work done = the electrical energy produced. No free lunch.

Counterexample thought experiment: "What if Lenz's Law were backwards?" Then you could build a perpetual motion machine: drop a magnet through a coil, the current attracts it, the magnet accelerates, more current, more attraction, perpetual acceleration, infinite energy. The fact that such machines have never been built (and can't be) is why Lenz's sign is what it is.

Here's a Lenz problem that fools most students on their first try: a copper ring is dropped OVER a strong vertical bar magnet, sliding down it. Which way does the induced current flow, and what does it do to the ring's motion?

Answer: as the ring slides down past the N pole toward the S pole, flux through the ring changes direction. Initially (above the magnet) it points up (N to S externally), then zero at the middle, then down (N to S going down). So there's always a change, and there's always an induced current fighting it.

Physically: the induced current creates a force that opposes the motion. So the falling ring experiences an upward force from the magnetic field — it falls slower than gravity alone would predict. If you drop a copper ring over a neodymium magnet, you can actually see it slow down and float briefly before resuming its fall. This is called eddy current braking.

Industrial application: roller coasters use this principle for "magnetic brakes" at the end of the ride. Powerful fixed magnets on the track combined with copper fins on the cars create induced currents that generate strong opposing forces — slowing the car smoothly with no physical contact. No friction, no wear, no noise, and no moving parts.

The rod in this setup is literally a battery. With ε = BLv volts across its ends, it drives current through the external resistor just like a chemical battery would. The difference: instead of chemistry, it's mechanical motion through a magnetic field that provides the energy.

Where does the energy come from? From whoever or whatever is pushing the rod. By Lenz's Law, the induced current creates a drag force F = BIL that opposes the motion. To keep the rod moving at constant v, you must push with an equal force. The power you supply, P = F·v = (BIL)·v = (BLv)·I = εI, exactly equals the electrical power dissipated in the resistor. Mechanical work in = electrical energy out.

This is the fundamental principle of EVERY generator. From a bicycle dynamo to a hydroelectric turbine to a wind turbine to a 3,000 MW nuclear plant generator, the underlying physics is identical: conductors move through magnetic fields, charges inside feel qv × B, and that creates a voltage that drives current through an external circuit.

Slide-wire generator is the simplest form, but the physics is universal. When you flip on a light switch, a rotating coil in a generator 100 miles away converted mechanical torque into BLv-style EMF, which ran down transmission lines, stepped through transformers (Slide 13), and finally ended up in your lightbulb.

In the US, wall outlets deliver AC at exactly 60 Hz — meaning the voltage oscillates 60 full cycles per second, or 120 zero-crossings per second. That's because all US power plants spin their generators at a constant rate locked to 60 Hz. If a generator drifts off frequency, it falls out of sync with the grid and must be disconnected before it tears itself apart.

Europe uses 50 Hz instead, which is why power adapters often need to convert frequencies (not just voltages) when you travel internationally. The difference is historical: German engineers standardized on 50 Hz in the 1890s, while American engineers picked 60 Hz a few years later. Both systems work; once chosen, switching would require replacing every generator and motor in the country.

Peak vs. RMS: the "120 V" number for US outlets is actually the RMS (root mean square) voltage — a DC-equivalent measure. The true peak is 120·√2 ≈ 170 V. Similarly, European "230 V" has a peak of 325 V. When physicists say ε_max = NBAω, they mean the peak, not the RMS.

Example 21.4 — a slide-wire generator (from the book) shows that these same formulas apply to a rotating rod or any rotating conductor. The sinusoidal output is universal because the geometry is — cos and sin are what you get when you project a rotating vector onto a fixed axis.

A transformer core is made of iron (to concentrate B field), but a solid iron block would have huge eddy currents swirling inside it as the AC flux changes. These eddies would dissipate energy as heat — a massive loss for a power transformer.

The solution: build the core out of many thin, insulated iron sheets stacked together. Each sheet is only a fraction of a millimeter thick, so the eddy-current loops that can form inside each sheet are tiny. Because P = I²R and the eddy current in each small loop is small, the total loss is vastly reduced. You'll see this "laminated core" construction in every transformer.

Induction cooktop trick: an induction cooktop has a coil beneath the glass surface carrying ~25 kHz AC. This creates a rapidly oscillating B field. When you place a ferromagnetic pan on top, eddy currents are induced in the pan — and because the pan is designed to have high resistance (e.g. stainless steel alloys), the I²R losses dump substantial heat directly into the pan metal, not into the glass. Result: the glass stays cool while the pan gets red-hot. Energy efficiency is ~90% (vs. ~40% for gas stoves).

Eddy current braking on roller coasters: powerful fixed magnets along the track combined with copper fins on the train cars. As the car approaches the magnets, B changes through the copper, inducing eddies that create opposing forces. The car slows smoothly without any physical contact. No brake pads to wear out, no friction, no noise — just F = I²R·... dissipation of the car's kinetic energy into heat in the copper.

Nikola Tesla understood mutual inductance better than anyone of his era. His famous "Tesla coil" consisted of a primary coil with high current and low voltage, tightly coupled (high M) to a secondary coil with many more turns. A rapidly interrupted current in the primary induced huge voltages (tens of thousands of volts) in the secondary — enough to produce dramatic lightning-like sparks through the air.

The modern version is wireless charging. Your phone sits on a charging pad; the pad has a primary coil with AC current, and your phone has a secondary coil just under the back glass. Mutual inductance transfers energy across the air gap — no wires, no physical contact. The mutual inductance depends on how close and aligned the coils are, which is why phone chargers are fussy about placement.

M is symmetric: the mutual inductance from coil 1 to coil 2 equals the mutual inductance from coil 2 to coil 1. This is a deep fact — it comes from the reciprocity of electromagnetism. You can drive either coil, and the coupling to the other is the same constant M.

Two factors determine M: (1) how much flux from coil 1 actually threads coil 2 (depends on geometry and proximity), and (2) the number of turns in each coil. Tightly wound coaxial coils have very high M. Two coils far apart or at 90° angles have M ≈ 0.

Here's the deepest analogy in all of circuit analysis: an inductor is the electrical equivalent of mass. Just as a massive object resists changes in velocity (F = ma → to change v quickly you need a large F), an inductor resists changes in current (ε = L dI/dt → to change I quickly you need a large ε). This is why inductors are sometimes called "current inertia" elements.

Energy stored in an inductor: U = ½LI². This looks identical to kinetic energy ½mv², and the analogy is exact: the energy stored in an inductor's magnetic field is like the kinetic energy of a moving mass. To "stop" the current (like braking a car), you have to somewhere absorb that ½LI² of energy.

Why you get a spark when you open an inductive switch: if you interrupt current through an inductor suddenly (say, flipping a switch), dI/dt becomes enormous (close to infinite). By ε = L dI/dt, a huge voltage develops across the inductor — often thousands of volts — which can arc across the switch contacts. That's the spark you see when unplugging a motor or a fluorescent light. Industrial relays need "snubber" circuits to absorb this energy and protect the contacts.

Example 21.10 — toroidal solenoid: for a solenoid with N = 200, A = π(0.04)² = 5.03×10⁻³ m², ℓ = 0.25 m, then L = μ₀N²A/ℓ = (4π×10⁻⁷)(40000)(5.03×10⁻³)/0.25 ≈ 1.01 × 10⁻³ H = 1 mH. Very typical inductor value.

In the book example, a coffee maker runs on 120 V AC but needs 12 V for a low-voltage heating element. Use a step-down transformer with N₁/N₂ = 10. If the heating element draws 5 A at 12 V, the primary side sees 5/10 = 0.5 A at 120 V. Same power: 12 × 5 = 60 W = 120 × 0.5.

Why the current goes UP when voltage goes DOWN: energy conservation. P = VI must be the same on both sides (for an ideal transformer). If V drops by a factor of 10, I must rise by a factor of 10 to keep the product the same. This is why the secondary wire of a step-down transformer is often thicker — it must carry more current without overheating.

Transformers ONLY work with AC. If you plug a transformer into DC (say, a battery), the flux through the core is constant, dΦ/dt = 0, and the secondary produces no voltage at all. This is one of the big reasons AC beat DC in the "War of the Currents" of the 1890s — Tesla's AC could be transformed to high voltages for efficient long-distance transmission, while Edison's DC could not.

Real transformers have losses: eddy currents in the core (minimized by lamination), I²R losses in the windings, hysteresis losses in the iron. A well-built modern transformer can be 98–99% efficient. Power grid transformers are among the most efficient engineered devices in all of human technology.

Here's a subtle but important point: the energy in an inductor is NOT stored in the moving charges — it's stored in the magnetic field itself. This is why the formula u = B²/(2μ₀) exists: it tells you the energy per unit volume anywhere there's a magnetic field, regardless of what's producing it.

Fields are real, physical things. They carry energy, they carry momentum, and they can exchange these with matter. A magnetic field in empty space (like the interior of a solenoid) contains real energy you could, in principle, extract.

Compare to electric fields: the energy density of an electric field is u_E = ½ε₀E². So we have:

• Electric field energy density: u_E = ½ε₀E²
• Magnetic field energy density: u_B = B²/(2μ₀)

Both scale as (field squared), which is exactly how kinetic energy scales with velocity (½mv²). In all three cases, the factor of ½ comes from integrating a linear quantity (F = kx, F = ma, F = qE) that grows from zero to its final value.

Electromagnetic waves carry energy in both E and B fields. When sunlight hits your skin, the energy arrives in photons — each of which has both an E field and a B field oscillating together. The total energy density of the wave includes contributions from both. This is the deep reason E and B are truly the same field, viewed from different frames.

RC circuits have τ = RC (chapter 19). RL circuits have τ = L/R. Notice they're inverses in how R enters: adding more resistance speeds up an RL circuit but slows down an RC circuit.

Why: in an RL circuit, R limits the steady-state current (Ohm's Law). More R means less steady I, which means the inductor's stored energy (½LI²) is smaller, which means less energy to shuffle around, which means faster response. Counterintuitive but true.

Why inductors fight sudden changes: at t = 0 (just after closing the switch), the current through an inductor must be zero (it can't jump instantly because dI/dt would be infinite, requiring infinite voltage). So I starts at 0 and grows exponentially to its steady-state value ε/R.

Real-world application — ignition coil: a car's spark plug needs ~30 kV to jump the gap and ignite fuel. An inductor in the ignition circuit stores energy at low voltage (12 V battery). When the switch opens suddenly, dI/dt is huge, so L·dI/dt produces a massive voltage spike — enough to create a spark. This is exactly the "inductive kick" problem discussed on Slide 12, but harnessed for useful purposes.

An L-C circuit is the electrical equivalent of a frictionless pendulum. Energy sloshes back and forth between two forms:

• Pendulum: kinetic energy (½mv²) ↔ gravitational PE (mgh)
• L-C circuit: inductor energy (½LI²) ↔ capacitor energy (½CV²)

At any instant, the total is conserved (for an ideal L-C with no resistance). The oscillation frequency is ω = 1/√(LC), the electrical analog of ω = √(g/ℓ) for a pendulum.

Radio tuning: every AM/FM radio has an L-C circuit. The capacitance (or inductance) is tunable with a knob. When you turn the dial, you change L or C, which shifts the resonant frequency. When the resonant frequency matches the broadcast frequency of a radio station, the circuit amplifies that specific signal while ignoring all others. This is the basis of every radio receiver, TV tuner, and wireless device's channel selector.

Real circuits have resistance, which damps the oscillations. Over time, the energy dissipates as I²R heat and the amplitude dies out — exactly like a pendulum with air friction. To keep the oscillation alive, you need an active "gain" element (transistor, vacuum tube) to replace the lost energy — this is the basis of every oscillator circuit in every electronic device on Earth.

The LC discovery was key to understanding light: when Maxwell worked out the equations of electromagnetism and derived the speed of propagation, he got c = 1/√(μ₀ε₀) — structurally identical to ω = 1/√(LC). The similarity was no coincidence. Light is electromagnetic waves oscillating through "the inductance and capacitance of empty space."

Chapter 21 is the keystone that ties together chapters 17–22. Here's how:

• Ch 17 (electric fields): Faraday's Law shows that a changing B creates a circulating E field. This is the missing piece that turned electrostatics into electrodynamics.
• Ch 18 (capacitors): the RC time constant τ = RC from ch. 19 parallels the RL time constant τ = L/R here. Both are first-order exponential responses.
• Ch 19 (circuits): Faraday gives you an EMF source; Kirchhoff tells you how to solve the circuit it drives. Transformers, motors, and generators all combine both.
• Ch 20 (magnetism): you produce B fields with currents (ch. 20), but you also produce currents with changing B fields (ch. 21). The loop is closed.
• Ch 22 (AC circuits): the L-C circuit at the end of ch. 21 is the gateway to all of ch. 22's AC analysis. Resonance, impedance, and power factor all rest on what you learn here.
• Ch 23 (EM waves): combining Faraday's Law with Ampère's Law (with Maxwell's displacement current) gives the wave equation. Light, radio, X-rays, microwaves — all children of ch. 21.

Faraday's 1831 notebook entry was the moment electrodynamics was born. Everything that follows is elaboration.

1. Measuring angle from the plane instead of the normal. The angle φ in Φ = BA cos φ is always from the normal. "B is 30° above the plane of the loop" means φ = 60° from the normal.
2. Forgetting the minus sign (Lenz's Law). When the problem asks for direction, you need the sign. For magnitudes, just take the absolute value.
3. Computing Δ incorrectly. ΔΦ = Φ_final − Φ_initial. If flux flips from +0.5 Wb to −0.5 Wb, ΔΦ is NOT zero — it's −1.0 Wb. The field had to pass through zero and reverse.

If you keep these three pitfalls in mind, you can solve any induction problem in the book.