Faraday's Law, Lenz's Law, Induced EMF & Transformers

Magnetic flux Φ is not a thing you can touch — it's a count of how many magnetic field lines pass through a surface. Imagine field lines as invisible arrows; flux measures how many thread through your loop. More arrows = more flux. The unit is the Weber, where 1 Wb = 1 T·m².

Why the cosine? If the loop faces the field head-on (θ = 0°, normal parallel to B), every field line passes through it → maximum flux, BA. Tilt the loop 90° so it's edge-on to the field (θ = 90°) → the field lines slide past without crossing the loop → zero flux. The cosine smoothly interpolates between these extremes: cos 0° = 1 (max), cos 90° = 0 (none), cos 60° = 0.5 (half).

Always measure θ from the NORMAL, not from the loop's face. This is the #1 source of mistakes. The "normal vector" sticks straight out perpendicular to the loop's surface like a little flag on the coil's face. The angle you want is between this flag and the direction of B.

Only three knobs exist: Φ = B·A·cos θ has exactly three variables, so only three things can possibly induce an EMF — changing B, A, or θ. Every induction problem in this course is one of these three. If you can identify which knob is turning, you've already done half the work.

Faraday's Law is the most important equation in this unit, and the thing students miss most often is this: a large flux produces zero EMF if it isn't changing. A powerful magnet sitting motionless next to a coil induces nothing. The EMF depends only on how fast the flux changes — ΔΦ/Δt — not on how much flux there is.

The anatomy of ε = −N·ΔΦ/Δt:

• N = number of turns in the coil. Double the turns → double the EMF. This is why coils have so many windings.
• ΔΦ = change in flux per turn = Φ_final − Φ_initial (in Webers, Wb)
• Δt = time over which that change happens (in seconds)
• − = Lenz's Law sign; nature opposes the change

Why N multiplies: each turn of the coil is its own little loop, and each loop sees the same ΔΦ/Δt. The EMFs add in series like batteries stacked end-to-end, so total EMF = N × (EMF per turn). A 1000-turn coil generates 1000× more voltage than a single turn in the same field change.

Units check: [Wb/s] = [T·m²/s] = [V]. Weber per second is volts. Faraday's constant (−1) just carries the direction.

Students get Lenz wrong constantly because they say "the induced field opposes the external field." That's not quite right. The induced field opposes the change in flux — not the field itself.

Example to see the difference: suppose B points into the page and is growing stronger. Induced B points out of the page (opposing the increase). Now suppose B points into the page and is getting weaker. Induced B points into the page (supporting the dying field). In both cases the external B is "into the page," but the induced B points opposite directions — because in one case Φ is rising, in the other it's falling.

Memory aid — the bar-magnet push test: push the north pole of a magnet toward a loop. The loop "sees" an approaching N pole and doesn't want it, so it induces a current that makes the loop itself act like a north pole facing the magnet (so N–N repels). Pull the magnet away? The loop makes itself a south pole to attract it back. The loop is always fighting the change.

Right-hand rule for the induced current: once you know which way the induced B points, curl the fingers of your right hand in that direction inside the loop. Your thumb gives the current direction. Practice this until it's automatic — Lenz problems become trivial.

The key insight of Lenz's Law: the induced current always fights whatever you're doing. Push a magnet in? Current flows to push back. Pull it out? Current flows to pull it back. Shrink the loop? Current flows to support the dying field. The universe, essentially, does not like change.

Why it must be this way — energy conservation. Imagine Lenz's Law were backwards and induced currents helped the change. Push a magnet in → current creates a field that pulls it in harder → magnet accelerates → stronger change → more current → more pull — a runaway loop producing infinite energy from nothing. That's impossible, so Lenz must oppose. The minus sign in Faraday's law isn't a convention; it's thermodynamics.

The two patterns to memorize (everything else is just variations):

• Φ increasing some direction → B_induced points the opposite way → fingers of right hand curl that way → thumb gives current direction
• Φ decreasing some direction → B_induced points the same way (to support) → same RHR trick

Test yourself: a loop sits in a B field pointing out of the page. The field is being turned off. Which way does induced current flow? (Answer: CCW — it tries to keep the outward flux alive.)

Everything we've seen so far involved motion — a magnet moving, a rod sliding, a coil shrinking. But these examples show induction happening with nothing moving at all. Just a current changing in one wire produces a voltage in a completely separate circuit nearby. This is mutual induction, and it's how wireless chargers and RFID tags work.

The chain: changing I in wire 1 → changing B field in space → changing flux Φ through loop 2 → induced EMF in loop 2 (via Faraday) → induced current in loop 2 (which by Lenz opposes the flux change).

Which way does induced current flow? Use Lenz's 4-step walk-through:

• Closing the switch → outer current ramps up from 0 → Φ through inner loop grows
• Outer current CCW → B (by RHR curl) points out of page inside the inner loop → Φ_ext is increasing out of page
• Inner loop opposes increase → its induced B must point into page
• RHR: fingers curling into page means thumb points to a CW current → inner I is clockwise

The induction is transient. Once the switch has been closed for a while and the outer current is steady, dI/dt = 0, so the induced current in the inner loop dies. You only get induction while something is changing.

Every Faraday problem — no matter how it's worded — boils down to three questions. Answer them in order and you can solve any of these:

1. What's changing? B, A, or θ? Only one (usually) is changing; the rest are constants.
2. What's ΔΦ? Compute Φ_final − Φ_initial. Keep B, A, cos θ that don't change outside the subtraction.
3. Plug into ε = −N·ΔΦ/Δt. The N is the number of loops, not turns per meter.

This problem: the B field changed (not A, not θ), so ΔΦ = ΔB · A · cos θ. With A = π(0.135)² and ΔB = 11.32 T, we get 0.648 Wb per turn. One loop only (N=1), so ε = 0.648/0.023 ≈ 28 V. Then Ohm's Law gives I = ε/R.

Once you have ε, the circuit just obeys Ohm: the induced EMF behaves exactly like a battery. Current is ε/R, power is εI = ε²/R, and Lenz's Law only tells you direction — not magnitude.

Trap #1 — Flux reversal. If a loop starts with flux +0.50 Wb pointing up and ends with flux −0.50 Wb (pointing down), the change is not zero — it's ΔΦ = −0.50 − (+0.50) = −1.00 Wb. The field had to pass through zero and go the other way. Students commonly write ΔΦ = 0 here; that's a 50-point mistake.

Trap #2 — Stretching changes area. If you pull a coil so its width increases from w₁ to w₂ while height stays fixed, ΔA = h·(w₂ − w₁). The B field never changed; only A did. Same Faraday equation, different variable inside the Δ.

Trap #3 — "30° to the plane" ≠ θ for the formula. The angle in Φ = BA·cos θ is always measured from the normal to the loop surface, not from the loop's face. "30° above the plane" means θ = 90° − 30° = 60° from the normal. Draw the normal first, always.

Why the sign of ε matters: the minus in Faraday's law is Lenz's Law in disguise. If you get a positive ε, the induced current flows one way; negative ε means it flows the opposite way. For magnitudes (which problems usually ask for), just take the absolute value — but the sign carries the physics of "opposition."

Flux is Φ = BA·cos θ, which means there are three ways to change it: change B, change A, or change θ. Any one of them produces an EMF. Which one a problem uses tells you which variable goes inside the Δ — and which ones come outside as constants.

This problem: B is constant (3 T), A is constant (0.03 m²), only the angle rotates from 70° → 30°. So ΔΦ = B·A·(cos 30° − cos 70°). The B and A come out of the Δ because they don't change. Only the cos term gets subtracted.

Common mistake: students compute Δθ = 40° and try to put it inside the cosine as "cos(40°)". That's wrong — you must evaluate cos at the two angles separately and subtract: cos(θ_f) − cos(θ_i).

Why the sign is negative: going from 70° to 30° means the normal is rotating to align more with B, so flux is increasing (cos 30° > cos 70°). By Lenz's Law, the induced EMF opposes this growth — hence the minus sign. The magnitude is what matters for the numeric answer; the sign just tells you which direction current flows.

Motional EMF is a real battery made of nothing but a moving wire in a magnetic field. Here's what's happening at the particle level: the rod contains free electrons. When the rod moves with velocity v through a field B, each electron feels the Lorentz force F = qv×B. This force pushes electrons along the length of the rod, piling them up at one end.

The pile-up creates an electric field inside the rod, which eventually balances the magnetic force. At equilibrium, one end of the rod is at higher potential than the other — by exactly ε = BLv volts. That's a voltage source, identical from the outside to a battery.

Where does the energy come from? You do! To keep the rod moving at constant v against the magnetic drag force F = BIL, you must push with an equal external force. The power you supply, P = F·v = BIL·v = (BLv)·I = εI, equals the electrical power dissipated in the resistor. Mechanical work in = electrical energy out. This is the underlying principle of every generator — from a bicycle dynamo to a 500-ton turbine at a hydro dam.

Two paths, one answer: the Faraday derivation (flux sweeping out area) and the Lorentz derivation (force on charges) give the exact same ε = BLv. Physics is self-consistent like that — two totally different approaches land on identical predictions.

A generator is just a rotating coil in a magnetic field. As the coil spins, the angle θ between the loop's normal vector and B changes continuously: θ = ωt. So the flux through the loop is Φ = BA·cos(ωt) — a cosine that oscillates between +BA and −BA.

Where the sine comes from: Faraday says EMF = −N·dΦ/dt. Take the derivative of cos(ωt) and you get −ω·sin(ωt). So ε(t) = NBAω·sin(ωt). The peak value is ε_max = NBAω — that's where the formula comes from. It's not memorization; it's calculus meeting geometry.

Why AC is sinusoidal: every wall outlet in your house runs on this equation. In the US, ω = 2π·60 Hz ≈ 377 rad/s, which is why the peak voltage is ~170 V (and the RMS value — the "effective" DC-equivalent — is 120 V).

Why faster = bigger voltage: spinning the coil twice as fast doubles ω, which doubles both the rate of flux change AND the angular frequency. So ε_max scales linearly with RPM. A 700-RPM generator at 2400 RPM puts out exactly 2400/700 = 3.43× more voltage.

A transformer has no electrical connection between its two sides. Current on the primary side never crosses into the secondary. So how does energy get across? Through the shared magnetic flux in the iron core.

The chain of events: primary AC current → changing B field in the core → changing flux Φ passes through both coils → Faraday's Law induces an EMF in the secondary. The iron core is there only to concentrate and guide the magnetic flux so almost none of it leaks out.

Transformers only work on AC. If you plugged one into a DC battery, the flux would be constant, ΔΦ/Δt = 0, and the secondary would produce zero voltage. This is one of the practical reasons AC beat DC in the "War of the Currents" in the 1890s — Tesla's AC could be easily transformed to high voltages for long-distance transmission; Edison's DC could not.

The turn ratio is the voltage ratio: because each turn of wire sees the same dΦ/dt, the total EMF is (turns)·(EMF per turn). So V_P/V_S = N_P/N_S exactly. Power in ≈ power out (ideal transformer), so currents go the other way: more turns means higher voltage but lower current.

A step-down transformer is the everyday device hiding in phone chargers, laptop bricks, and doorbell wiring. Wall outlets in the US deliver ~120 V AC, but a phone battery wants ~5 V DC. The transformer is the first stage: it chops the voltage down to something safe.

The trick of the 4:1 ratio: if the primary has 80 turns and the secondary has 20, then each turn on both sides sees the same flux Φ (they share a common iron core). Faraday says ε per turn = −ΔΦ/Δt — the same number on both sides. So the total voltage just scales with turn count.

Why current goes UP when voltage goes DOWN: energy conservation. Power in = power out (ideally), so V_P·I_P = V_S·I_S. If voltage drops by 4×, current must rise by 4×. This is why the secondary wire is usually thicker — it carries more amps.

Real-world tie-in: the national grid runs at hundreds of thousands of volts precisely because high V means low I, and P_loss = I²R in the wires. Halving the current quarters the loss. Every power pole you see has a transformer stepping down from distribution voltage to your house's 240 V.

An inductor is any coil that generates a magnetic field when current passes through it. When you try to change that current, the coil's own changing flux induces an EMF back on itself — this is called self-induction. The minus sign in ε = −L·ΔI/Δt is just Lenz's Law applied to the coil itself: it fights any change you try to make.

Mental model: think of an inductor as the electrical equivalent of mass. A massive object resists changes in velocity (inertia); an inductor resists changes in current. That's why you can't switch current through an inductor instantly — doing so would require infinite voltage.

Energy stored: U = ½LI² looks exactly like kinetic energy ½mv². This isn't a coincidence — current "flowing" through an inductor carries the same kind of stored kinetic-like energy, but in the magnetic field rather than in moving mass.

Why L = μ₀N²A/ℓ: more turns (N²) means more flux linkage; bigger area (A) means more flux per turn; longer solenoid (ℓ in denominator) means a weaker field per amp. Double the turns → quadruple the inductance.