Faraday's Law of Induction

On 29 August 1831, Michael Faraday wound two coils of wire onto opposite sides of an iron ring, connected one to a battery and the other to a galvanometer, and saw something maddening: the needle kicked only at the instant he closed the switch, and kicked the other way when he opened it. In between — with a healthy current flowing steadily through the first coil — nothing. A magnetic field alone, however strong, induced no current at all. Only change did. In that flicker of a needle lives essentially every watt of electricity humanity has ever generated. Ten years earlier, Ørsted had shown that electricity makes magnetism, and the obvious question — does magnetism make electricity? — had defeated a decade of attempts. Everyone, Faraday included, had been looking for a steady effect: park a magnet next to a wire and wait for current. The answer, when it came, was a law about a rate. The previous module built the exact quantity that matters — the magnetic flux $\Phi = BA\cos\theta$ threading a circuit — and Faraday's law is one line about it: the electromotive force induced in a circuit equals the rate at which its flux linkage changes. No change, no EMF, no exceptions. This module is the law at exam grade: the formula $\varepsilon = -N\,\Delta\Phi/\Delta t$ and what each piece buys, the three discovery experiments every board loves to describe, the graph-reading craft (flux–time slopes are the language JEE and A-Level set questions in), and the classic numericals — ramped fields, flipped coils, a coil wound around a solenoid. Master this one rate and the rest of the chapter is applications: Lenz's Law signs it, Motional EMF builds it from forces, and the AC generator spins it into the waveform in your wall socket.

Key takeaways

Faraday's law: the induced EMF equals the rate of change of flux linkage, $\varepsilon = -N\,\dfrac{\Delta\Phi}{\Delta t}$ — an $N$ -turn coil multiplies the effect $N$ times.
The circuit is a rate-meter, not a flux-meter: a colossal steady flux induces nothing; a small fast wobble induces plenty.
Flux $\Phi = BA\cos\theta$ (previous module) changes by three levers — $B$ , $A$ , or $\theta$ — and the law bills all three identically.
The minus sign is the direction rule: the induced effects oppose the change. Its full doctrine is the next module, Lenz's Law.
Induced EMF drives induced current through whatever resistance the circuit offers: $I = \varepsilon/R$ ; on a $\Phi$ – $t$ graph, EMF is the slope.
Key idea: $\varepsilon = -N\,\Delta\Phi/\Delta t$ — change the threading count of a circuit and nature pays out a voltage proportional to how fast the count changes.

The core insight

Faraday's law of induction: a changing magnetic flux through a circuit induces an electromotive force equal to the rate of change of flux linkage, $\varepsilon = -N\,\Delta\Phi/\Delta t$ — the faster the change and the more turns that link it, the larger the EMF, with the minus sign recording that induced effects oppose the change.

Where this lives today

Roughly 95 percent of the world's electricity still arrives by this module's equation: in every thermal, hydro, nuclear and wind plant, something spins flux past a many-turn winding, and $N\,\Delta\Phi/\Delta t$ appears on the wires. The machine anatomy — why the waveform is a sine, slip rings versus commutators — is The AC Generator, later in this chapter; the law is the part that never changes. Off the grid, the induction economy is in your pocket and kitchen: a phone's wireless charger is two coils trading a kilohertz flux (Mutual Inductance builds the formalism), an induction cooktop drives its pan directly with a changing field (Eddy Currents explains why the pan and not the glass gets hot), and a credit-card reader's swipe is a magnetized stripe playing its flux pattern past a pickup coil — Faraday's ring, miniaturized and put to work at a checkout counter. The law's reach goes far beyond machines. The Earth's own field, changing as molten iron churns, induces measurable currents in pipelines and power grids during solar storms — grid operators run Faraday's law as a risk model. And in the physics to come, the law is one of the four pillars: it is a Maxwell equation, the statement that changing magnetism creates electric fields even in empty space — the half of the light-wave engine this track completes in EM Waves.

The physical picture

Picture the field lines threading a wire loop like strands of spaghetti through a ring — the count from Magnetic Flux. The loop watches that count, and it is completely indifferent to the count's value: park it in a uniform three-tesla field forever and it sits inert. But the moment the count starts changing, the loop answers with a voltage — an electromotive force, the same per-charge push a battery supplies (EMF, Terminal Voltage & Internal Resistance built that idea) — and if the circuit is closed, a current flows for exactly as long as the change lasts. Flux behaves as if the loop resents losing or gaining it; the induced current is the protest. Faraday established this with three experiments in the autumn of 1831, and board examiners still ask for them by name:

The ring (29 August). Two coils on one soft-iron ring: a primary with a battery and switch, a secondary with only a galvanometer. Close the switch — the needle kicks and returns to zero. Open it — the needle kicks the other way. Steady current: nothing. What the secondary feels is the ring's field building up at make and collapsing at break; in between, flux is large but constant, and constant does not pay. (Faraday had built, in all but name, the first transformer.)

The magnet and the coil (17 October). Thrust a bar magnet into a coil: the galvanometer kicks. Hold it still, deep inside, field at full strength: nothing. Pull it out: opposite kick. Push it in faster: a bigger kick. The response is to motion of the flux, and it scales with the rate.

The coil and the coil. Replace the magnet with a second, current-carrying coil and move that instead: same kicks. Then hold both coils still and merely vary the current in one: kicks again. The flux doesn't care whether it changes because something moved or because a field died — and neither does the law.

Figure

Faraday's ring — the needle kicks only at make and break, never during steady current

All three experiments — and every induction device since — pull one of the three levers catalogued in Magnetic Flux: change $B$ (the ring, the varied current), change the geometry's effective count via motion (the travelling magnet), or, in the machines to come, change $\theta$ by spinning the coil. The law you are about to write down does not distinguish between them. It reads the count's rate of change and pays accordingly.

The law

Everything Faraday's galvanometer recorded compresses into one line. For a coil of $N$ turns, each threaded by flux $\Phi$ ,

\varepsilon = -N\,\frac{\Delta\Phi}{\Delta t} \qquad (1)

where: $\varepsilon$ is the induced EMF in volts, $\Delta\Phi$ the change in flux through one turn (webers), $\Delta t$ the time the change takes, and $N$ the number of turns — the flux-linkage multiplier from the previous module, each turn contributing its own EMF in series like stacked batteries. Read the structure carefully: $\Phi$ itself never appears, only its change per time. A weber per second is a volt — which is why the weber's alternative dress is the volt-second. Written over an interval, equation (1) gives the average EMF; that average is what CBSE, ISC and NEET numericals ask for almost exclusively, and it is exact whenever the flux changes at a steady rate. What if the rate isn't steady? Then the honest statement is about the instant: the EMF right now is the slope of the tangent to the $\Phi$ – $t$ graph right now — written $\varepsilon = -N\,d\Phi/dt$ in the calculus notation CBSE and JEE display. You never need to compute a derivative in this module: every graph question hands you straight-line segments (constant slope — read it as rise over run) or asks where the EMF is biggest (steepest tangent) or zero (flat graph). Slope-of-tangent is the whole calculus you need, used as a reading skill. The minus sign is the law's conscience. It records the direction of the induced EMF: always the one that opposes the change producing it — push flux in and the induced current fights the rise; let flux die and the induced current props it up. That single sign carries an entire module's worth of doctrine — direction algorithms, pole rules, and the energy-conservation argument for why nature had no other choice — and it is the very next article, Lenz's Law & Energy Conservation. In this module we compute magnitudes: state $|\varepsilon| = N\,\Delta\Phi/\Delta t$ , and when a question asks for direction, answer with Lenz by name. Close the circuit through a total resistance $R$ and the EMF becomes a current, exactly as any battery would drive it:

I = \frac{\varepsilon}{R} = \frac{N}{R}\,\frac{\Delta\Phi}{\Delta t} \qquad (2)

where: $R$ is the whole loop's resistance, coil included. This chain — flux change → EMF → current — is the standard two-step of every numerical. One corollary is worth a single line here because it is a whole problem industry later: over a complete change, the total charge driven is $q = N\Delta\Phi/R$ , independent of how fast the change happened — fast flips drive big brief currents, slow ones small long currents, same charge. The flux-change problem gym (coming up in this chapter as Flux-Change Numericals) mines that theorem; here, recognize it and move on.

Derivation AAverage EMF from the flux ledger — a ramped field through a fixed coil

The goal is to see equation (1) work end-to-end on the cleanest possible case — a stationary coil in a field that ramps at a steady rate — and to establish the bookkeeping discipline (count before, count after, divide by time, multiply by turns) that every induction numerical reuses. The strategy is pure ledger-keeping with the flux formula from Magnetic Flux; nothing moves, so the only lever pulled is $B$ . Watch the units flow: tesla times square metre per second lands on volts, which is the dimensional heart of the law.

Step 1. The setup. A flat circular coil of $N = 200$ turns and radius $r = 5.0$ cm sits face-on ( $\theta = 0°$ ) in a uniform field that grows steadily from $B_1 = 0.10$ T to $B_2 = 0.35$ T in $\Delta t = 0.40$ s. Area per turn: $A = \pi r^2 = \pi (0.050)^2 = 7.85 \times 10^{-3}\ \mathrm{m^2}$ .

Step 2. Count before and after. Face-on means $\cos\theta = 1$ , so $\Phi_1 = B_1 A = 0.10 \times 7.85\times10^{-3} = 7.85\times10^{-4}$ Wb and $\Phi_2 = B_2 A = 2.75\times10^{-3}$ Wb. The change per turn:

\Delta\Phi = (B_2 - B_1)A = 0.25 \times 7.85\times10^{-3} = 1.96\times10^{-3}\ \mathrm{Wb}.

Why per turn: the ledger tracks one turn's flux; the turns multiply at the end. Keeping $N$ out until the last line is the habit that prevents double-counting when problems mix linkage and per-turn flux.

Step 3. Apply the law. The ramp is steady, so the average EMF is also the EMF throughout:

|\varepsilon| = N\,\frac{\Delta\Phi}{\Delta t} = 200 \times \frac{1.96\times10^{-3}}{0.40} = 0.98\ \mathrm{V}.

Step 4. Read the anatomy of the answer. Each of the four inputs is a lever an examiner can pull: double the turns → double the EMF; double the radius → quadruple it (area goes as $r^2$ ); same field change in half the time → double; and if the coil were tilted at $60°$ the whole answer would halve (the count carries $\cos\theta$ ). One computation, a family of MCQs. Verified: units, $\mathrm{Wb/s} = \mathrm{T\,m^2/s} = \mathrm{V}$ ✓; magnitude recomputed, $200 \times 1.963\times10^{-3} / 0.40 = 0.982$ V ✓; sanity — a fraction of a volt from a lab-scale coil and a quarter-tesla ramp is the right scale (real generators reach hundreds of volts with thousands of turns and faster rates).

Reading the graphs

Half the competitive questions on this law never mention a number — they hand you a graph of flux against time and ask about the EMF. The dictionary is one line: EMF is proportional to the negative slope of the $\Phi$ – $t$ graph. Steep means large EMF; flat means zero; a corner where the slope jumps means the EMF jumps; and where the flux peaks — slope momentarily zero — the EMF crosses zero even though the flux is at its largest. That last reading is the rate-meter lesson in graph form, and setters return to it every season.

Figure

Flux–time graph to EMF–time graph — the slope dictionary

Limiting cases

case	result	meaning
steady flux, any size	$\varepsilon = 0$	the rate-meter reads nothing — the 1831 lesson
$N$ turns	$\varepsilon$ multiplied by $N$	series turns, stacked batteries
same $\Delta\Phi$ , half the time	double the EMF	rate, not amount
same $\Delta\Phi$ , any speed	same total charge $q = N\Delta\Phi/R$	the charge corollary — the gym module's theorem
flux at a maximum on the graph	$\varepsilon = 0$ at that instant	slope zero at the peak
open circuit	EMF present, current zero	induction makes EMF; current needs a closed path
coil edge-on throughout	$\varepsilon = 0$	no flux to change ( $\cos 90° = 0$ )

Common trapA strong magnetic field through a coil induces a large EMF

Only a changing flux induces anything. This misconception survives because it feels economical — big cause, big effect — but the 1831 experiments killed it on day one: Faraday's secondary coil sat silent while a strong steady current magnetized the ring, and kicked only at make and break. The exam versions are everywhere. "A coil rests in a uniform 2 T field; find the induced EMF" — zero, whatever the area and turns. "At the instant the flux through a rotating coil is maximum, the EMF is…" — zero: the graph is momentarily flat at a peak. "Doubling the field doubles the induced EMF" — only if the rate of change doubles too; a doubled but equally steady field still induces nothing. Train the reflex: before touching the formula, ask what is changing, and how fast? If the answer is "nothing," the answer is zero, full stop.

Worked examples

Board level — the ramped field (CBSE/ISC pattern, 3 marks)

A circular coil of 200 turns and radius 5.0 cm is placed face-on in a uniform magnetic field. The field increases steadily from 0.10 T to 0.35 T in 0.40 s. Calculate (a) the change in magnetic flux through one turn, (b) the magnitude of the induced EMF, and (c) the induced current if the circuit's total resistance is 2.0 Ω.

Step 1. Area: $A = \pi r^2 = \pi(0.050)^2 = 7.85\times10^{-3}\ \mathrm{m^2}$ . (Radius in metres first — the classic slip.)

Step 2. Part (a): $\Delta\Phi = \Delta B \times A = (0.35 - 0.10) \times 7.85\times10^{-3} = \boxed{1.96\times10^{-3}\ \mathrm{Wb}}$ per turn.

Step 3. Part (b), Faraday's law: $|\varepsilon| = N\,\dfrac{\Delta\Phi}{\Delta t} = 200 \times \dfrac{1.96\times10^{-3}}{0.40} = \boxed{0.98\ \mathrm{V}}$ .

Step 4. Part (c): $I = \varepsilon/R = 0.98/2.0 = \boxed{0.49\ \mathrm{A}}$ . The mark scheme wants the chain stated in order — flux change, then EMF, then current — and the direction, if asked, answered in one sentence: "by Lenz's law, the induced current opposes the increase in flux" (the next module's algorithm turns that sentence into a definite arrow).

Competitive level — the coil around the solenoid (JEE-Main pattern, ~2.5 minutes)

A long solenoid with 800 turns per metre and cross-sectional radius 1.5 cm carries a current that rises steadily from 2.0 A to 8.0 A in 0.30 s. A 150-turn coil is wound snugly around the solenoid's middle. Find (a) the EMF induced in the coil and (b) the induced current if the coil's circuit resistance is 2.0 Ω. (c) Why does the coil's own larger radius not enter the calculation?

Step 1. The field lives inside the solenoid: $B = \mu_0 n I$ (Solenoids & Toroids), uniform across the bore and essentially zero outside. Field change: $\Delta B = \mu_0 n\,\Delta I = 4\pi\times10^{-7} \times 800 \times 6.0 = 6.03\times10^{-3}$ T.

Step 2. Flux change per turn of the outer coil — through the solenoid's cross-section, since that is where the field is: $\Delta\Phi = \Delta B \times \pi (0.015)^2 = 6.03\times10^{-3} \times 7.07\times10^{-4} = 4.26\times10^{-6}$ Wb.

Step 3. Faraday: $|\varepsilon| = N\,\dfrac{\Delta\Phi}{\Delta t} = 150 \times \dfrac{4.26\times10^{-6}}{0.30} = \boxed{2.1\ \mathrm{mV}}$ . Then $I = \varepsilon/R = \boxed{1.1\ \mathrm{mA}}$ .

Step 4. Part (c), the concept the problem exists to test: the outer coil's own area is irrelevant because the field is confined to the solenoid's bore — the flux through a 5 cm outer turn and a 1.6 cm outer turn is identical, namely $B$ times the solenoid's area. Notice the quiet marvel here (it returns as Mutual Inductance): the outer coil sits in a region of essentially zero field, yet responds to the changing flux threading it. Timing note: two and a half minutes. The speed tricks are pre-computing $\mu_0 n \Delta I$ in one line and refusing the bait of the outer coil's radius. The MCQ variant asks only "which area?" — solenoid's, always. If both radii appear in your working, you have the wrong physics, not the wrong arithmetic.

Exam traps

Using $\Phi$ instead of $\Delta\Phi$ . The law bills the change. A question giving initial and final fields wants $(B_2 - B_1)A$ , not $B_2 A$ . For a coil flipped 180° face-on, the flux goes from $+BA$ to $-BA$ : $\Delta\Phi = 2BA$ , and forgetting the factor 2 is the single commonest flip-problem error.

Dropping $N$ — or double-counting it. EMF questions need $N\,\Delta\Phi/\Delta t$ with $\Delta\Phi$ per turn. If you tracked flux linkage ( $N\Phi$ ) from the start, the $N$ is already inside — multiplying again is the double-count.

The angle, again. $\Phi = BA\cos\theta$ with $\theta$ from the normal (Magnetic Flux prosecutes this fully). Induction problems inherit the trap wholesale.

Average vs instantaneous. $N\Delta\Phi/\Delta t$ over an interval is the average EMF. If the flux changes non-uniformly, the instantaneous EMF differs moment to moment — it is the tangent slope. Questions saying "calculate the EMF" over an interval want the average; graph questions asking "when is the EMF greatest" want the steepest segment.

Magnitude vs direction. When asked for magnitude, compute $N\Delta\Phi/\Delta t$ and drop the sign — writing a bare minus number as "the EMF" costs presentation marks. When asked for direction, one sentence of Lenz (next module) is the expected currency; hand-waving about "the minus sign" is not.

Units. Wb/s = V exactly; radius in cm must become m before squaring; and mV-scale answers (like the solenoid example) are normal — do not "fix" a small answer by hunting for a lost factor of 1000.

Boards and depth. CBSE/ISC/NEET: statement, the three experiments, average-EMF numericals (ISC states the law with both Faraday's laws — the "first law" being that a changing flux induces an EMF at all, the "second" the rate proportionality). JEE-Main/Advanced: graphs, combined setups (solenoid-fed flux, then Lenz and force follow-ups). A-Level 9702: "recall and use Faraday's and Lenz's laws" plus flux-linkage language. AP-2: conceptual flux + Faraday reading (their derived rod-on-rails equation is Motional EMF's business). IGCSE: qualitative only — induction happens when a conductor cuts field lines or the field linking it changes, and the EMF grows with speed, field strength and turns; no formula. IB-HL: $\varepsilon = -N\,\Delta\Phi/\Delta t$ as stated here.

Origins · 1 minMichael Faraday and how this was discovered

Michael Faraday

1791–1867 · Royal Institution, London

A blacksmith's son apprenticed to a bookbinder at fourteen, Faraday taught himself science from the volumes he stitched, talked his way into Humphry Davy's laboratory as a bottle-washer, and became the greatest experimentalist of his century without ever learning much mathematics. He hunted the converse of Ørsted's discovery for a decade — if currents make magnetism, magnetism must make currents — and found it on 29 August 1831 with the iron ring, two coils and a flickering galvanometer. Within months he had the travelling-magnet version, the first generator, and the quantitative flux dependence, established with nothing but 1830s instruments and relentless method. His deeper gift was the concept that outlived the apparatus: the field, pictured as lines of force filling space, with induction as their counting rule — the very lines this chapter counts. He declined a knighthood, twice declined the Royal Society presidency, gave the Christmas Lectures for children for decades, and left notebooks containing not a single equation. Maxwell, who turned those lines into the equations that carry this law today, insisted Faraday "was in reality a mathematician of a very high order." (The 1851 iron-filing field maps that visualize wire-system patterns appear in his portrait in Net Fields of Wire Systems; this module's Faraday is the discoverer at the ring.)

History

1820

— Ørsted's compass twitches beside a current: electricity makes magnetism, and the decade-long hunt for the converse begins

1831

— 29 August: the iron ring; October: the travelling magnet and the first dynamo. Joseph Henry finds induction independently in Albany

1832

— Faraday establishes the quantitative flux dependence with ballistic charge measurements

1834

— Lenz publishes the direction rule — the minus sign gets its doctrine

1866

— practical self-excited dynamos arrive; electricity becomes an industry within two decades

1882

— Edison's Pearl Street station lights Manhattan; the 1890s AC grid — generators and transformers, pure Faraday — wins the world

Going further

Texts. NCERT Physics Class 12, Ch. 6 (§6.3–6.5) · HC Verma Vol. 2, Ch. 38 · SL Arora, ISC Physics XII, electromagnetic induction · CIE 9702 topic 20.5.

Provenance. All results standard. A verifier should recompute: ramped-coil example $A = 7.85\times10^{-3}\ \mathrm{m^2}$ , $\Delta\Phi = 1.96\times10^{-3}$ Wb, $|\varepsilon| = 0.98$ V, $I = 0.49$ A; solenoid example $\Delta B = \mu_0 \times 800 \times 6.0 = 6.03\times10^{-3}$ T, $\Delta\Phi = 4.26\times10^{-6}$ Wb, $|\varepsilon| = 150 \times 4.26\times10^{-6}/0.30 = 2.1$ mV, $I = 1.1$ mA; flip factor $\Delta\Phi = 2BA$ .

Prerequisites. Magnetic Flux (the count this law differentiates). Leads to. Lenz's Law & Energy Conservation (the minus sign's doctrine, next) · Motional EMF (the law built from forces) · Flux-Change Numericals — Pulls, Flips & Ramps and The AC Generator (coming up in this chapter) · Faraday's Law of Induction (UG deep dive).

Why does only a changing flux induce an EMF? What's physically special about change?

Because the pushing agent only exists during change. In a stationary coil, the charges aren't moving with the wire, so a static magnetic field exerts no net driving force along it — a parked flux is invisible to them. When the flux changes, something genuinely new appears: a changing magnetic field creates an electric field, circulating in loops around the change, and that electric field is what shoves the charges around the circuit. The coil is really a detector for this circulating field, which exists whether or not a wire is present. That statement — changing $B$ makes $E$ — is one of the four fundamental equations of electromagnetism, and its partner (changing $E$ makes $B$ ) completes the pair that lets light travel through vacuum. The full field story is UG territory; at exam level, "no change, no EMF" plus the rate law is the complete working truth.

What exactly is an EMF here — is it a voltage, a force, or what?

Same species as a battery's EMF: energy given to each coulomb of charge that traverses the source, measured in volts (EMF, Terminal Voltage & Internal Resistance built the concept). A battery's push comes from chemistry; an induced EMF's push comes from the changing flux. The name "force" is a two-century-old fossil — nothing in newtons here. Operationally, treat an induction coil exactly as a battery of EMF $N\Delta\Phi/\Delta t$ with the coil's own resistance as its internal resistance: connect a load, apply the circuit chapter, done. One honest subtlety: with an open circuit the EMF still exists (a voltmeter across the coil's ends reads it) but no current flows — induction manufactures the push, not the path.

Where does the induced current's energy come from? It looks like electricity from nowhere.

From whatever changes the flux — always, and exactly. Push a magnet toward a coil and you feel a resistance (the induced current's field pushes back — Lenz's Law makes this precise); the work your hand does against that resistance is the electrical energy delivered, joule for joule. In a power station, the turbine does work against the same opposition, scaled up: a loaded generator is physically harder to turn than an idle one, and the extra shaft work is your city's consumption. Nothing is free. Indeed the direction rule exists because nothing is free: if induced currents aided the change instead of opposing it, flux would amplify itself and mint energy from nowhere — the reductio the next module runs in full.

Does the wire have to physically "cut" field lines for induction to happen? My textbook keeps saying "cutting."

"Cutting" is the picture for one family of cases — a conductor sweeping through a field region — and IGCSE-level treatments lean on it because it is vivid. But Faraday's ring breaks the picture: nothing moves, nothing cuts, and induction happens anyway because the field through the stationary loop changes. The flux formulation swallows both cases: any change in $\Phi = BA\cos\theta$ , by motion or by field variation, induces. Use "cutting" as intuition for motional setups (Motional EMF makes it quantitative as $\varepsilon = BLv$ ), and trust the flux ledger for everything.

Why do more turns give more EMF? The same flux goes through each turn, so shouldn't it count once?

Follow one charge around the circuit. In an $N$ -turn coil, its path threads the changing flux region $N$ separate times before returning to the start, and the induced push acts on it during each pass — energy per coulomb $N$ times one turn's worth. The right mental model is $N$ identical batteries in series: voltages add. This is also why the multiplier is engineering gold: turns cost copper, not physics, so a generator winding with 2000 turns delivers 2000 single-loop EMFs — the difference between millivolts and mains voltage. The bookkeeping name for "flux counted with multiplicity" is flux linkage $N\Phi$ , from the previous module; Faraday's law differentiates linkage.

My problem says "calculate the EMF" but the flux change isn't steady. Which EMF am I computing?

$N\Delta\Phi/\Delta t$ over an interval always gives the average EMF for that interval — a perfectly well-defined and frequently-asked quantity. The instantaneous EMF is the slope of the $\Phi$ – $t$ tangent at one moment, and it only equals the average when the graph is a straight line. Board questions signal which they want: "in 0.4 s" or "during the change" → average; "at $t = 2$ s" with a graph → read the slope of that segment; "maximum EMF" → find the steepest part. If a question gives a smooth curve and asks for an instant without giving you a slope to read, it is a calculus question in disguise and belongs to a later course — at this level, someone will always have drawn you straight segments.

Is the induced EMF the same everywhere in the coil, or does each turn have its own?

Each turn contributes its share — one turn's $\Delta\Phi/\Delta t$ — and the shares add in series, so the terminal EMF is the sum, $N\Delta\Phi/\Delta t$ . There is no single point where "the EMF lives": the push is distributed along the winding, unlike a battery's, which sits between two plates. For circuit purposes the difference never matters — a distributed EMF drives current exactly as a lumped one — but it explains an odd fact (cut the coil anywhere and each fragment shows a proportional share of the voltage) and previews a later-course warning: with distributed EMFs, "the voltage between two points" needs careful path bookkeeping — a subtlety the UG deep dive treats properly.