Preview

Electromagnetic Induction

Q: A single-turn coil of area 0.01 m² rotates at ω = 10 rad/s in B = 0.5 T. What is the peak EMF?

You can work it out this way: use ε_peak = NBAω with N = 1.

Q: What physical law does Lenz's Law ultimately conserve?

You can work it out this way: think about what would happen if induced current aided the flux change.

Q: If the number of turns N is doubled, how does the peak EMF change?

You can work it out this way: look at the formula ε_peak = NBAω.

See how changing magnetic flux generates EMF — Faraday's and Lenz's Laws in action

Two labs ago you watched charges price empty space with static vectors — the electric field just sat there, frozen between the red and blue spheres. This lab breaks the stillness. Take away the charges entirely: one bar magnet, one coil of wire, and a startling rule — the coil only notices the magnet while something is CHANGING. Park the magnet right inside the coil, its field threading every turn at full strength, and the voltmeter reads exactly zero. Move it, and the same coil lights up with a voltage that prices nothing but the SPEED of the change. Faraday caught this in 1831: push a magnet into a coil and the meter kicks; pull it out and the meter kicks the other way; hold it still — nothing, no matter how strong the magnet. His law writes the kick as ε = −N·dΦ/dt: the number of turns times the rate of change of flux. This bench hands you three levers on that rate — how fast the magnet moves, how many turns listen, how much area each turn presents — plus a live trace of the EMF as the magnet plunges through the coil. And the minus sign is not bookkeeping: the induced current always fights the change that feeds it, because if it helped instead, the magnet would fall in and hand you current for free. Lenz's law is energy conservation wearing a minus sign. Watch the glow ring around the coil flash red, then blue, as the flux change flips sign. Your first job is to predict how the kick scales with speed — before you touch the slider.

What you'll be able to do

Close Faraday's law on the panel's own rows at the reference pass (markup slider defaults v = 2.0 m/s, N = 100, A = 50 cm²; magnet crossing z = 25 cm above the coil plane): the engine's on-axis dipole model (B = (μ₀/4π)·2m/z³, m = 1.5 A·m²) gives Φ = N·A·B = 9.600 μWb exactly on the Φ row, and ε = −N·A·(dB/dz)·v = C·N·A·v/z⁴ with C = 3·(μ₀/4π)·2m·0.5 = 4.5×10⁻⁷ (the 0.5 scene scale) gives 1.152×10⁻⁴ V = 0.1152 mV on the ε row (panel '0.12'), with I_ind = ε/R = 0.0115 mA through the fixed 10 Ω coil — 'ε = −N dΦm/dt' (OpenStax Eq. 13.3) and Example 13.1's I = ε/R run as live arithmetic
Price the rate, not the field: holding the same pass (z = 25 cm, so the Φ row reads 9.600 μWb both times), dragging Velocity −1.0 → −2.0 m/s doubles the ε row −0.06 → −0.12 mV — 'The faster the motion, the greater the emf' and 'there is no emf when the magnet is stationary relative to the coil' (OpenStax §13.1 verbatim); the turns and area levers are the same linear bookkeeping (ε ∝ N, 'the net magnetic flux through the circuits is N times the flux through one turn' Eq. 13.3; ε ∝ A through Φ = B·A), machine-certified by two full-grid monotonic sweeps and the exact doubling-ratio bounds
Read Lenz's law off the sim's observable outputs: the ε sign, the polarity row, and the glow-ring/particle color (red = ε > 0, blue = ε < 0) track the SIGN of dΦ/dt — dragging Velocity through zero flips ε's sign at unchanged magnitude (a −1 exact ratio, machine-certified), the visible form of 'Emfs of opposite signs are produced by motion in opposite directions' (§13.1) and 'always oppose the change in magnetic flux that causes the emf' (§13.2). Of the three routes to changing flux — change B, change A, change θ — this bench operates the first two (magnet motion prices dB/dt; the Area slider prices A) while θ is FIXED (coil axis locked to the magnet's traverse axis, no angle control anywhere in the HTML — θ ≡ 0, cos θ ≡ 1); the ε_max = NBAω rotating-coil regime is NOT this sim (ghost-world formula, mismatch registry #3)

Formulas

\Phi = B \cdot A \cdot \cos(\theta)

Magnetic Flux

\varepsilon = -N\frac{d\Phi}{dt}

Faraday's Law of Induction

B_{axis} = \frac{\mu_0}{4\pi} \cdot \frac{2m}{r^3}

Dipole field on the axis (the bar magnet's field in this simulation)

\text{Lenz's Law: induced current opposes change in flux}

Direction of induced current

Make a prediction

Set Turns N = 100 and Coil area A = 50 cm², then drag Velocity to −1.0 m/s: the magnet plunges from above down through the coil. As it passes z = 25 cm above the coil plane, the Φ row reads 9.600 μWb and the ε row reads −0.06 mV. Now reset, drag Velocity to −2.0 m/s — same magnet, same coil, same pass, twice the speed — and watch the ε row at that same position z = 25 cm (the Φ row reads 9.600 μWb again). What does the ε row read?

No grading here — commit to a guess, then scroll down and test it yourself.

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Run the gate live: with N = 100, A = 50 cm², plunge at Velocity −1.0 m/s and note the ε row at z = 25 cm (−0.06 mV) and the red-to-blue trace spike on the EMF graph; reset and plunge at −2.0 m/s — same position, doubled deflection (−0.12 mV), the spike twice as tall at every height. Then pause mid-pass (Space) and compare: the Φ row freezes with the magnet, and the textbook's sentence 'there is no emf when the magnet is stationary' is one slider-drag-to-zero away (registered engine quirk: the panel keeps showing the slider's phantom ε while paused — see review pack, assembly note).
Flip the polarity with your own hand: mid-plunge, drag the Velocity slider slowly through 0 from −2.0 to +2.0 m/s and watch the ε row's sign, the polarity-row text, the glow ring, and the particle stream swap blue ↔ red together — the magnitude at a given height never changes, only the sign of dΦ/dt does. 'Emfs of opposite signs are produced by motion in opposite directions' (§13.1), machine-certified at exactly −1.
Give the geometry levers their turn: hold Velocity at −2.0 m/s and drag Turns 10 → 500 — the coil helix visibly re-wraps (the renderer caps at 15 drawn turns while the physics uses every turn) and the ε row climbs linearly, ×50 end to end; then Coil area 1 → 100 cm² for another clean ×100. The HTML's own quiz Q3 ('Doubling the number of coil turns N… doubles the induced EMF') verified with your own hands.
Walk the preset ladder one click at a time: Slow Move (ε at z = 25 cm: 0.0288 mV, panel '0.03') → Fast Plunge (0.553 mV — exactly 19.2× Slow at the same height: ×2 turns, ×1.6 area, ×6 speed compounded; note it is the shipped init state, and its positive velocity drifts the magnet AWAY — drag v negative to actually plunge) → Oscillating Magnet (shuttle ±3 m through the coil plane, the trace pulsing at every pass — the bench's standing AC generator).

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Check

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Your prediction

You skipped the prediction — jump back to the preview and commit to a guess first; the comparison is the whole point.

Answer: −0.12 mV — same sign, doubled magnitude: the position sets the flux (9.600 μWb both times), but the EMF prices the RATE dΦ/dt, and doubling the magnet's speed doubles the rate

Faraday's law prices a rate: ε = −N·dΦ/dt (OpenStax Eq. 13.3). At z = 25 cm the flux is identical on both passes — 9.600 μWb, fixed by position alone — but the engine's closed form ε = C·N·A·v/z⁴ (C = 4.5×10⁻⁷) is strictly proportional to the speed: −5.76×10⁻⁵ V at −1.0 m/s (panel '−0.06' mV) and −1.152×10⁻⁴ V at −2.0 m/s (panel '−0.12' mV). The textbook states it flat: 'The faster the motion, the greater the emf, and there is no emf when the magnet is stationary relative to the coil' (OpenStax §13.1, verified). Option B is the documented flux-for-flux-rate conflation — it reads the law as ε ∝ Φ and concludes that equal flux means equal EMF; the Φ row really does read 9.600 μWb both times, which is exactly why this option seduces, and exactly what the experiment refutes. Option C grafts a direction error onto the right magnitude: the sign of ε tracks the sign of the flux CHANGE, and both passes are the same downward plunge with the same increasing-flux direction — reversal of polarity comes from reversing the MOTION (the −1 antisymmetry gate), not from speeding it up. The lab's machine gates bracket the doubling ratio at exactly 2 and the reversal ratio at exactly −1.

Quiz (0/3)

A single-turn coil of area 0.01 m² rotates at ω = 10 rad/s in B = 0.5 T. What is the peak EMF?

What physical law does Lenz's Law ultimately conserve?

If the number of turns N is doubled, how does the peak EMF change?

You can now

Close Faraday's law on the panel's own rows at the reference pass (markup slider defaults v = 2.0 m/s, N = 100, A = 50 cm²; magnet crossing z = 25 cm above the coil plane): the engine's on-axis dipole model (B = (μ₀/4π)·2m/z³, m = 1.5 A·m²) gives Φ = N·A·B = 9.600 μWb exactly on the Φ row, and ε = −N·A·(dB/dz)·v = C·N·A·v/z⁴ with C = 3·(μ₀/4π)·2m·0.5 = 4.5×10⁻⁷ (the 0.5 scene scale) gives 1.152×10⁻⁴ V = 0.1152 mV on the ε row (panel '0.12'), with I_ind = ε/R = 0.0115 mA through the fixed 10 Ω coil — 'ε = −N dΦm/dt' (OpenStax Eq. 13.3) and Example 13.1's I = ε/R run as live arithmetic
Price the rate, not the field: holding the same pass (z = 25 cm, so the Φ row reads 9.600 μWb both times), dragging Velocity −1.0 → −2.0 m/s doubles the ε row −0.06 → −0.12 mV — 'The faster the motion, the greater the emf' and 'there is no emf when the magnet is stationary relative to the coil' (OpenStax §13.1 verbatim); the turns and area levers are the same linear bookkeeping (ε ∝ N, 'the net magnetic flux through the circuits is N times the flux through one turn' Eq. 13.3; ε ∝ A through Φ = B·A), machine-certified by two full-grid monotonic sweeps and the exact doubling-ratio bounds
Read Lenz's law off the sim's observable outputs: the ε sign, the polarity row, and the glow-ring/particle color (red = ε > 0, blue = ε < 0) track the SIGN of dΦ/dt — dragging Velocity through zero flips ε's sign at unchanged magnitude (a −1 exact ratio, machine-certified), the visible form of 'Emfs of opposite signs are produced by motion in opposite directions' (§13.1) and 'always oppose the change in magnetic flux that causes the emf' (§13.2). Of the three routes to changing flux — change B, change A, change θ — this bench operates the first two (magnet motion prices dB/dt; the Area slider prices A) while θ is FIXED (coil axis locked to the magnet's traverse axis, no angle control anywhere in the HTML — θ ≡ 0, cos θ ≡ 1); the ε_max = NBAω rotating-coil regime is NOT this sim (ghost-world formula, mismatch registry #3)