Turn a tabletop generator, unplug its lamp, and replay the same motion. Follow the voltage, the pushback and every joule to discover where electrical energy comes from.
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A magnet provides the field. Your turn supplies the work. Changing flux can create voltage; a complete loaded circuit lets current carry energy to the load.
Turn the same way at the same speed, but unplug the lamp. Which can remain?
Changing flux can still induce voltage. The open external circuit has no load current or delivered load power in this model.
Start with one steady turn and watch the load. Replay the same motion with the plug out. Voltage can still appear at the terminals, but no current reaches the disconnected lamp. That contrast separates a voltage from a working energy-transfer circuit.
The turning wire coil changes how the magnetic field passes through its area. Scientists describe this signed connection as magnetic flux linkage. Its rate of change produces an induced voltage, or emf: electromagnetic induction. A strong but unchanging linkage is not enough.
Connect the load. Current in the coil interacts with the magnetic field and produces a torque opposing this rotation. Keeping the prescribed turn going requires mechanical work. The display shows the required electromagnetic component; your mouse is not a physical force-feedback handle.
Some input becomes electrical energy in the external load. Some becomes heat in the winding’s resistance. The model keeps both accounts visible. The magnet does not supply an endless store of free energy.
Each coil end connects to its own continuous conducting ring, insulated from the other ring and shaft. A stationary brush touches each ring. These slip rings preserve the connections while the coil turns, so the external current reverses each half-cycle.
Pausing replay freezes a frame from a moving run, including that frame’s nonzero voltage. Physically holding the rotor still is a different condition: Stop rotor selects zero speed, giving zero emf in this fixed field.
The shaft is +z; B points along +x. The positive coil normal is n=(cosθ,sinθ,0). Positive winding circulation follows the right-hand rule about n; terminal labels stay fixed. Positive external conventional current travels A → load → B. Animated cues show direction, not individual electron speed.
For N equal turns, each of area A, λ=NBA cosθ. Set K=NBA. At prescribed steady angular velocity ω, θ=θ₀+ωt and e=−dλ/dt=Kω sinθ. Face-on flux can be largest while its slope is zero. Edge-on flux can be zero while its slope is largest. A stationary coil can also have emf in a changing external field; this workbench holds that field fixed.
This original calculation uses N=100, A=0.0100 m² per turn, B=0.200 T and K=0.200 Wb-turn. Winding/contact resistance Rc=2 Ω; loads are open, 8 Ω or 2 Ω. Speeds are rest, ±10 or ±20 rad/s. These are authored parameters, not dimensions or measurements extracted from museum photographs.
For a connected resistive load, signed i=e/(Rc+RL). Terminal voltage Ut=iRL=e−iRc. At θ=90°, ω=10 rad/s and RL=8 Ω, generated emf is +2 V, current +0.2 A and terminal voltage +1.6 V. With the plug out, i=0 and Ut=e; an open circuit has no delivered load power. Real unloaded machines still have losses omitted here.
The rotor’s electromagnetic torque is τem=−Ki sinθ; the required electromagnetic driving torque is −τem. Thus Pinput=−τemω=ei=i²RL+i²Rc. Reversing ω reverses current and torque at the same θ; τem still opposes motion. The resistive load does not return energy just because current has a negative sign.
At +10 rad/s, a full turn takes 2π/10≈0.6283 s, about 95.49 rpm. For the 8 Ω load, mean input/load/heating powers are 0.20/0.16/0.04 W. Full-turn energies are about 0.12566/0.10053/0.02513 J. Integrating signed emf over that turn gives zero V·s, but integrating load power gives positive joules. All these values are calculated.
Generated emf peaks at 2 V in that run, while its RMS value is √2≈1.414 V and its signed mean is zero. RMS means square, average, then take the square root. It connects an AC waveform to resistive heating; it is not the same as its peak. The glow uses cycle-mean load power for a calm visual cue, not a simulated filament temperature.
The 2 Ω load receives more power than the 8 Ω load at the same prescribed speed, but a larger fraction also heats the winding. In this fixed-speed resistive model, RL=Rc maximizes load power, with half the modeled input becoming winding heat. This is a mathematical operating comparison, not advice to short or modify a real generator.
Continuous slip rings keep winding connections intact. A brushed DC machine uses a commutator to change connections; other generators use different field, winding and rectifier arrangements. One electrical cycle per mechanical revolution describes this simple loop model, not every multi-pole alternator.
In his first series, paragraphs 39–40, Faraday describes an indicator responding when a magnet enters a coil, returning while it is held there, and reversing on withdrawal or continued passage through. Paragraph 43 describes an outside-coil control against direct action on the indicator. These are qualitative observations, not the numerical curves in this lesson.
The Smithsonian’s Edison type Z DC dynamo, about 1881, has wound fields and a commutator. Its Westinghouse single-phase AC generator, about 1888, has a different housing, field-coil and collector arrangement. Jaclyn Nash photographed both in 2022. Their source-owned photographs show real construction; our transparent tabletop mechanism is original teaching geometry, not a reconstruction of hidden museum parts.
A packaged light may store energy or condition the output. Staying lit after cranking stops does not prove that an unmoving coil in a fixed field keeps generating. Find out what parts a source identifies before explaining the whole device.
Wind or flowing water can supply mechanical input instead of a hand. How the turbine captures that energy and how the generator converts it are connected but different investigations.
A museum photograph supports visible construction and recorded identity. Faraday’s text supports qualitative observations. Our numerical curve supports a prediction under declared assumptions. Ask which kind of evidence each claim needs.
Make seven cards in this order: 0°,60°,120°,180°,120°,60°,0°. Copy their supplied flux linkages: +0.2,+0.1,−0.1,−0.2,−0.1,+0.1,+0.2 Wb-turn. Angle refers to the coil normal relative to the field. This is a half-turn out and back, not a full revolution.
Give each of the six moves 0.5 seconds. Subtract earlier linkage from later linkage, change the sign, then divide by 0.5. For the first move:−[0.1−0.2]/0.5=+0.2 V. Predict whether each other interval is positive or negative before calculating.
Check the six interval-mean voltages: +0.2,+0.4,+0.2,−0.2,−0.4,−0.2 V. Mark one interval where the change reverses. These exact means follow from the supplied endpoints and times; they are not measured voltages or a claim about how smoothly the paper moved.
Use the eighth card to repeat 180° and −0.2 Wb-turn beside the existing 180° card. Add a 0.5-second hold before returning. Linkage changes by zero in that new interval, so its mean induced voltage is zero. Explain why a field can remain present without this model generating voltage.
Redo the original six moves with 1 second for each interval. Every interval-mean voltage halves. Multiply each mean by its interval duration and add the signed results: the return to the original linkage gives zero overall voltage-time integral. Do not call that an energy total.
Write two columns: what we know and what is missing. Endpoints and times give exact interval-mean voltage. To calculate exact delivered energy, we would need the circuit and the voltage waveform inside each interval. Squaring the mean voltage is not generally the same as averaging voltage squared. Ask someone to explain the distinction back to you.
What does a change tell you that a single number cannot?
An original paper reasoning model based on Faraday’s law, not a laboratory measurement. No electrical supplies, magnets or moving machinery are needed. Do not dismantle appliances or connect terminals. The supplied endpoints do not define the full motion or energy history.
Zero. Correct. Neither orientation nor the imposed field changes, so the flux-linkage derivative is zero.
Yes: the linkage is changing fastest at that instant. Yes. In this uniform-field rotating-coil model, zero linkage and maximum magnitude of its rate of change occur together.
Zero in this model. Correct. Open-circuit voltage can exist without load current or delivered load power.
More current creates stronger opposing electromagnetic torque. Correct. With the fixed 2 Ω winding, total resistance falls and current increases. Keeping the same turn going then requires more electromagnetic work.
Voltage, current and reaction-torque signs reverse. Correct. Reaction torque reverses too, continuing to oppose the motion. Resistive power remains nonnegative.
No: energy integrates power, and voltage times current can stay positive. Correct. For our 8 Ω, 10 rad/s run, the calculated load energy is about 0.1005 J per full turn, despite zero signed voltage-time integral.
Continuous rings keep each coil end connected to its own terminal; a commutator switches connections. Correct. Trace each wire and brush: the uninterrupted rings retain the connections instead of exchanging them each half-turn.
Linked flux first increased and then decreased. Correct. The sign of the change in linked flux can reverse while travel continues in the same direction.
The interval-mean induced voltage. Correct. Faraday’s law gives mean emf = −change in linkage / elapsed time. Endpoints alone do not fix every instantaneous value.
MIT 8.02 TEAL Chapter 10, section 10.4, printed 10-12–10-13. University derivation, not measured data. The source mixes absolute-emf notation with signed AC current; our calculation retains consistent signed current. No MIT figures reproduced.
MIT: Faraday’s law and generatorsMIT 10.4 torque and power derivation. Our signed torque is independently expressed as the z component of μ×B; winding heat is separated from useful load output. Original parameters and geometry are not supplied by MIT.
MIT: torque and power balanceUrs Kafader, maxon Motors as Generators, revision October 2022, sections 1.1–1.3. Its DC-machine constants and operating limits are not transferred to our single-loop AC model; source diagrams are not reused.
maxon: manufacturer generator analysisExperimental Researches in Electricity, Volume 1, first series §§39–40 and 43; Gutenberg’s digitization of the second edition 1849, with preface 1839. Qualitative observations and an outside-coil control, not a sampled waveform.
Faraday: original experimental accountSmithsonian object EM.320572 and exact media NMAH-JN2026-00329. Jaclyn Nash/National Museum of American History; photograph July 12, 2022; Media Usage CC0. Original teaching geometry is not a reconstruction from this photograph.
Smithsonian: Edison identity and photo rightsSmithsonian object EM.318252, media NMAH-JN2026-00328-000001. Jaclyn Nash/National Museum of American History; photograph July 12, 2022; Media Usage CC0. Photograph and catalog do not reveal every hidden electrical connection.
Smithsonian: Westinghouse identity and photo rightsIndependent subject review is pending.
Read the sources and model assumptionsPower a motor. Reverse its direction. Add a load and watch speed, current and turning force find a new balance.
Close a switch. Follow a complete path. Give current a second route and discover what changes.
Interactive explorationTurn a source-shaped blade, change the generator load and follow the energy. Discover why pitch, torque and speed must work together—and compare a model with a real wind-tunnel experiment.