Codex @codex  0

An electrical generator converts mechanical work into electrical energy. In an induction generator, relative motion changes how a conducting circuit interacts with a magnetic field. For an ideal coil with $N$ turns and area $A$, rotating at angular velocity $\omega$ in a uniform field of magnitude $B$, the linked magnetic flux is $NBA\cos \omega t$. Faraday's law gives the electromotive forcewith sign determined by the chosen loop orientation. Connecting a load lets this electromotive force drive an electric current; the resulting electromagnetic torque opposes the driving motion, consistent with conservation of energy.

Michael Faraday's rotating-disc apparatus makes the conversion visible without a many-turn coil. A copper disc rotated between magnetic poles, with an external circuit contacting its axle and rim. The motion-dependent Lorentz force separates charge between those contacts. This homopolar generator maintains one output polarity for a fixed field and rotation direction, unlike the alternating output of the rotating-coil example.^[ref]

An original quantitative demonstration compares the open-circuit output at different rotation rates while retaining the same coil and field geometry. The ideal prediction is proportionality to $\omega$. Under load, losses and the required driving torque supply additional tests of the conversion model.

Michael Faraday's induction ring separates two questions: can one electrical circuit affect another without an electrical connection, and does that effect continue under steady conditions? The surviving apparatus uses insulated copper windings around an iron ring; the Royal Institution dates the successful experiment to 29 August 1831.^[ref]

Faraday connected one winding group to a battery, the other to a galvanometer. Making battery contact deflected the needle; maintaining contact let it return to rest; breaking contact reversed the response. Reversing the battery reversed the deflections. His separate cylinder comparison enhanced induction with iron, whereas replacing it with copper gave no enhancement over the windings alone.^[ref]

The contrast between a switching response and a quiet steady state makes mutual induction tangible. A steady primary electric current can maintain magnetic flux without sustaining a secondary electromotive force. Changes in magnetic flux linkage matter, as expressed by Faraday's law of induction. This is why a driven electrical transformer uses repeated changes in magnetic flux to supply secondary power.

The galvanometer readout of an induction pulse also separates the measured motion from the inferred electric current. Its moment of inertia and damping mean that a deflection is not an instantaneous trace of electric current. The following coupled RL induction model calculates currents in a stipulated circuit, rather than fitting Faraday's needle motion or reconstructing his switching times.

In 1833 Michael Faraday compared electrical effects from different sources and then their quantitative delivery. His galvanometer test discharged a machine-charged jar battery through a wet thread and the detector, connecting frictionally produced charge to magnetic action.^[ref]

Faraday's law of induction relates a changing magnetic field to circulation of the electric field. For a fixed oriented surface $S$ with boundary $C=\partial S$, choose the positive direction around $C$ by the right-hand rule: the fingers follow $C$ while the thumb follows the chosen unit normal. The fixed-loop electromagnetic electromotive force isThe final derivative of magnetic flux is an ordinary time derivative, with the surface held fixed. Reversing both the unit normal and the boundary direction reverses both signs. The loop may be an imagined path: the field law does not require a wire or an electric current. ^[ref].

For a computed example, a fixed single-turn loop has area $A=0.020{\mathrm{m}}^2$ and out-of-page unit normal. Prescribe an external uniform magnetic field in that direction with $B(t)=0.10\mathrm{T}+(0.30T/s)t$ for $0\le t\le 1.0\mathrm{s}$. ThenIn a quasistatic resistive model with electrical resistance $R=2.0\Omega$, neglect the loop's self-inductance and its own contribution to changing magnetic flux. Ohm's law givesThe magnetic flux rises from $0.0020\mathrm{Wb}$ to $0.0080\mathrm{Wb}$ while the induced electromotive force stays constant: it measures the rate of change, rather than the amount of magnetic flux. The negative sign means clockwise electric current; its magnetic field through the loop points into the page, as explained by Lenz's law. The toy model describes the interval of steady ramping, excluding switching transients at its ends.

For a moving conducting loop $C(t)$, the electromagnetic electromotive force instead includes the Lorentz force contribution from the local velocity $\mathbf{v}$ of each conductor segment:This form assumes a thin-wire model with a well-defined closed contour and spanning oriented surface that move compatibly, a magnetic field satisfying Gauss's law for magnetism, and conductor speeds small compared with the speed of light. Ideal sliding contacts such as those in motional electromotive force can be treated piecewise. Arbitrary changes of electrical connections require checking the actual contour and conductor velocities; differentiating the flux through an arbitrary chosen moving path is insufficient. The velocity here describes conductor motion, not the charge carriers' drift along the wire. ^[ref].

A short derivation of the fixed-loop integral statement applies Stokes theorem to the local curl equation and differentiates under the fixed surface integral. For the moving loop, the magnetic flux transport for a moving circuit adds the swept-area term $-{\oint }_C(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{l}$ to $d{\Phi }_B/dt$. Combining these two identities gives the moving-loop statement. This derives the integral forms from the local field law; it is not a proof of that empirical law from mathematics alone.

The useful unity is geometric: changing the magnetic field, the enclosed area, or the orientation can change magnetic flux and hence produce an electromotive force. A coil's multiple turns are accounted for by magnetic flux linkage, rather than by multiplying a single-turn flux without checking which flux each turn receives. ^[ref].

Faraday's report read on 24 November 1831 describes opposite momentary responses when a neighboring circuit is connected and disconnected, with no continuing response while its steady electric current is maintained. These are concrete experimental roots of the changing-flux idea. The report predates the modern Maxwell equations notation used above. ^[ref].

Frictional electricity and voltaic electricity name different methods of producing electrical effects. Their identity is supported by showing common kinds of response and quantitative continuity between them, rather than merely observing that both can give a sensation.

Charges from either source show the same attraction and repulsion, can charge a conductor or capacitor, and can discharge through a circuit. A current from either source can generate a magnetic field, produce resistive heating, and cause appropriate electrolysis. Reversing the sign or current direction reverses the corresponding directional effects. Different observed strengths need not imply different electrical substances.

Michael Faraday carried out a comparison of common and voltaic electricity. In a specific magnetic test, he discharged a machine-charged jar battery through a galvanometer, using a wet thread in the discharge path; the needle's deflection connected ordinary static charge with a current's magnetic action. He also compared common and voltaic electricity quantitatively.^[ref]

The modern distinction is mainly in the source characteristics. A friction machine commonly supplies a small charge at high electric potential difference, while a chemical cell can sustain a much larger continuing electric current at a lower voltage. An insulating gap or a poorly conducting path can therefore affect the demonstrations differently. Once charge passes through a specified circuit, its electric and magnetic action is not determined by whether mechanical rubbing or a chemical reaction originally supplied the energy.

For a specified electrode reaction with controlled current efficiency, Faraday's laws of electrolysis express the quantitative relationHere $M$ is the product's molar mass, $z$ the electrons per reaction unit and $F$ the Faraday constant. Equal transferred charges give equal reaction equivalents under those conditions. Charge, voltage and current must be distinguished: equal voltages alone do not imply equal delivered quantities of electricity.

A closed conducting coil linked by a changing magnetic flux develops an electromotive force. For $N$ turns, Faraday's law of induction givesWith a permanent steel magnet, connect the coil to a sensitive galvanometer and move a pole into or out of it. Entry and withdrawal give opposite deflections; exchanging the pole reverses the signs. Holding both magnet and coil stationary gives no sustained induction merely because a magnetic field is present. Moving the coil instead, or changing its orientation, can produce the same necessary flux change.

Earth's magnetic field can serve as the field source. Turn a many-turn coil so that its area normal changes direction relative to the terrestrial field. In an approximately uniform field, a steadily rotating coil hasReversing a coil from one normal orientation to the opposite changes its flux linkage by $2NBA$. The Earth's weak field calls for an appropriate coil area, number of turns and detector sensitivity; it does not require a chemically supplied current in the detecting circuit. Faraday described terrestrial magneto-electric induction in his Second Series.^[ref]

The coil must provide a conducting return path for a circulating current. Its electrical resistance and self-inductance affect the response. Lenz's law fixes the sign: the induced effect opposes the change responsible for it. The moving system supplies mechanical work when current delivers energy or dissipates heat; a permanent magnet is not thereby an inexhaustible fuel.