# Electromagnetic induction
**Electromagnetic induction** is the production of an electromotive force (emf, a voltage) in a conductor when the magnetic flux through it changes. [Michael Faraday](https://www.edgepedia.org/michael-faraday) performed the first productive experiments on the effect in 1831, pushing a bar magnet into and out of a coil and observing that a current flowed only while the field at the coil was changing; a stationary magnet produced none[1](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law). The phenomenon underlies electric generators, transformers, wireless charging, induction cooktops, and electromagnetic braking.
| Key fact | Detail |
|---|---|
| Discovery | First productive experiments on time-varying magnetic fields performed by Michael Faraday in 1831; Joseph Henry (1799–1878) discovered induced emf independently[1](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) • [2](https://openstax.org/books/university-physics-volume-2/pages/14-1-mutual-inductance) |
| Faraday's law | The induced emf equals the negative rate of change of magnetic flux: \( \epsilon = -\dfrac{d\Phi_{\mathrm{m}}}{dt} \); for a coil of \( N \) turns, \( \epsilon = -N\,d\Phi_{\mathrm{m}}/dt \)[1](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) |
| Flux and its unit | Magnetic flux \( \Phi_{\mathrm{m}} = \int_S \vec{B} \cdot \hat{n}\,dA \), measured in webers; 1 Wb = 1 T·m²[1](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) |
| Direction rule | Lenz's law, named for Heinrich Lenz (1804–1865): the induced current opposes the change in flux that produces it[3](https://openstax.org/books/college-physics-2e/pages/23-2-faradays-law-of-induction-lenzs-law) |
| Transformer relation | \( v_{\mathrm{S}} = (N_{\mathrm{S}}/N_{\mathrm{P}})\,v_{\mathrm{P}} \); stepping voltage up steps current down in the same ratio[4](https://openstax.org/books/university-physics-volume-2/pages/15-6-transformers) |
| Generator output | A coil of \( N \) turns and area \( A \) rotating at angular velocity \( \omega \) in field \( B \) produces \( \epsilon = NBA\omega \sin(\omega t) \)[5](https://openstax.org/books/university-physics-volume-2/pages/13-6-electric-generators-and-back-emf) |
| Eddy-current control | Slots or thin insulated laminations break the current loops and greatly reduce eddy-current heating in motor and transformer cores[5](https://openstax.org/books/university-physics-volume-2/pages/13-6-electric-generators-and-back-emf) • [6](https://openstax.org/books/university-physics-volume-2/pages/13-5-eddy-currents) |
## Faraday's law
Faraday's experiments showed that the induced emf depends on only a few factors: it is directly proportional to the change in flux \( \Delta\Phi \), inversely proportional to the time \( \Delta t \) over which that change occurs, and multiplied by \( N \) for a coil of \( N \) tightly wound turns, each of which experiences the same flux[1](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law) • [3](https://openstax.org/books/college-physics-2e/pages/23-2-faradays-law-of-induction-lenzs-law). In integral form,
\[ \epsilon = -\frac{d}{dt}\int_S \vec{B} \cdot \hat{n}\,dA = -\frac{d\Phi_{\mathrm{m}}}{dt}, \]
where \( S \) is any open surface bounded by the circuit; the flux is the same for all such surfaces[1](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law). The flux can change because the field strength changes, because the circuit moves or changes orientation in the field, or because the current in a neighboring circuit changes; it is the relative motion and the change that matter, not which element moves[1](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law).
A worked example shows the scale: a 200-turn square coil of side 0.25 m in a field decreasing at 0.040 T/s develops an emf of 0.50 V and, with a 5.0 Ω resistance, a circulating current of 0.10 A[1](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law).
## Lenz's law and energy conservation
The minus sign in Faraday's law carries physical content. It means the emf drives a current whose own magnetic field opposes the change in flux, a rule stated so clearly by Heinrich Lenz that it bears his name; Faraday was aware of the direction, but Lenz is credited for its formulation[3](https://openstax.org/books/college-physics-2e/pages/23-2-faradays-law-of-induction-lenzs-law). Induction opposes any change in flux: thrust a magnet's north pole into a coil and the induced current makes the coil face a north pole that repels it; withdraw the magnet and the induced current attracts it[3](https://openstax.org/books/college-physics-2e/pages/23-2-faradays-law-of-induction-lenzs-law).
This opposition is what makes induction consistent with energy conservation. In Faraday's original two-coil experiment, closing the switch in the first loop produced a transient current in the second loop in the opposite direction, trying to keep the flux through it at zero; opening the switch produced a transient in the same direction, trying to keep the flux constant. In a real loop the induced current decays exponentially with a time constant \( \tau = L/R \) set by the loop's self-inductance and resistance[7](https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/097c0d0ffd19513067e485f2b4d50ef7_MITRES_6_002S08_chapter6.pdf). If the induced current aided the change instead of opposing it, a small initial change would amplify itself and create energy from nothing.
## Inductance
Induction between circuits is quantified by **mutual inductance** \( M \), defined for two coils as \( M = N_2\Phi_{21}/I_1 = N_1\Phi_{12}/I_2 \). Like capacitance, it is a geometric quantity, depending on the shapes and relative positions of the coils and not on the currents. Its SI unit is the henry (H), 1 H = 1 V·s/A, named for [Joseph Henry](https://www.edgepedia.org/joseph-henry), the American scientist who discovered induced emf independently of Faraday[2](https://openstax.org/books/university-physics-volume-2/pages/14-1-mutual-inductance). Combining the definition with Faraday's law gives the induced emf in coil 2 as \( \epsilon_2 = -M\,dI_1/dt \)[2](https://openstax.org/books/university-physics-volume-2/pages/14-1-mutual-inductance).
A large mutual inductance is desirable in a transformer but not everywhere. The heating coils of an electric clothes dryer can induce a dangerous emf on the metal case, so the coils are counter-wound so their fields cancel; digital circuits likewise counter-wind to prevent induced emfs from flipping 1s and 0s in neighboring conductors[2](https://openstax.org/books/university-physics-volume-2/pages/14-1-mutual-inductance).
## Transformers and generators
A transformer consists of two windings on a soft iron core. The alternating primary current creates a changing shared flux \( \Phi \), and Faraday's law gives \( v_{\mathrm{P}} = -N_{\mathrm{P}}\,d\Phi/dt \) and \( v_{\mathrm{S}} = -N_{\mathrm{S}}\,d\Phi/dt \), so the voltage ratio equals the turns ratio, \( V_{\mathrm{S}}/V_{\mathrm{P}} = N_{\mathrm{S}}/N_{\mathrm{P}} \). [Energy conservation](https://www.edgepedia.org/energy-conservation) fixes the current in the inverse ratio, \( i_{\mathrm{S}} = (N_{\mathrm{P}}/N_{\mathrm{S}})\,i_{\mathrm{P}} \)[4](https://openstax.org/books/university-physics-volume-2/pages/15-6-transformers). A utility-pole transformer stepping 12 kV down to 240 V has a 1:50 turns ratio and multiplies the input current fiftyfold[4](https://openstax.org/books/university-physics-volume-2/pages/15-6-transformers).
Generators invert the arrangement: a coil of \( N \) turns and area \( A \) rotating at constant angular velocity \( \omega \) in a uniform field \( B \) produces a sinusoidal emf \( \epsilon = NBA\omega \sin(\omega t) \)[5](https://openstax.org/books/university-physics-volume-2/pages/13-6-electric-generators-and-back-emf). The same physics runs a motor in reverse. When a motor's coil turns, the changing flux induces a **back emf** that opposes the driving voltage; the motor draws maximum current at startup, when the back emf is zero, and less current at speed. A motor with 0.400 Ω coils driven at 48.0 V draws 120 A and dissipates 5.76 kW at startup, but only 20 A and 160 W at operating speed with a 40.0 V back emf[5](https://openstax.org/books/university-physics-volume-2/pages/13-6-electric-generators-and-back-emf).
## Eddy currents
When a bulk conductor experiences a changing flux, induced currents circulate through its volume in swirling loops called **eddy currents**, named by analogy with water eddies[6](https://openstax.org/books/university-physics-volume-2/pages/13-5-eddy-currents) • [8](https://www.khanacademy.org/science/electromagnetism/x4352f0cb3cc997f5%3Ahow-wireless-charging-and-transformers-work/x4352f0cb3cc997f5%3Ahow-induction-cooktops-work/v/eddy-currents-their-applications-how-to-reduce-them). They produce useful effects and unwanted losses.
**Magnetic damping and braking.** A metal pendulum bob swinging between magnet poles is quickly slowed, because the [Lorentz force](https://www.edgepedia.org/lorentz-force) on the eddy currents opposes the motion on both entry and exit; a slotted plate swings much longer, since the slots limit the size of the current loops and adjacent loops carry opposing currents that cancel[6](https://openstax.org/books/university-physics-volume-2/pages/13-5-eddy-currents). The same principle brakes roller coasters, whose copper fins pass rows of rare-earth magnets, and it separates metals from other trash at recycling centers, where conductors are slowed by magnetic drag while nonmetals move on; the method works for all metals, not just ferromagnetic ones[6](https://openstax.org/books/university-physics-volume-2/pages/13-5-eddy-currents).
**Induction heating and cooktops.** An induction cooktop's electromagnets vary the field rapidly, driving eddy currents in a ferromagnetic pot base and heating the pot directly; the cooktop surface itself stays cool because it is a poor conductor in which only negligible eddy currents arise[6](https://openstax.org/books/university-physics-volume-2/pages/13-5-eddy-currents) • [8](https://www.khanacademy.org/science/electromagnetism/x4352f0cb3cc997f5%3Ahow-wireless-charging-and-transformers-work/x4352f0cb3cc997f5%3Ahow-induction-cooktops-work/v/eddy-currents-their-applications-how-to-reduce-them). A lecture demonstration transfers power through a single-turn secondary: a 250-turn primary coil induces enough current in a lead wire loop seated on an iron ring to melt it in about 30 seconds[9](https://physicslabs.colorado.edu/demos/electricity-and-magnetism/electromagnetic-induction/transformers/induction-heating/).
**Levitation.** An aluminum ring above an alternating-current electromagnet carries eddy currents whose field, by [Lenz's law](https://www.edgepedia.org/lenzs-law), opposes the coil's field; when the repulsion exceeds gravity the ring levitates, the principle behind maglev trains[8](https://www.khanacademy.org/science/electromagnetism/x4352f0cb3cc997f5%3Ahow-wireless-charging-and-transformers-work/x4352f0cb3cc997f5%3Ahow-induction-cooktops-work/v/eddy-currents-their-applications-how-to-reduce-them).
**Losses.** Where eddy currents are not wanted, they are suppressed by increasing the effective resistance: motor and transformer cores are built from thin, electrically insulated sheets, which cut the resistive heating considerably while hardly affecting the magnetic properties[5](https://openstax.org/books/university-physics-volume-2/pages/13-6-electric-generators-and-back-emf) • [6](https://openstax.org/books/university-physics-volume-2/pages/13-5-eddy-currents).
## Place in electromagnetic theory
Faraday's law became one of [Maxwell's equations](https://www.edgepedia.org/maxwells-equations), describing how changing magnetic fields produce electric fields; paired with Maxwell's displacement current, which describes the reverse, it supplies the symmetry from which electromagnetic waves follow[10](https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves). [Heinrich Hertz](https://www.edgepedia.org/heinrich-hertz) exploited induction in his 1887 experiments, using a resonant [RLC circuit](https://www.edgepedia.org/rlc-circuit) whose inductor formed a transformer primary; high voltages induced across a spark gap in the secondary loop generated the waves, and a tuned receiving loop across the laboratory produced induced sparks that detected them[10](https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves). Consumer wireless charging pads work the same way in miniature: a coil carrying alternating current induces an emf in a coil in the smartphone, charging its battery[11](https://openstax.org/books/university-physics-volume-2/pages/14-introduction).
## References
1. [Faraday's Law, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law)
2. [Mutual Inductance, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/14-1-mutual-inductance)
3. [Faraday's Law of Induction: Lenz's Law, College Physics 2e, OpenStax](https://openstax.org/books/college-physics-2e/pages/23-2-faradays-law-of-induction-lenzs-law)
4. [Transformers, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/15-6-transformers)
5. [Electric Generators and Back Emf, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/13-6-electric-generators-and-back-emf)
6. [Eddy Currents, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/13-5-eddy-currents)
7. [Electromagnetic Field Theory: A Problem-Solving Approach, Chapter 6, MIT OpenCourseWare](https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/097c0d0ffd19513067e485f2b4d50ef7_MITRES_6_002S08_chapter6.pdf)
8. [Eddy currents and their applications, Khan Academy](https://www.khanacademy.org/science/electromagnetism/x4352f0cb3cc997f5%3Ahow-wireless-charging-and-transformers-work/x4352f0cb3cc997f5%3Ahow-induction-cooktops-work/v/eddy-currents-their-applications-how-to-reduce-them)
9. [Induction Heating demonstration, Physics Undergraduate Labs, University of Colorado Boulder](https://physicslabs.colorado.edu/demos/electricity-and-magnetism/electromagnetic-induction/transformers/induction-heating/)
10. [Maxwell's Equations and Electromagnetic Waves, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves)
11. [Ch. 14 Introduction, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/14-introduction)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Electromagnetic induction and time-varying fields*
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