induction | Physics Library

Induction in Physics: Electromagnetic Induction, Faraday’s Law, and Inductance

In Electromagnetism, induction is the production of an electromotive force, and possibly an electric current, through a change in magnetic environment. The change may come from a magnetic field that varies with time, from motion of a Conductor through a magnetic field, or from a combination of both effects.

The central physical principle is Faraday’s law: changing magnetic flux through a circuit is associated with an induced electromotive force. The direction of the induced response is described by Lenz’s law. These ideas underlie electric generators, transformers, induction motors, wireless charging, pickup coils, many sensors, and the behavior of inductors in electrical circuits.

The word induction is also used in other contexts. Electrostatic induction refers to redistribution of Electric Charge caused by an external electric field, while mathematical induction is a method of proof. Unless otherwise stated, this article uses induction to mean electromagnetic induction.

1 Magnetic flux

Electromagnetic induction is most naturally expressed using magnetic flux. Consider an oriented surface S with unit normal n. The vector area element is

dA = ^n dA.

The magnetic flux through the surface is

∫ Φ = B ⋅ dA. B S

For a uniform magnetic field over a flat surface of area A,

ΦB = BA cos𝜃,

where 𝜃 is the angle between B and the chosen surface normal.

The SI unit of magnetic flux is the weber:

1 Wb = 1 T m2.

Figure 1. Magnetic flux measures the component of the magnetic field passing through an oriented surface. The sign depends on the chosen surface normal.

Magnetic flux is not simply “the amount of magnetic field.” It combines field strength, surface area, orientation, and spatial variation. A flux can change because B changes, because the area changes, because the surface rotates, or because the circuit moves into a region where the field is different.

2 Electromotive force

The quantity induced around a closed conducting path is the electromotive force, abbreviated emf and commonly written ℰ. Despite its historical name, emf is not a force. It is energy transferred per unit charge around a circuit:

W ℰ = ---. q

Its SI unit is the volt:

1V = 1J∕C.

For a closed stationary path C, the emf associated with an electric field is

∮ ℰ = E ⋅ dl. C

This expression is a line integral. In electrostatics the electric field is conservative, so the closed-loop integral vanishes. An induced electric field produced by changing magnetic flux is different: it can have a nonzero circulation around a closed path.

3 Faraday’s law

For a fixed conducting loop bounding a surface S, Faraday’s law states

ℰ = − dΦB--. dt

For a coil of N identical turns linked by the same magnetic flux,

ℰ = − N dΦB--. dt

The law says that it is the rate of change of flux, not merely the presence of magnetic field, that produces the induced emf. A large steady magnetic field can produce zero induced emf in a stationary loop, while a much smaller field changing rapidly can produce a substantial emf.

For a uniform field through a flat loop,

ΦB = BA cos𝜃,

so induction can result from changes in any of the quantities B, A, or 𝜃.

4 Lenz’s law and the minus sign

The minus sign in Faraday’s law expresses Lenz’s law: the induced effect acts in a direction that opposes the change in magnetic flux that produced it.

Suppose the magnetic flux through a loop is increasing in one direction. The induced current, if the circuit is closed, creates its own magnetic field in the opposite direction. If the original flux is decreasing, the induced current tends to reinforce the original direction and resist the decrease.

Figure 2. Lenz’s law. The induced current produces magnetic flux that opposes the change in the original flux. The opposition is to the change, not necessarily to the original magnetic field itself.

Lenz’s law is closely related to energy conservation. If the induced current assisted the flux change that produced it, a small change could reinforce itself and generate energy without an external source. Instead, mechanical work or another energy source is required to sustain the change.

5 A simple worked example

Consider a single circular loop of radius

r = 0.10 m

in a uniform magnetic field perpendicular to the loop. Suppose the field increases from

0.20 T

0.80 T

Δt = 0.15 s.

The loop area is

A = πr2 = π(0.10)2 = 3.1416 × 10− 2m2.

Because the field is perpendicular to the loop,

Δ ΦB = AΔB.

Thus

− 2 −2 Δ ΦB = (3.1416 × 10 )(0.60) = 1.885 × 10 Wb.

The magnitude of the average induced emf is therefore

|Δ ΦB | 1.885 × 10−2 |ℰ| = -------= -------------= 0.126 V. Δt 0.15

Hence

|ℰ | ≃ 0.126 V.

The current direction is then determined by Lenz’s law after the direction of the original increasing field is specified.

6 Induction by changing field: transformer emf

A time-varying magnetic field can induce an electric field even when the conducting loop itself does not move. For a fixed contour, Faraday’s law can be written

∮ d ∫ E ⋅ dl = −-- B ⋅ dA. C dt S

This is sometimes called transformer induction. The electric field is not electrostatic: its circulation around a closed path need not vanish.

The local, differential form is the Maxwell-Faraday equation,

∂B-- ∇ × E = − ∂t .

This equation says that a time-varying magnetic field is associated with a circulating electric field. It is one of Maxwell’s Equations and is therefore more general than a circuit-only description.

7 Induction by motion: motional emf

Induction can also occur when a conductor moves through a magnetic field. A charge q moving with velocity v in a magnetic field experiences the magnetic Lorentz force

FB = qv × B.

If a conducting rod of length ℓ moves with speed v perpendicular to a uniform magnetic field, charges in the rod are pushed toward opposite ends. For the simple perpendicular geometry, the resulting motional emf has magnitude

ℰ = Bℓv.

Figure 3. Motional induction. Charges in a moving conductor experience the magnetic part of the Lorentz force, producing charge separation and an emf.

For a moving conducting path, a useful general expression is

∮ ℰ = (E + v × B ) ⋅ dl. C

The E term describes force per unit charge from the electric field, while the v × B term accounts for magnetic force on charges carried with the moving conductor.

The familiar flux rule

dΦB-- ℰ = − dt

can describe both transformer and motional induction when the changing geometry and sign conventions are handled consistently. The underlying force mechanisms, however, need not be identical.

8 Induced electric fields are not electrostatic fields

A static electric field generated by fixed charges satisfies

∮ E ⋅ dl = 0 C

for any closed path in an electrostatic region.

An electric field induced by a changing magnetic field can instead satisfy

∮ E ⋅ dl ⁄= 0. C

Therefore it cannot generally be represented everywhere by a single-valued electrostatic potential V with

E = − ∇V.

This is an important conceptual change from electrostatics. Induction introduces electric fields whose geometry naturally involves circulation and curl.

9 Self-induction

A changing current produces a changing magnetic field. That changing magnetic field can change the flux through the same circuit that carries the current. The resulting effect is called self-induction.

For a coil with N turns, define the flux linkage

λ = N ΦB.

In a linear magnetic system, the flux linkage is proportional to current:

λ = LI,

where L is the self-inductance.

The SI unit of inductance is the henry:

1 H = 1 Wb ∕A = 1 V s∕A.

Faraday’s law then gives the induced emf

ℰ = − L dI- L dt

when L is constant.

The minus sign again represents Lenz’s law: the induced emf opposes the change in current responsible for the changing flux.

In circuit analysis, the passive-sign-convention voltage across an ideal inductor is usually written

dI vL = L --. dt

This is compatible with the induced-emf expression; the apparent sign difference comes from which direction is chosen as the circuit voltage reference.

Figure 4. Self- and mutual-induction chain. Changing current produces changing magnetic field and flux, which in turn produces an induced emf.

10 Mutual induction

A changing current in one circuit can produce changing magnetic flux through another circuit. This is called mutual induction.

For two circuits in a linear system, the flux linkage of circuit 2 due to current I₁ may be written

λ21 = M I1,

where M is the mutual inductance.

The emf induced in circuit 2 is then

ℰ2 = − M dI1- dt

for constant M.

This is the basic physical principle behind a transformer. An alternating current in the primary winding produces changing magnetic flux, and that changing flux induces an emf in the secondary winding.

11 Energy stored in an inductor

Building current in an inductor requires work because the induced emf resists the increase in current. For an ideal linear inductor, the energy stored in its magnetic field is

1 2 UB = 2LI .

This energy can later be returned to the circuit as the current decreases. Thus an inductor is an energy-storage element, just as a capacitor stores electric-field energy.

The energy viewpoint reinforces Lenz’s law: induction resists abrupt changes because changing the current requires changing stored field energy.

12 Induction and generators

A generator converts mechanical work into electrical energy through electromagnetic induction. A loop rotating in a magnetic field has time-varying flux. If

ΦB = BA cos(ωt),

then Faraday’s law gives

ℰ = BA ω sin(ωt)

for a single turn, up to the chosen orientation convention.

A coil with N turns gives

ℰ = N BA ω sin (ωt ).

The mechanical torque required to keep the generator rotating provides the energy that appears electrically. This is another direct manifestation of Lenz’s law and energy conservation.

13 Induction and transformers

In an ideal transformer, both windings link approximately the same time-varying magnetic flux. Faraday’s law gives

dΦB-- ℰ1 = − N1 dt

and

dΦB-- ℰ2 = − N2 dt .

Therefore

ℰ2-= N2-. ℰ1 N1

The voltage ratio is set by the turns ratio in the idealized model. Real transformers additionally involve winding resistance, leakage flux, finite permeability, hysteresis, eddy currents, and frequency-dependent losses.

14 A broader field viewpoint

At an introductory circuit level, induction is often summarized as “changing magnetic flux produces emf.” At a deeper level, the electromagnetic field viewpoint is more fundamental.

For a stationary contour,

∂B ∇ × E = − ---- ∂t

relates local field variation to electric-field circulation.

For moving matter, the Lorentz force introduces the additional magnetic-force term

qv × B.

The complete physical description therefore involves fields, moving charges, geometry, and the choice of circuit path. The flux rule is an extraordinarily useful compact result, but the Maxwell-Lorentz description reveals the local physics behind it.

15 Common misconceptions

A magnetic field by itself does not guarantee induction

A stationary loop in a steady magnetic field can have nonzero flux but zero induced emf. Induction depends on changing flux or motion through the field.

The induced current does not always oppose the magnetic field

It opposes the change in flux. If the original field is decreasing, the induced field can point in the same direction as the original field.

Emf is not a mechanical force

Emf has units of volts, or energy per charge. The name is historical.

Flux is not a substance flowing through the loop

Magnetic flux is a surface integral of the magnetic field. It is a mathematical measure of field crossing an oriented surface.

Motional and transformer induction are related but not identical mechanisms

A moving conductor in a static magnetic field can acquire emf through the magnetic Lorentz force. A stationary loop in a changing magnetic field experiences a circulating induced electric field.

16 Connections to other PhysicsLibrary topics

Electromagnetic induction naturally connects several topics:

magnetic flux and surface integrals;
line integrals and circulation;
the Lorentz force;
Maxwell’s equations;
inductors and RL circuits;
LC oscillations;
mutual inductance and transformers;
electric generators and motors;
electromagnetic energy and the Poynting vector;
electromagnetic waves.

In particular, the Maxwell-Faraday equation provides one half of the feedback structure needed for electromagnetic wave propagation: changing magnetic fields generate circulating electric fields, while the Ampere-Maxwell law relates changing electric fields to magnetic fields.

17 Summary

Electromagnetic induction is the production of emf through a changing magnetic environment. The basic flux definition is

∫ ΦB = B ⋅ dA. S

Faraday’s law gives

dΦ ℰ = − ---B-, dt

and for N turns,

dΦB-- ℰ = − N dt .

Lenz’s law determines the direction of the induced effect: it opposes the change in flux.

For a moving conductor, magnetic Lorentz force contributes through

∮ ℰ = (E + v × B ) ⋅ dl. C

For a stationary contour, Faraday’s law is equivalent to

∂B-- ∇ × E = − ∂t .

Self-induction and mutual induction are summarized by

dI- ℰL = − L dt

and

dI1- ℰ2 = − M dt .

These relations connect basic field physics directly to generators, transformers, inductors, oscillating circuits, and electromagnetic waves.

References

[1] M. Faraday, Experimental Researches in Electricity, Taylor and Francis, collected papers originally published beginning in 1832.

[2] J. C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon Press, 1873.

[3] D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Pearson, 2013.

[4] E. M. Purcell and D. J. Morin, Electricity and Magnetism, 3rd ed., Cambridge University Press, 2013.

[5] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II, Basic Books, New Millennium ed., 2011.

[6] J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999.

"induction" is owned by bloftin.
(view preamble)