Electromagnetic induction and inductance

A changing linkage is a physical question

A magnet held still beside an ordinary stationary coil need not drive a current. Moving the magnet, moving the coil or changing a source current can produce an emf. These observations connect the magnetic field to electric circulation and mechanical motion. The useful question is not simply whether a strong field is present, but which field or circuit quantity changes and what supplies the energy.

This lesson uses modern SI throughout: magnetic flux Φ is measured in weber, emf in volt, flux linkage λ in weber-turn and inductance in henry. A turn count is dimensionless, so weber-turn records a winding convention rather than a new SI dimension. Positive emf means circulation in the chosen direction; it does not mean that every endpoint voltage has that sign.

Four original models keep the assumptions visible: a moving rectangular material loop in a changing field, a stationary circular contour around a long ideal solenoid, a reciprocal two-coil inductance matrix and a reversible nonlinear coil. Each model answers a different question. An imposed field, an induced circuit current, a terminal voltage and stored energy are related quantities, not interchangeable labels.

Choose the first and advanced paths

For a first reading, fix an oriented contour, work out the sign of its flux change and separate the electric and motional contributions to emf. Then connect a winding’s flux linkage to current and calculate a linear inductor’s stored energy. The sliding-rod example closes the mechanical and electrical power balance.

Return to the advanced sections for a transported-surface derivation, contour-dependent voltage measurements, reciprocity and positive energy, reversible nonlinear inductance, coenergy and quasistatic limits. These refinements explain when the familiar shortcuts apply. They do not replace Faraday’s law with a different law every time the apparatus changes.

Keep the preceding distinctions between B and H, free currents and material response, and fixed-moment energy versus source work. Here B is the field entering flux. A coil containing a magnetic material can have a linkage that depends on its constitutive state and history; geometry alone does not guarantee a constant inductance.

Orient a contour and its surface

Choose a closed contour C and a spanning surface S. A right-hand normal n̂ fixes the positive direction around C. Define Φ_B = ∫_S B·n̂ dA. Reversing the contour and normal reverses the signed flux and the signed emf together. A physical prediction cannot depend on an unannounced reversal of one sign convention.

For a uniform field and planar surface, Φ_B = BA cosθ, where θ is the angle between B and the chosen normal. The angle to the plane is the complementary angle. The shortcut also assumes that the relevant B is uniform over the surface; it cannot replace a nonuniform integral merely because one field sample is available.

The absence of magnetic charge makes flux through any closed surface zero. Consequently, two appropriate surfaces with the same oriented boundary give the same flux when the field and domain are defined on the region between them. This is why a flux calculation can use a convenient spanning surface. It is not a claim that flux is a substance flowing through the conductor.

Keep the units and signs together

An emf is an oriented integral of force per unit charge along a specified circuit at an instant. Its name does not make it a force measured in newton. In the electromagnetic part of a thin moving-wire model, ℰ = ∮_C(E + u_w × B)·dl, where u_w is the local velocity of the wire or material boundary used in that model. Chemical sources, if present, require their own non-electromagnetic contribution.

The carrier’s full laboratory velocity includes its drift relative to the wire. For a thin wire the drift is taken tangent to the wire, so its own v_d × B has no tangential component in this integral. This explains the appearance of u_w in the motional term. Extended conductors, moving contacts and distributed currents can require a more detailed choice of path and local velocities.

Do not interpret this instantaneous circuit integral as the work along an arbitrary charge trajectory completed over a long time in a changing field. The fields and apparatus can change during that journey. An instantaneous emf, a two-terminal reading and the net mechanical work of a full experiment have different definitions.

Start with a fixed curve

For a fixed contour and fixed spanning surface, Faraday’s law is ∮_C E·dl = −d/dt ∫_S B·n̂ dA. The local differential statement is ∇ × E = −∂B/∂t. Stokes’ theorem connects them when the fields and surfaces have the required regularity; ideal sheet or impulsive limits need their integral or distributional interpretation.

An electric field with this nonzero circulation cannot be represented everywhere in that region by one single-valued electrostatic potential alone. Surface charges may add a conservative component, so Faraday’s law fixes the curl rather than every component of E. Boundary data, charge conservation and material response supply the remaining information.

The law does not require a physical wire on C. An imaginary stationary circular contour in empty space still has a well-defined electric circulation. A wire is needed to turn the field into a particular conduction-current problem; the presence of that wire can also alter the fields and source response.

Derive the circular induced field

Take a long ideal solenoid with axial field B_z(t) approximately uniform for r < a and negligible B outside. In a bounded observation region, assume axial symmetry and a magnetoquasistatic regime. Around a stationary circle of radius r, the induced azimuthal field has the same E_φ at each point. Faraday’s law then reads 2πr E_φ = −dΦ_B/dt.

For r < a, Φ_B = πr²B_z, so E_φ = −r dotB_z/2. For r > a, the enclosed flux is πa²B_z, giving E_φ = −a² dotB_z/(2r). A growing positive-z field therefore gives negative azimuthal electric circulation when positive azimuth means counterclockwise viewed from +z. No charge sign has been used to reverse the contour convention.

E_φ is regular and zero at the centre. Both one-sided limits at r = a agree, although the radial derivative changes in this idealized field profile. Outside the field-filled cylinder, E can be nonzero even though local ∂B/∂t is negligible. Its nonzero circulation encloses changing flux; a curl-free local exterior field need not have a single-valued scalar potential around a linked hole.

Read an exterior circulation

Set ρ = r/a and dotB_z = B₀/T₀ > 0. The electric-field scale is aB₀/T₀. Our table follows the sign of E_φ, rather than plotting only its positive magnitude. At ρ = 1/2 and ρ = 2 the normalized fields happen to have the same value, but the two circles have different circumferences and enclosed fluxes.

This is a local long-solenoid approximation. An infinite solenoid with arbitrary time dependence is not a complete radiating solution of Maxwell’s equations. Require characteristic dimensions small compared with cT₀, remain away from finite-coil ends and restore displacement-current, propagation and source details when those assumptions fail. The curve ends at ρ = 3 rather than claiming validity to infinite radius.

Let the material contour move

For a smooth moving material loop and a compatible moving spanning surface, combine Faraday’s electric circulation with the motional term. The result is ℰ = ∮(E + u_w × B)·dl = −dΦ_B/dt, where the derivative follows that same loop. The total derivative includes both an explicit change in B and a change in the surface’s position, shape or orientation.

If a magnetic field is steady, the electric contribution around a stationary closed contour vanishes, while a moving rod can still have a motional contribution. If a loop is fixed and B changes, the motional term vanishes while the electric circulation need not. An apparatus that does both can require both terms at once.

The transport identity assumes that the moving contour and its velocities describe the circuit being analyzed. A rotating conducting disk with sliding contacts is not one fixed set of charges following one closed material boundary. Integrating the local Lorentz-force contribution on its actual conductor and return path is safer than differentiating an arbitrarily drawn area. Faraday’s field equation remains valid even when a careless flux shortcut fails.

Differentiate every changing factor

In our rectangular loop, positive C is counterclockwise with normal +z. The left edge and rails are fixed, the right bar moves along x, and its height is ℓ = 2L₀/3. Write u = t/T₀, x_right/L₀ = 1 + u/3 + u²/12 and B_z/B₀ = 1 + u/2 for 0 ≤ u ≤ 2. The fields are prescribed locally; no induced-current law has yet been imposed.

The flux is B_z ℓ x_right. Its derivative gives ℰ = −ℓx_right dotB_z −B_zℓ dotx_right. The first term is the electric circulation and the second is the motional contribution. Each has units volt. In particular, substituting B at the current time into −Bℓv and calling it the whole emf misses the field-change term.

One compatible local electric field is E = (dotB_z y/2, −dotB_z x/2, 0). Its curl is −dotB_z ẑ and its loop integral is −dotB_zℓx_right. On the right bar, u_w × B points along −y; integrating along that bar’s positive +y direction gives −B_zℓ dotx_right. These independent contour calculations explain the signs in the product derivative.

Compare the two contributions

Normalize flux by B₀L₀² and emf by B₀L₀²/T₀. The two contributions are separately negative throughout the chosen interval. That is a result of this geometry and time dependence, not a rule that transformer and motional emfs must always reinforce one another. Different signed changes can partially cancel.

The table specifies the exact rational controls used for the original curves. It predicts emf under the stated prescribed fields. Computing current requires resistance, self-inductance, source loading and an initial state as appropriate. The manuscript does not quietly insert ℰ/R into every time-dependent induction problem.

Trace where the energy comes from

A sliding rod in a steady uniform B provides a direct energy check. Let the positive contour traverse the right rod along +y, let the rod move along +x and let B point along +z. The motional emf is −Bℓv. In a purely resistive quasistatic approximation with negligible self-inductance, the signed current is I = −Bℓv/R.

The magnetic force on the rod is F_x = IℓB = −B²ℓ²v/R. Maintaining the speed needs an external positive-x force. Its power is −F_xv = B²ℓ²v²/R = I²R. The apparatus converts the translating agent’s work into resistive heating. There is no source of free electrical energy hidden in the static magnet.

The magnetic Lorentz force on a point charge is perpendicular to that charge’s full velocity, so its instantaneous magnetic work is zero. In the rod, the mechanical constraint and the electric field exchange energy among the conductor, charges and external agent. The circuit power balance does not contradict that local perpendicular-force fact. With appreciable self-inductance or changing external sources, include storage and source terms before identifying mechanical input with heating.

Apply Lenz’s sign with the right state

The minus sign in Faraday’s law constrains the induced emf relative to the signed flux change. In a simple resistive loop, the resulting current produces flux opposing that change. If positive flux is decreasing, a positive induced emf can drive a current that supports the existing positive flux. Saying that induction always opposes the existing field gives the wrong direction in that case.

In a circuit with stored magnetic energy, current is a dynamical state and need not instantly have the sign of the newly imposed external emf. An earlier current can persist or feed energy back into a source. Use the full circuit equation and initial conditions, with Lenz’s sign in its induced-emf term, rather than replacing dynamics by a verbal direction rule.

Eddy currents in a bulk conductor form distributed loops and can produce drag and heat. Laminations or slots can interrupt some paths, but their effect depends on orientation, frequency and geometry. A quasistatic damping example is useful; it is not a universal statement that every moving conductor always experiences a force opposite to every component of its velocity.

Sum the linkage before defining inductance

For a winding, define signed flux linkage λ = Σ_k Φ_k with every turn’s orientation tied to a declared current and terminal convention. The shortcut λ = NΦ assumes equal signed flux through all N turns. Fringing, leakage and a nonuniform winding can make the individual linkages different without invalidating their sum.

For fixed geometry and a linear, reversible response with no permanent bias, λ = LI. Then L has units Wb/A, or henry. A long solenoid with N turns, area A, length l and approximately uniform linear permeability has L ≈ μN²A/l when end effects and winding details are negligible. The square in N comes from a source field proportional to N and a linkage sum over N receiving turns.

A finite wire has internal as well as external magnetic storage depending on its current distribution. An ideal zero-radius filament’s self-inductance is singular without a cutoff or return-path specification. A quoted inductance therefore belongs to a complete specified geometry, material regime and frequency range; it is not an isolated length of abstract wire with its surroundings erased.

Separate induced emf from terminal drop

With a fixed winding orientation, its self-induced circulation is ℰ_self = −dλ/dt. For an ideal lumped inductor using the passive terminal convention, current enters the terminal labeled positive and the associated voltage drop is v_L = +dλ/dt. For constant L, v_L = L dI/dt. The different signs describe two roles of the same induction term within a declared circuit convention.

A real winding also has resistance and can link an externally imposed flux. With consistent orientations, a useful quasistatic terminal description is v = RI + dλ_total/dt, where λ_total contains the specified self and external contributions. Do not count the same external induction once inside λ_total and again as an added source.

An ideal voltage impulse can change current through an ideal constant-L inductor by ΔI = (1/L)∫v_L dt. Under finite bounded voltage, its current is continuous. Opening a current-carrying real coil can create a large voltage until a capacitance, arc, protection path or another mechanism carries the energy. The ideal open-circuit limit alone does not specify a finite peak voltage.

Couple two windings with declared ports

In a fixed linear two-coil model, λ₁ = L₁I₁ + MI₂ and λ₂ = MI₁ + L₂I₂ when reciprocity and the chosen orientations permit one signed M. The induced emfs are the negative time derivatives of these linkages. A changing current in coil 1 can therefore affect coil 2 even without a conducting connection between them.

Reversing the reference orientation of coil 2 changes the signs of I₂, λ₂ and M consistently. It leaves the physical energy invariant. The dot convention used in circuit diagrams is a compact way of tracking such references; it cannot be inferred from an unlabeled pair of arrows or from the word transformer alone.

For our original model, L/L★ = [[4, 3/2], [3/2, 9/4]], with L★ > 0. The determinant is 27L★²/4 and the coupling magnitude is |M|/√(L₁L₂) = 1/2. This is an illustrative passive reciprocal matrix. A measured device additionally has winding resistance, leakage, frequency dependence and possible core nonlinearity. For the separate illustrative current path I₁/I★ = u and I₂/I★ = (1−u)/2, define u = t/T★ with T★ > 0 in seconds. Its port-power curves are normalized by L★I★²/T★ in watts; they represent I_j dλ_j/dt, rather than a physical power inferred from a dimensionless parameter alone.

Derive linear energy and its cross term

At fixed geometry in a reversible lossless model, electrical power supplied through the ports is Σ_j I_j dλ_j/dt. Integrating from the zero-current reference for a symmetric constant L gives W_mag = ½IᵀLI. With two coils, W_mag = ½L₁I₁² + MI₁I₂ + ½L₂I₂². The mutual term has no extra factor of one half after both symmetric off-diagonal entries are included.

The cross term can be negative for one current orientation while the total stored energy remains nonnegative. For a passive lossless linear two-port, require L₁ ≥ 0, L₂ ≥ 0 and M² ≤ L₁L₂. With strictly positive diagonal terms and determinant, the energy is positive for every nonzero current vector. A signed mutual coefficient is compatible with these conditions; an excessive magnitude is not.

The equality |M| = √(L₁L₂) gives a singular ideal matrix with a zero-energy current combination. It can be a useful perfect-coupling limit, but does not imply two independent strictly positive energy modes. Real leakage usually removes that degeneracy. When the assumptions of a passive reciprocal lumped model fail, its simple energy matrix must not be imposed without justification.

Advanced return: transport the surface

For a smoothly transported oriented surface with ∇·B = 0, dΦ_B/dt = ∫_S ∂B/∂t·n̂ dA −∮_C(u_w × B)·dl. The boundary term accounts for swept area. Combining this with ∮E·dl = −∫_S ∂B/∂t·n̂ dA yields ∮(E + u_w × B)·dl = −dΦ_B/dt. The signs follow from a single common orientation.

A small outward displacement of the right edge of our +z rectangle sweeps positive area and increases flux in positive B_z. On that edge u_w × B points along −y while dl points along +y, so its integral is negative. The minus sign on the transport boundary term therefore makes the flux derivative positive. This geometric check is a useful guard against a remembered sign error.

A slip-contact apparatus may not supply a single smooth material contour to transport. A closed circuit can still be analyzed by explicit segment velocities and electric fields, but a changing drawn area is not automatically the surface swept by those segments. Also distinguish the velocity of a mathematical contour from the conductor velocity in the force law; they coincide only in the model that identifies them.

Advanced return: measurement paths and gauge

Two measurement lead paths between the same locations can surround different time-varying magnetic flux. Their electric line integrals can then differ. A reported voltage in an induction experiment needs a specified lead arrangement and instrument model; treating every pair of locations as if it had one electrostatic potential difference can conceal the apparatus dependence.

Faraday’s law remains valid while surface charges adjust to conductor boundaries. Splitting E into a conservative part and a rotational part is often useful for solving the field problem, but the split must respect those boundaries. The electric field outside an ideal solenoid is a particularly clear example of how local curl and global circulation can convey different information.

Writing E = −∇φ −∂A/∂t and B = ∇ × A is consistent with these facts. A gauge change redistributes the two potential contributions while preserving the physical E, B and measured circuit result when treated consistently. Full gauge choices and Maxwell-potential equations belong to the next lesson; induction here gives a concrete reason that a scalar electrostatic potential alone is insufficient.

Advanced return: reciprocity and positive energy

If a lossless reversible multiport has a twice-differentiable energy W_mag(λ,x) at fixed geometry x, then I_j = ∂W_mag/∂λ_j. Its Legendre transform W′(I,x) = Σ_j I_jλ_j −W_mag gives λ_j = ∂W′/∂I_j where the transform is well defined. Equality of mixed derivatives then gives ∂λ₁/∂I₂ = ∂λ₂/∂I₁. In the constant linear case this is the usual mutual-inductance reciprocity.

The smooth reversible energy assumptions matter. A hysteretic material, time-varying active modulation, nonreciprocal bias mechanism or unmodeled additional state can prevent this simple terminal description. A measured apparent mutual response is not automatically a symmetric real frequency-independent M in every regime.

For a specified symmetric matrix, completing the square gives 2W_mag = L₂(I₂ + MI₁/L₂)² + (L₁ −M²/L₂)I₁² when L₂ > 0. This derives the bound on |M| rather than guessing it from a picture of linked field lines. Substituting M/L★ = 4 into our diagonal values gives a negative coefficient and a negative-energy direction; that altered matrix cannot describe the assumed passive lossless system.

Advanced return: nonlinear linkage

A nonlinear reversible coil needs its full single-valued linkage relation. At fixed geometry, stored energy is W_mag = ∫₀^λ i(λ′)dλ′, while coenergy is W′ = ∫₀^i λ(i′)di′ when the chosen reference has zero linkage and energy. They satisfy W_mag + W′ = iλ. The expression ½iλ equals each only for a linear relation through that reference.

Our original illustrative law is λ/(L₀I₀) = s + s³/3 with s = i/I₀ and −2 ≤ s ≤ 2. Its differential inductance is dλ/di = L₀(1 + s²), whereas its secant ratio λ/i is L₀(1 + s²/3), with the continuous zero-current limit L₀. The differential quantity multiplies di/dt in v_L for this fixed-geometry memory-free model.

Integrating i dλ gives W_mag/(L₀I₀²) = s²/2 + s⁴/4. Integrating λ di gives W′/(L₀I₀²) = s²/2 + s⁴/12. Their sum is s² + s⁴/3 = iλ/(L₀I₀²), as required. The positive increasing differential inductance is part of this chosen example; it is not a measured saturation curve or a universal law for ferromagnetic cores.