Eddy current
Eddy currents, also known as Foucault currents, are closed loops of electric current induced within the bulk of an electrical conductor when it is exposed to a changing magnetic field, according to Lenz's law.[1] These currents flow in swirling patterns perpendicular to the magnetic field lines and can generate significant heat due to resistive losses in the conductor.[2] The phenomenon was first observed in 1855 by French physicist Léon Foucault, building on Michael Faraday's 1831 discovery of electromagnetic induction, when Foucault noted the increased resistance to rotation of a copper disk in a magnetic field.[3] Eddy currents arise either from the relative motion between a conductor and a magnetic field or from time-varying fields in stationary conductors, and their magnitude depends on factors such as the conductor's conductivity, the rate of magnetic flux change, and the material's geometry.[4] While they often represent energy losses in transformers and motors—mitigated by laminating cores to reduce cross-sectional paths—these currents also enable practical technologies.[1] Key applications harness eddy currents for electromagnetic braking in trains, roller coasters, and vehicles, where they produce drag without physical contact by inducing opposing magnetic fields.[5] In non-destructive testing (NDT), they detect surface and subsurface flaws in metals, such as cracks or corrosion, by analyzing changes in electrical impedance.[6] Additionally, eddy currents power induction heating in furnaces and cooktops, where rapid heating melts metals or boils water efficiently, and they underpin metal detectors and speedometers by sensing conductivity variations.[5] These uses highlight eddy currents' role in modern engineering, balancing their dissipative effects with controlled benefits in energy conversion and inspection.[1]History and Terminology
Origin of the term
The term "eddy" originates in fluid mechanics, where it denotes a small whirlpool or swirling motion in a turbulent fluid flow, such as the localized turbulence created when an oar is dragged breadthwise through water, giving rise to secondary circulating flows. This concept was borrowed to describe electromagnetic induction phenomena because the induced electric currents form similar closed, looping paths within conductors subjected to varying magnetic fields, resembling the rotational patterns of fluid eddies.[7][8] François Arago first observed these currents in 1824 while experimenting with the interaction between rotating conductors and magnetic needles. He reported that a suspended magnetic needle would rotate in the same direction as an underlying copper disk set in motion, an effect he attributed to some form of induced action in the metal.[4]Historical development
The understanding of eddy currents emerged from early 19th-century advancements in electromagnetism. Following Hans Christian Ørsted's 1820 observation of the magnetic effects produced by electric currents, André-Marie Ampère rapidly developed a comprehensive theory of electrodynamics, demonstrating how currents generate magnetic fields and laying indirect groundwork for comprehending induced currents within conductors.[9] In 1824, French scientist François Arago conducted a pivotal demonstration in which a copper disk rotated when suspended above a revolving magnetic needle, revealing induced motion in metals exposed to changing magnetic fields; this effect, known as Arago's rotations, represented the first observed manifestation of eddy currents.[10][4] Michael Faraday advanced this field through his 1831 experiments on electromagnetic induction, where he showed that a varying magnetic field induces electric currents in nearby conductors; notably, Faraday observed significant heating in solid metal cores during these tests, which he linked to the circulation of induced currents within the material.[11][12] Heinrich Lenz contributed crucially in 1834 by formulating Lenz's law, which specifies that an induced current flows in a direction opposing the magnetic flux change that generated it, thereby explaining the resistive and damping nature of eddy currents against motion in magnetic fields.[13][14] In 1855, French physicist Léon Foucault provided the first definitive demonstration of eddy currents by rotating a copper disk in a strong magnetic field and observing that the rotational resistance increased significantly compared to rotation without the field. He attributed this to closed loops of induced current within the disk, naming the phenomenon "Foucault currents," which are also known as eddy currents.[15] Late 19th-century progress included American inventor David E. Hughes' 1879 experiments, in which he applied eddy current principles to sort metals by conductivity in metallurgical assays, marking one of the earliest industrial uses.[4][16] Entering the 20th century, eddy currents gained prominence in metallurgy through the commercialization of induction heating in the 1920s, enabling efficient melting and forging of metals via controlled current induction in workpieces.[17][11] Following World War II, eddy current techniques advanced rapidly in non-destructive testing during the 1950s and 1960s, with innovations in instrumentation allowing precise detection of defects in aircraft and industrial materials, spurred by postwar demands for reliable quality control.[18][19]Theoretical Foundations
Basic principles
Eddy currents are closed loops of electric current induced within the bulk of an electrical conductor when it is exposed to a time-varying magnetic field. These currents form swirling patterns, analogous to eddies in a flowing fluid, as free electrons in the conductor respond to the induced electric fields generated by the changing magnetic flux.[20] The generation of eddy currents is governed by Faraday's law of electromagnetic induction, which quantifies the electromotive force (EMF) induced in the conductor as the negative rate of change of magnetic flux \Phi_B through it: \mathcal{E} = -\frac{d\Phi_B}{dt} Here, \Phi_B = \int \mathbf{B} \cdot d\mathbf{A} represents the magnetic flux, with \mathbf{B} as the magnetic field and d\mathbf{A} as the differential area element. This law establishes that any change in magnetic flux—whether due to motion of the conductor in a static field or variation of the field itself—produces an induced EMF that drives the currents.[21] Lenz's law determines the direction of these induced currents, stating that the currents will produce their own magnetic field opposing the original change in flux, thereby resisting the flux variation that induced them. This directional opposition is a direct consequence of the conservation of energy, as allowing the flux to change without resistance would imply energy creation or destruction.[22] For eddy currents to occur, the material must be electrically conductive, enabling free charge carriers like electrons to flow under the induced EMF, following Ohm's law I = \mathcal{E}/R, where R is the resistance of the current path. Basic familiarity with static magnetic fields and conductors is assumed, as these provide the environment for flux changes to induce the looping currents.[23] A qualitative example illustrates this: consider a flat metal sheet moving perpendicularly into a uniform magnetic field. As the leading edge enters the field, the increasing flux through the sheet induces counterclockwise eddy currents (viewed from above), generating a magnetic field that repels the sheet and slows its entry, per Lenz's law. These currents form closed loops parallel to the sheet's edges, diminishing as the entire sheet fully enters the uniform region.[20]Mathematical formulation
The mathematical formulation of eddy currents is grounded in Maxwell's equations, which govern the interaction between electric and magnetic fields in conducting media. Central to this is Faraday's law of electromagnetic induction in differential form, which quantifies the generation of an electric field by a time-varying magnetic field: \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} This equation indicates that a changing magnetic flux density \mathbf{B} induces a circulatory electric field \mathbf{E} within the material.[24][25] Within a conductor, the induced electric field drives free charge carriers, resulting in a current density \mathbf{J} according to Ohm's law: \mathbf{J} = \sigma \mathbf{E} where \sigma is the material's electrical conductivity. Substituting this into Faraday's law yields the expression for the induced current density: \nabla \times \mathbf{J} = -\sigma \frac{\partial \mathbf{B}}{\partial t} (assuming uniform \sigma). This induced \mathbf{J} generates its own magnetic field, which, by Lenz's law, opposes the original change in \mathbf{B}, thereby altering the total field distribution inside the conductor.[24][26] For quasi-static conditions where charge accumulation is negligible, the continuity equation enforces steady-state current flow: \nabla \cdot \mathbf{J} = 0 Combined with \mathbf{J} = \sigma \mathbf{E}, this implies \nabla \cdot \mathbf{E} = 0 in the conductor (for constant \sigma), ensuring that the induced currents form closed, looping paths rather than diverging or converging. These loops, perpendicular to the inducing \mathbf{B} field, are the hallmark of eddy currents and arise naturally from the rotational nature of the induced \mathbf{E}.[26][25] To illustrate field penetration in simple geometries, consider an infinite conducting slab of thickness $2d occupying -d < z < d, subjected to a uniform, time-varying external magnetic field \mathbf{B}_\text{ext} = B_0(t) \hat{x} parallel to the slab faces (infinite extent in x-y plane). Boundary conditions require continuity of the tangential \mathbf{E} and normal \mathbf{B} at the interfaces z = \pm d, with \mathbf{B}(z = \pm d) = B_0(t) \hat{x} imposed externally. Inside the slab, symmetry dictates \mathbf{E} = E_y(z, t) \hat{y} and \mathbf{B} = B_x(z, t) \hat{x}, satisfying Faraday's law as: \frac{\partial E_y}{\partial z} = \frac{\partial B_x}{\partial t} along with \mathbf{J} = \sigma E_y \hat{y}. Solving with the continuity condition \partial J_y / \partial y = 0 (by uniformity) confirms closed current loops in the y-z plane, with field penetration governed by these coupled equations under the specified boundaries.[24][26]Diffusion equation
The magnetic diffusion equation arises from combining Maxwell's equations with Ohm's law in conducting media under the quasi-static approximation, where the displacement current is neglected due to low frequencies compared to relevant time scales.[27] Starting with Ampère's law in the form \nabla \times \mathbf{B} = \mu \mathbf{J} and Ohm's law \mathbf{J} = \sigma \mathbf{E}, the electric field is expressed as \mathbf{E} = \frac{1}{\mu \sigma} \nabla \times \mathbf{B}. Substituting into Faraday's law \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} yields \nabla \times \left( \frac{1}{\mu \sigma} \nabla \times \mathbf{B} \right) = -\frac{\partial \mathbf{B}}{\partial t}. For constant permeability \mu and conductivity \sigma, and using the identity \nabla \times (\nabla \times \mathbf{B}) = -\nabla^2 \mathbf{B} (since \nabla \cdot \mathbf{B} = 0), the equation simplifies to the diffusion form: \frac{\partial \mathbf{B}}{\partial t} = \frac{1}{\mu \sigma} \nabla^2 \mathbf{B}. This derivation highlights the time-dependent penetration of magnetic fields into conductors, essential for understanding eddy current dynamics.[24] Physically, the equation describes how magnetic fields propagate into a conductor not instantaneously, but through a diffusive process analogous to heat conduction, limited by the finite conductivity that induces opposing eddy currents.[27] The propagation speed is governed by the magnetic diffusivity \eta = \frac{1}{\mu \sigma}, which quantifies the material's resistance to field penetration; higher \sigma or \mu reduces \eta, slowing diffusion. For typical metals like copper (\sigma \approx 6 \times 10^7 S/m, \mu \approx \mu_0), \eta \approx 1.3 \times 10^{-2} m²/s, meaning fields diffuse over distances on the order of millimeters in microseconds.[24] For simple geometries, exact solutions illustrate this behavior. Consider a semi-infinite conductor occupying z > 0, with a uniform magnetic field B_0 suddenly applied parallel to the surface at z = 0 for t > 0. The field inside decays as: B(z, t) = B_0 \operatorname{erfc}\left( \frac{z}{\sqrt{4 \eta t}} \right), where \operatorname{erfc} is the complementary error function. This solution shows exponential-like penetration, with the field reaching B_0 near the surface and approaching zero deep inside, demonstrating the diffusive spread over a characteristic distance \sqrt{\eta t}. In alternating current (AC) fields, the time-dependent nature introduces frequency-dependent diffusion time scales, \tau \approx L^2 / \eta, where L is a characteristic length. When the period $2\pi / \omega exceeds \tau, the field penetrates fully before reversal; otherwise, penetration is limited, affecting eddy current distribution without altering the overall diffusive framework.[24]Physical Properties
Power dissipation
The power dissipation due to eddy currents arises from Joule heating, as the induced circulating currents encounter the electrical resistance of the conductor material. The total power loss P is given by the volume integralP = \int \frac{\mathbf{J}^2}{\sigma} \, dV ,
where \mathbf{J} is the eddy current density, \sigma is the electrical conductivity of the material, and the integral is over the volume of the conductor.[28] In cases involving thin conducting sheets subjected to a uniform alternating magnetic field, where the frequency is sufficiently low to neglect skin effect, a simplified expression can be derived by assuming a linear variation of the induced electric field across the sheet thickness. For such a thin sheet, the approximate power loss per unit length along the direction perpendicular to both the field and the current flow is
P \approx \frac{\pi^2 B^2 f^2 t^3 d}{3 \rho} ,
where B is the root-mean-square magnetic field strength, f is the frequency of the field variation, t is the sheet thickness, d is the width of the sheet in the direction of the induced current path, and \rho = 1/\sigma is the material resistivity. This formula results from averaging the squared current density over the cross-section and applying the time-averaged value of the squared rate of change of the magnetic field.[29] Several key factors govern the magnitude of this dissipation: the material's resistivity \rho (higher values reduce losses by limiting current magnitude), the conductor geometry (notably thickness t, with losses scaling as t^2 for power density or total losses in fixed-volume laminated structures), and the applied field's properties (strength B^2 and frequency f^2, both quadratically increasing the induced electromotive force and thus the currents). These dependencies highlight why materials with moderate conductivity and geometries optimized for minimal loop sizes are preferred in designs prone to eddy currents.[30] In engineering contexts like transformers and electric motors, eddy current power dissipation contributes substantially to operational inefficiencies, manifesting as core heating that necessitates enhanced cooling and derating of devices. For instance, in transformer cores, these losses form a major component of no-load iron losses, potentially comprising up to several percent of input power at standard 50–60 Hz frequencies and reducing overall energy conversion efficiency.[31]