---
title: Eddy Currents
module: Electromagnetic Induction
moduleNumber: 8
lessonNumber: 5
order: 805
summary: >
  A wire carries current along one path; a solid block of metal offers a continuum
  of them, and any changing flux threading that block sets charge circulating in
  closed loops it chooses for itself. We ask what those eddy currents do — where
  they heat, where they drag, and how Lenz's law fixes their direction — and why the
  same circulation is a feature in an induction furnace and a loss to be suppressed
  in a transformer core. From a representative-loop estimate we get the scaling
  (heating grows with the square of frequency and flux rate) and the two design
  levers, lamination and resistivity, that break the paths a solid conductor would
  otherwise hand the current.
topics: [Electromagnetic Induction]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 28 — Magnetic Induction; §28-5 Eddy Currents"
---

Eddy currents are closed circulating currents induced inside a bulk conductor. A thin
wire constrains charge motion to one obvious path. A metal plate, disk, or transformer
core contains many possible closed paths, each of which can enclose changing magnetic
flux. Faraday's law produces an emf around those paths; finite conductivity allows the
emf to drive local current loops.

The induced loops follow Lenz's law. Their magnetic fields oppose the change in external
flux through the local material. A changing field through a stationary plate can heat the
plate. A plate moving into or out of a field region can experience a magnetic drag force.
The electrical energy comes from the varying field source or from the mechanical agent
that moves the conductor. The local current patterns can be complicated because current
paths depend on conductor shape, field geometry, and conductivity.

## Current Paths and Heating

Consider a wide metal plate in a magnetic field directed into the page. If the field
magnitude increases, the into-page flux through every small loop drawn in the plate
increases. The induced currents circulate so their magnetic contribution points out of
the page. Viewed from the page, the conventional current is counterclockwise around
each local loop. The loops overlap in a real sheet; the diagram represents the direction
of the distributed current density rather than separate insulated wire rings.

$$
% caption: An increasing into-page $B$ through a conducting sheet induces distributed counterclockwise eddy-current loops. Each local circulation produces an out-of-page magnetic contribution opposing the growing external into-page flux. The crossed circles mark the into-page field; the arrows indicate current-density circulation within one continuous sheet.
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$$

If the into-page field decreases, the current circulation reverses. The induced magnetic
contribution then points into the page and tends to preserve the waning external flux.
The direction decision uses the field change, rather than the field value. A large steady
field through a stationary plate can have zero eddy current after transients die out.

The circulation can also arise from conductor motion. Pull a metal sheet rightward out
of the gap between magnet poles. A material loop carried with the sheet has decreasing
flux through the field region. Its induced current produces into-page magnetic field and
experiences a magnetic force leftward in the gap. The force opposes the sheet motion and
converts mechanical work into Joule heating in the sheet.

$$
% caption: A conducting sheet leaves a finite into-page $B$ region (shaded, crossed circles). The field-threaded portion of a material loop shrinks as the sheet moves right. The induced current in the gap circulates so the magnetic drag force points left, opposing the imposed motion.
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$$

### Current density and resistive heating

At each point in an ordinary isotropic conductor, the local current density satisfies

$$
\vec J=\sigma\vec E_{\rm eff},
$$

where $\sigma$ is conductivity and $\vec E_{\rm eff}$ includes the induced driving
field and, for moving material, the magnetic-force contribution per unit charge. A
moving-medium expression is

$$
\vec J=\sigma\left(\vec E+\vec v\times\vec B\right)
$$

when charge accumulation and material properties permit a local ohmic model. The
current density closes through the conductor so that charge does not accumulate
indefinitely at interior points.

Joule heating per unit volume is

$$
p=\vec J\cdot\vec E_{\rm ohmic}=\frac{J^2}{\sigma}.
$$

The total eddy-current power is the volume integral of this nonnegative density. Large
conducting loops, high conductivity, strong field changes, and low-resistance paths can
therefore create substantial heat. A material's magnetic permeability can also alter the
field distribution and change the induced-current pattern, especially in ferromagnetic
cores.

The heating direction does not reverse with current direction. Reversing an eddy loop
reverses $\vec J$, while $J^2/\sigma$ remains positive. An alternating magnetic
field drives alternating eddy currents and deposits heat during both halves of the
cycle. This property supports induction heating and also creates unwanted core loss in
transformers and motors.

## Magnetic Damping

Magnetic damping uses eddy currents to oppose motion without mechanical contact. A
conducting plate attached to a pendulum can swing through the gap of a permanent magnet.
As the plate enters the gap, material loops acquire increasing flux. As it exits, the
same loops acquire decreasing flux. In both intervals, induced currents produce magnetic
forces opposite the plate velocity. The plate loses mechanical energy, and the sheet
warms by a small amount.

$$
% caption: A conducting plate on a pendulum swings toward a magnetic gap. Eddy currents circulate in the portion of the plate near the pole faces. Their interaction with the applied $B$ produces a force opposite the instantaneous velocity on entry and exit, reducing the oscillation amplitude.
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$$

For sufficiently slow motion and a fixed geometry, the eddy-current drag often has the
approximate form

$$
F_{\rm eddy}\approx-cv,
$$

where the coefficient $c$ depends on field strength, plate dimensions, conductivity,
and the distribution of accessible current loops. The approximation follows because
the flux rate is proportional to speed, the induced current is proportional to emf, and
the magnetic force on that current is proportional to current. At larger speed or high
frequency, field redistribution, inductance, skin effects, and nonuniform geometry can
alter the simple linear relation.

The mechanical power removed by a linear magnetic damper is

$$
P_{\rm mech}=F_{\rm eddy}v=-cv^2.
$$

The negative sign refers to mechanical energy of the moving object. The matching
positive thermal power in the conducting plate is $cv^2$ under the ideal damping model.
The sign relation prevents a magnetic brake from accelerating a freely moving plate;
the induced-current force must oppose the relative motion that created it.

$$
% caption: Force and power for an ideal eddy-current damper. The drag force $F=-cv$ is a straight line through the origin with negative slope, so it opposes the motion at every speed. The thermal power $cv^2$ is a parabola: positive for either direction of motion and zero only at zero speed.
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$$

In a damped oscillator with mechanical restoring force, eddy-current damping decreases
the amplitude on successive cycles. The damping force changes direction with velocity,
so it removes energy on both halves of the motion. It does not permanently shift the
equilibrium position of an ideal symmetric oscillator. A strong enough damper can make
the return to equilibrium slow without oscillation; the exact underdamped or overdamped
condition depends on the mass, restoring constant, and damping coefficient.

$$
% caption: Oscillation records with and without eddy-current damping. The undamped trace retains its amplitude in the ideal model. The damped trace loses amplitude because each passage through the magnetic gap drives resistive current in the conductor and dissipates mechanical energy.
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$$

## Brakes and Core Loss

Eddy-current brakes use the same drag without brake pads pressing on a moving surface.
A conducting disk passing between magnet poles, an aluminum track under a moving magnet,
or a vehicle rail beneath an energized electromagnet can provide braking force. The
brake becomes weak at low relative speed because the flux rate and induced current
decrease. Mechanical friction brakes supply holding force near a complete stop, while
eddy brakes supply contactless braking at moderate and high speed.

The dissipated heat appears in the conductor that carries the eddy loops. A rail brake
can heat the rail; a disk brake can heat the disk. Thermal design must account for the
heat capacity, conduction away from the active region, and repeated braking cycles.
Slots or laminations that reduce unwanted transformer-core loss would also reduce the
current paths needed for a strong eddy-current brake. The desired design depends on
whether the current loops are intended to dissipate energy or must be suppressed.

$$
% caption: A rotating conducting disk passes through a magnetic gap near its rim. The local material speed is largest near the rim, where the flux-changing motion drives eddy currents and a tangential drag torque. The disk loses rotational kinetic energy as heat in the conducting material.
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$$

### Suppressing eddy-current loss in cores

Transformer cores, motor stators, and inductors often carry a changing magnetic flux
through ferromagnetic material. The high permeability of the core guides magnetic flux,
but the metal can also provide broad low-resistance paths for eddy currents. A solid
core behaves like a large conducting plate: changing flux drives loops through its bulk,
and their Joule heating wastes energy from the applied electrical source.

Laminating the core interrupts those broad loops. A laminated core is built from thin
metal sheets separated by insulating coatings. Each sheet can still support small eddy
loops within its own thickness, but the insulating layers block current that would cross
from one sheet into the next. The effective resistance of a large circulating path rises
sharply, and the path area available to a single loop falls. Both effects reduce the
eddy-current magnitude.

$$
% caption: Solid and laminated core cross sections. A changing $B$ through the solid core permits a large current loop spanning the full thickness. Insulating layers in the laminated core divide the metal into thin sheets, blocking cross-layer current and confining any loop to a much smaller area.
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Thin laminations are stacked parallel to the dominant magnetic field. That orientation
keeps the desired magnetic flux largely within the sheets while placing insulation across
the broad transverse current loops. A poor lamination orientation can increase magnetic
reluctance or leave large current paths available. Core design balances magnetic
permeability, saturation, mechanical strength, manufacturing cost, and the frequency of
the applied field.

At higher frequencies, thinner sheets are required because the induced emf rises with
the flux rate. Very high-frequency magnetic components may use ferrite or powdered-iron
cores. Ferrites have much higher electrical resistivity than steel, which strongly limits
eddy current. Powdered cores isolate small magnetic grains with insulating binder, again
breaking long current paths. The material choice modifies both magnetic and electrical
properties of the core.

Slots in a conducting plate produce a related effect. A disk with radial slots loses
the wide circular paths that would carry large eddy currents. The slots raise path
resistance and limit loop area. The same change reduces magnetic braking torque, so a
slotted disk reduces heating and drag where those effects are unwanted, while it is
unsuitable as the active element of an eddy-current brake.

## Circuit Models

A simple calculation estimates an eddy loop by replacing a distributed current path
with a representative loop of area $A$, resistance $R_{\rm loop}$, and approximately
uniform perpendicular field. If the field changes at rate $\d B/\d t$, Faraday's law gives

$$
|\mathcal E_{\rm loop}|\approx A\left|\frac{\d B}{\d t}\right|,
\qquad
I_{\rm loop}\approx\frac{A}{R_{\rm loop}}
\left|\frac{\d B}{\d t}\right|.
$$

The estimated heating is

$$
P_{\rm loop}\approx I_{\rm loop}^2R_{\rm loop}
=\frac{A^2}{R_{\rm loop}}\left(\frac{\d B}{\d t}\right)^2.
$$

Large low-resistance loops dissipate substantial power in a changing field. Doubling
loop area raises the emf by two and the resistive heating by four when
resistance is unchanged. Real eddy patterns contain many loops, shared current paths,
inductance, and nonuniform field. The expression gives a scaling estimate, not an
exact core-loss calculation.

A lamination of thickness $t$ has characteristic loop dimensions and resistance that
change with $t$. Decreasing thickness generally reduces the loop area faster than it
reduces resistance, lowering the loss. The detailed dependence includes material
resistivity and magnetic field distribution, but the direction of the design change
follows directly from the loop estimate.

## Frequency and Penetration

An alternating source often produces a field that is approximately sinusoidal in a core
region,

$$
B(t)=B_0\sin(\omega t).
$$

The field rate is

$$
\frac{\d B}{\d t}=\omega B_0\cos(\omega t).
$$

A representative current loop of area $A$ has induced-emf amplitude
$\omega AB_0$. Doubling the frequency doubles that emf amplitude. In the resistive
loop estimate, current amplitude also doubles, and average heating grows approximately
with the square of frequency. Doubling the field amplitude has the same quadratic effect
on this loss estimate.

$$
% caption: A sinusoidal magnetic source and its eddy-current heat rate. The field rate is largest when $B$ crosses zero, so the induced emf and current magnitude peak there. Eddy heating depends on current squared and gives a positive pulse during both signs of the alternating current.
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$$

The simple square-frequency relation applies only while the field penetrates the
conductor and the local resistive response follows the source without large phase delay.
At sufficiently high frequency, induced currents alter the interior field distribution.
Current can concentrate near a surface, and the representative-loop estimate becomes
inadequate. Magnetic devices then require electromagnetic field analysis, material data,
and thermal measurements rather than a single core-loss coefficient.

At fixed frequency, a waveform with sharp transitions has large $\d B/\d t$ over short
intervals. Such transitions can drive strong transient eddy currents even when the
field's average value is small. Smooth sinusoidal excitation distributes the flux rate
through the cycle. The field rate, not the mean field, sets the induced emf around a
material path.

> **Worked example (Representative eddy loop in a copper plate).** Take a
> representative square loop of area $A=4.0\times10^{-4}\ \mathrm{m^2}$ and
> effective resistance $R_{\rm loop}=2.0\times10^{-3}\ \Omega$. The normal field
> rises uniformly from zero to $0.20\ \mathrm T$ in $5.0\times10^{-2}\ \mathrm s$,
> so the field rate is
>
> $$
> \frac{\d B}{\d t}=\frac{0.20\ \mathrm T}{5.0\times10^{-2}\ \mathrm s}=4.0\ \mathrm{T\,s^{-1}}.
> $$
>
> The loop emf and current follow from Faraday's law and Ohm's law:
>
> $$
> |\mathcal E|=A\left|\frac{\d B}{\d t}\right|
> =(4.0\times10^{-4})(4.0)
> =1.6\times10^{-3}\ \mathrm V,
> \qquad
> I\approx\frac{|\mathcal E|}{R_{\rm loop}}=\frac{1.6\times10^{-3}}{2.0\times10^{-3}}
> =0.80\ \mathrm A.
> $$
>
> The resistive heating in that representative loop is
>
> $$
> P\approx I^2R_{\rm loop}=(0.80)^2(2.0\times10^{-3})
> =1.3\times10^{-3}\ \mathrm W.
> $$
>
> This is an order-of-magnitude scale for one path, not a total-loss prediction for
> the sheet: neighbouring paths share metal and couple through their magnetic fields.
> Use it to fix the local emf, current, and heating before turning to a distributed
> field calculation.

$$
% caption: Elements of a representative-loop estimate. A changing normal $B$ (crossed circles) through the selected area gives an emf around the path. The path resistance $R$ converts that emf into an estimated loop current and heating. In a continuous sheet, many overlapping paths require a field-distribution calculation for exact total loss.
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$$

### Induction heating versus core loss

The local heating density follows from the current distribution,

$$
P_{\rm eddy}=\int_V\frac{J^2}{\sigma}\d V.
$$

- **Induction heater:** a driven coil places a changing field inside the selected
  workpiece volume. Coil shape, frequency, resistivity, thickness, and thermal contact
  determine where $J$ and deposited power are greatest.
- **Transformer core or motor housing:** the same circulation becomes a loss channel.
  Temperature rise changes resistance, magnetic properties, and insulation lifetime.
  Laminations, high-resistivity material, reduced field rate, and cooling reduce this
  contribution.
- **Thermal calculation:** combine the electrical loss with conduction through the
  object, convection and radiation at its surface, heat capacity, and duty cycle.
  A short high-power pulse and a continuous low-power drive can have different peak
  temperatures at the same average energy.
- **Direction and magnitude:** Lenz's-law direction fixes the local current sense and
  the force direction on moving material. The field solution provides $J$, heating, and
  force density. These are separate outputs of the same changing-flux source term.

### Geometry of distributed current paths

An eddy-current drawing should not be read as a circuit diagram with a single resistor.
The metal contains a continuum of possible paths. Current density must satisfy charge
conservation,

$$
\nabla\cdot\vec J=0
$$

inside a region where charge density is steady. At an insulating outer surface, the
normal component of current density is zero. These conditions bend current paths so they
close within the available conductor. Sharp corners, holes, slots, thin necks, and
changes in conductivity all reshape the loops.

A circular plate in a spatially uniform changing perpendicular field has circular
symmetry. The induced electric field and eddy current run along circles centered on the
plate axis. A rectangular plate breaks that symmetry: current follows rounded loops that
turn before reaching the boundaries. A hole forces the current to circulate around the
hole, often increasing local current density near its edge. The qualitative direction
still follows Lenz's law, while the magnitude requires a solution of the field and
current distribution.

$$
% caption: Geometry changes the available eddy-current paths. A round plate in a uniform changing $B$ supports circular current density around its center. A rectangular plate uses rounded loops near its boundary. A hole redirects the circulation around the opening and can concentrate current near the inner edge.
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  \draw[black, thick, fill=black!8, even odd rule] (3.15,0) circle (1.05) (3.15,0) circle (0.34);
  \draw[->, acc, very thick] (3.15+0.66,-0.15) arc[start angle=-13,end angle=255,radius=0.66];
  \node[below] at (3.15,-1.45) {with hole};
\end{tikzpicture}
$$

Thickness adds a second spatial coordinate to the current model.

- **Thin sheet at low frequency:** current density can be approximately uniform through
  the thickness, so a planar material-loop estimate can establish scale and direction.
- **Thick conductor or rapid drive:** the eddy-current field changes the internal
  applied-field distribution. Surface and interior regions require a spatial
  electromagnetic solution.
- **Moving sheet:** loops near the leading and trailing field boundaries have different
  flux histories. Current density follows the flux rate through each material loop.
- **Stationary sheet in a uniform ramp:** the geometry can support a more symmetric
  circulation, although holes, slots, and material variations still redirect paths.

## Energy and Force

Take a conducting disk of moment of inertia $I_{\rm rot}$ rotating with angular speed
$\Omega$ through a fixed magnetic gap. Suppose the low-speed braking torque is

$$
\tau_{\rm eddy}=-\gamma\Omega.
$$

Newton's rotational law gives

$$
I_{\rm rot}\frac{\d\Omega}{\d t}=-\gamma\Omega,
$$

so the angular speed decays as

$$
\Omega(t)=\Omega_0e^{-t/\tau_d},
\qquad
\tau_d=\frac{I_{\rm rot}}{\gamma}.
$$

The rotational kinetic energy is $K=I_{\rm rot}\Omega^2/2$. Its time derivative is

$$
\frac{\d K}{\d t}=\tau_{\rm eddy}\Omega=-\gamma\Omega^2.
$$

The positive heat rate in the disk equals $\gamma\Omega^2$ in the ideal model. A brake
that brings a disk from one angular speed to another converts the corresponding loss of
rotational kinetic energy into internal energy of the conducting disk and nearby parts.
No friction pad needs to touch the rotating surface.

The low-speed coefficient in $\tau_{\rm eddy}=-\gamma\Omega$ belongs to the complete
geometry rather than to the conductor alone.

| change | local consequence | measured effect |
| --- | --- | --- |
| larger conductivity | lower current-path resistance | larger current and possible torque increase |
| wider magnet gap | smaller field at the disk | reduced flux rate and braking |
| magnet closer to the rim | greater material speed through the gap | larger torque lever arm |
| greater thickness | more conducting volume but possible nonuniform penetration | geometry-dependent current and heating |

Record the field map, disk temperature, speed range, and magnet position before fitting
$\gamma$. A value measured at one gap or speed range cannot be transferred to a
different brake geometry without a new current-path model.

### Eddy currents versus other magnetic losses

Eddy-current loss is resistive heating from induced charge circulation. It changes when
laminations, slots, or high-resistivity material interrupt the available current paths.
Magnetic material can dissipate energy through a separate magnetization process. A
measured device loss can contain both channels together with winding resistance:

$$
P_{\rm input}=P_{\rm winding}+P_{\rm eddy}+P_{\rm magnetic}+P_{\rm mechanical}.
$$

Lamination is a differential test for the eddy-current term when the field waveform,
core geometry, and winding state are held fixed. Residual heating after lamination
points to magnetic-material loss, winding loss, thermal paths, or an incomplete
electrical isolation of the sheets.

### Force distribution near a field boundary

The local force density is

$$
\vec f=\vec J\times\vec B_{\rm ext}.
$$

Magnetic braking concentrates where material loops enter or leave a field region, or
where field strength changes across the conductor. A rigid plate fully inside a spatially
uniform region can contain local charge redistribution and circulating current patterns
with a small net translational force.

A wide sheet moving rightward out of an into-page field has a local loop near the
trailing boundary that loses into-page flux. The induced circulation produces into-page
field through that loop. The portion of the loop inside the gap has leftward
$\vec J\times\vec B_{\rm ext}$ force. Away from the boundary, paired local
contributions can reduce the net force density.

The same analysis applies to a moving magnet above a stationary conducting sheet. The
magnet's field pattern moves relative to the metal, so the flux through nearby material
loops changes. The induced currents produce fields that oppose that changing pattern.
The resulting interaction can exert a retarding force on the magnet, a force on the
sheet, or equal and opposite forces on both when the apparatus permits motion.

At very small separation between a magnet and a conductor, the field gradient can be
large and the current paths confined to a shallow region. The force becomes sensitive to
gap distance, plate thickness, and conductivity. A simple $F=-cv$ relation may hold over
a limited low-speed range, but the coefficient is a property of the full geometry rather
than a universal constant for the material.

### Energy loss in a damped pendulum

Suppose a pendulum plate has mass $m$, moves through a magnetic gap, and experiences an
approximately linear tangential damping force $-cv$. Its mechanical energy is

$$
E_{\rm mech}=\frac12mv^2+U(\theta).
$$

The rate of change of that energy from the eddy-current force is

$$
\frac{\d E_{\rm mech}}{\d t}=(-cv)v=-cv^2.
$$

The energy lost during one swing through the gap is

$$
\Delta E_{\rm mech}=-\int c v^2\d t.
$$

The corresponding positive thermal energy appears in the plate and magnet assembly.
The integral emphasizes that damping depends on the speed history. A fast swing loses
more energy than a slow swing through the same field region. A narrow magnet gap limits
the time within the field but can have a large field strength; the net damping requires
the actual velocity and force profile along the path.

Identify the source that maintains the flux rate before assigning the heating budget.

- **Freely decaying pendulum or disk:** mechanical energy becomes heat.
- **Stationary conductor in a driven electromagnet:** electrical input to the field
  source becomes heat.
- **Motor-driven disk in a permanent-magnet gap:** motor work becomes heat at
  constant speed.

The current paths and force distribution depend on geometry. The energy source follows
the actuator or field supply that maintains the changing flux.

### Field penetration and effective resistance

Eddy currents produce fields that oppose changes in the applied field inside a bulk
conductor. The penetration scale for a homogeneous linear material under sinusoidal
excitation is

$$
\delta=\sqrt{\frac{2}{\mu\sigma\omega}},
$$

where $\omega$ is angular frequency, $\sigma$ is conductivity, and $\mu$ is the
relevant permeability. The relation provides a regime check, not a complete geometry
solution.

| comparison | expected current distribution | design consequence |
| --- | --- | --- |
| thickness small compared with $\delta$ | current reaches much of the section | broad-distribution approximation can be tested |
| thickness comparable with or greater than $\delta$ | surface concentration and phase variation | calculate local heating and field penetration |
| thin insulated laminations | short transverse paths | lower eddy-current loop area and loss |

An effective eddy-path resistance summarizes one geometry and frequency. Material
conductivity, permeability, dimensions, boundary conditions, and field waveform can
change its fitted value. Use the lumped loop for sign and scale; use a field solution
when penetration or gradients control the result.

## Measurement and Design

Eddy currents are inferred from their power, force, or impedance consequences because
the circulation has no single wire terminal.

- **Force comparison:** measure the additional pull force on a conducting plate or the
  decay envelope of a damped vane, then subtract friction and air drag from a matched
  field-off control.
- **Thermal comparison:** record temperature or calorimetric power under a stated field
  waveform, duty cycle, cooling condition, and conductor geometry.
- **Electrical comparison:** measure coil voltage and current, or complex probe
  impedance, while controlling target spacing, conductivity, and drive frequency.
- **Geometry comparison:** compare solid, slotted, and laminated specimens at the same
  field map. Reversing field direction reverses local circulation while preserving the
  heating and drag magnitudes for an otherwise matched run.

### Measurement channels

Use at least two independent records for a quantitative eddy-current result. Force,
thermal, and electrical records constrain different parts of the same energy balance
and diagnose different systematic errors.

$$
F_{\rm eddy}(v)=F_{\rm field\,on}(v)-F_{\rm field\,off}(v),
\qquad
P_{\rm mech}=F_{\rm eddy}v.
$$

- **Force channel.** Measure force at several positive and negative speeds under
  matched field-on and field-off conditions. The subtraction removes bearing friction,
  aerodynamic drag, and a static offset only when those contributions remain unchanged.
  Reverse the motion direction and retain the sign convention. A drag force that
  reverses with velocity and remains opposite the motion supports the Lenz-law
  assignment. Plotting $F_{\rm eddy}/v$ over the intended speed range tests whether a
  linear damping coefficient is defensible.

- **Thermal channel.** A short early-time temperature record can be reduced with a
  lumped heat-capacity model,

$$
C_{\rm th}\frac{\d T}{\d t}
=P_{\rm eddy}-G_{\rm th}(T-T_{\rm amb}).
$$

  When the temperature rise is small and heat transfer is initially weak,
  $P_{\rm eddy}\approx C_{\rm th}\d T/\d t$. Longer runs require the thermal conductance
  $G_{\rm th}$, the mounting path, sensor location, and ambient condition. Surface
  temperature can differ from the hottest current-carrying region, especially when
  penetration concentrates loss near one face.

- **Electrical channel.** For a driven coil and target, record synchronized terminal
  voltage and current rather than rms magnitudes alone. The cycle-average input is

$$
P_{\rm in}=\left\langle v_{\rm coil}(t)i_{\rm coil}(t)\right\rangle.
$$

  Subtract measured winding loss, known mechanical output, and calibrated background
  loss before attributing the remaining power to the target. A phase shift between
  voltage and current changes average power even when both rms values remain fixed.
  Repeat the empty-coil measurement with the same drive waveform and coil temperature.

- **Impedance channel.** A probe-coil measurement should retain amplitude and phase:
  $Z=V/I$ is complex under sinusoidal drive. Target lift-off, orientation, thickness,
  conductivity, and crack geometry can each change the real and imaginary parts. Use
  reference specimens that vary one parameter at a time. A single amplitude reading
  cannot distinguish a material change from a spacing error.

- **Flux and geometry channel.** Record the field waveform at the conductor location,
  the region over which it acts, and the material speed or angular speed. A Hall probe,
  search coil, or calibrated drive coil can supply the field record. The relevant
  quantity is the local loop flux rate; a nominal magnet grade or source-current value
  alone does not determine the induced emf in the specimen.

- **Closure test.** At steady speed in a brake, compare $F_{\rm eddy}v$ with the
  measured thermal or electrical loss after background corrections. During a decay,
  integrate the force-power record and compare it with the loss of mechanical kinetic
  energy. Agreement within stated uncertainty supports the current-path and field-map
  model. A discrepancy can arise from unmeasured friction, temperature-dependent
  conductivity, an incomplete field map, or a sensor boundary that differs from the
  modeled object.

- **Uncertainty record.** Report force-sensor calibration, velocity uncertainty, field
  repeatability, temperature, specimen dimensions, target spacing, and background-loss
  subtraction. Evaluate the result over the same speed, frequency, and temperature range
  used for the stated model. A fitted damping coefficient without those conditions has
  limited predictive value because current paths and conductivity can change between
  measurements.

- **Control sequence.** Begin with a field-off run, then repeat with the same
  mechanical trajectory and field on. Reverse the field or motion for a sign check,
  repeat after thermal equilibrium for a temperature check, and compare a slotted or
  laminated specimen for a path-geometry check. The sequence separates a genuine
  induction response from a change in friction, alignment, or drive waveform.

### Design choices for cores, plates, and brakes

Select the current-path geometry from the intended effect.

| device | desired current-path condition | principal constraint |
| --- | --- | --- |
| transformer core | thin insulated laminations or resistive magnetic material | core heating and flux capability |
| magnetic brake | continuous disk, plate, or rail within a field gradient | force, thermal capacity, and low-speed performance |
| induction heater | current density in the selected workpiece volume | thermal uniformity, coupling, and material temperature |

Conductivity, thickness, and cooling have opposite roles across these devices. A
high-conductivity brake disk can provide strong torque but may need a heat path. A core
uses high resistivity and short transverse loop dimensions to reduce loss. A thicker
conductor adds heat capacity and available current area while also permitting nonuniform
penetration. State the operating frequency, field map, and thermal boundary before
selecting material thickness.

The placement of a conductor relative to a magnet changes the available flux gradient.
A disk centered far from the gap may have only a small portion within strong field. A
larger radius increases rim speed and can increase braking torque at the same angular
speed. A magnet pair placed symmetrically on both sides of a disk can provide a more
uniform gap field and reduce unwanted side forces. The resulting torque must still be
measured or modeled from the actual current distribution.

> **Worked example (Calibrated eddy-current brake).** A conducting plate moves
> through a magnet gap; a calibration gives an eddy-current drag coefficient
> $c=0.80\ \mathrm{N\,s\,m^{-1}}$ over the intended speed range. At
> $v=2.5\ \mathrm{m\,s^{-1}}$ the drag magnitude is
>
> $$
> F_{\rm eddy}=cv=(0.80)(2.5)=2.0\ \mathrm N,
> $$
>
> and the mechanical power converted into heat is
>
> $$
> P=cv^2=(0.80)(2.5)^2=5.0\ \mathrm W.
> $$
>
> Release the plate, mass $0.40\ \mathrm{kg}$, from that speed with no other force.
> Its speed decays as
>
> $$
> v(t)=v_0e^{-ct/m}=2.5\,e^{-2.0t}\ \mathrm{m\,s^{-1}},
> $$
>
> with mechanical time constant $m/c=0.50\ \mathrm s$; after one time constant the
> speed is $0.92\ \mathrm{m\,s^{-1}}$. The initial kinetic energy is
>
> $$
> K_i=\tfrac12mv_0^2=\tfrac12(0.40)(2.5)^2=1.25\ \mathrm J.
> $$
>
> Under the stated idealization the plate and its surroundings receive
> $1.25\ \mathrm J$ of heat as the speed approaches zero. The coefficient $c$ is
> measured, not derived, because one rectangular eddy loop cannot fix the full
> plate force; the geometry lives inside $c$.

### A direction checklist

An eddy-current question becomes manageable when the material loop is selected before
the current direction is drawn.

1. **Choose a material loop or a small surface patch.** Mark its normal and whether it moves with the conductor.
2. **Determine the external-flux change through that loop.** State whether the signed contribution becomes more positive or more negative.
3. **Draw the induced magnetic contribution.** It must oppose the signed change through the same local loop.
4. **Use the right-hand rule for conventional circulation.** In a bulk conductor, interpret the result as a local current-density loop.
5. **Apply a force check for moving material.** The current inside the external field must give force or torque opposite the imposed relative motion.
6. **Apply an energy check.** The induced-current heating comes from mechanical work or from the source that changes the magnetic field.

The current direction can reverse from one region of a sheet to another when local flux
histories differ. A single arrow around the outer boundary is insufficient for a large
object spanning a nonuniform field. Divide the sheet into regions, identify the local
field change, and then assemble the distributed current pattern.

## Limits and Checks

The quasistatic model uses a local constitutive relation,

$$
\vec J=\sigma\vec E_{\rm ind},
\qquad
p=\vec J\cdot\vec E_{\rm ind}
=\frac{J^2}{\sigma},
$$

after the induced electric field has been obtained from the changing magnetic flux.
It is appropriate when the field pattern and material response adjust on a time scale
short compared with the imposed mechanical motion or source waveform. The model gives
current direction, heating density, and magnetic-force direction without requiring a
single wire-like current path.

| condition | effect on a loop estimate | required treatment |
| --- | --- | --- |
| thin conductor and slow field change | current samples much of the thickness | distributed quasi-static current is often adequate |
| large thickness or rapid periodic drive | current crowds near surfaces and has phase lag | use a field-penetration or Maxwell boundary-value model |
| strong field gradient across the object | local loops have unequal flux rates | partition the conductor into local regions |
| temperature rise or ferromagnetic core | $\sigma$ or permeability changes during the run | record state and use a temperature- or field-dependent model |

The induced-current mechanism has an unambiguous energy record: changing flux drives
circulation, resistivity converts part of that response to heat, and the field-current
interaction produces force or torque. Geometry, conductivity, permeability, and the
field history determine the measured magnitudes. A large separation between magnet and
conductor can reduce the flux rate below a measurable level. A close gap can demand a
spatial field map rather than a uniform-field estimate.

### Solid versus laminated core

Compare a solid core and a laminated core at the same magnetic operating point. The
solid cross section permits broad circulation; insulated sheets break the transverse
electrical path while retaining a continuous magnetic path. Under a controlled
sinusoidal test, an energy balance for the measured assembly is

$$
P_{\rm core}
\approx \langle v_{\rm coil}i_{\rm coil}\rangle
-I_{\rm rms}^{\,2}R_{\rm winding}
-P_{\rm mechanical}.
$$

The residual includes eddy-current and magnetic-material loss. A lower residual for
the laminated core at matched flux amplitude and frequency supports the expected
reduction in eddy-current loss; it does not by itself isolate every microscopic loss
mechanism.

- **Match the magnetic state.** Use the same winding, core gap, excitation frequency,
  and flux waveform. A search-coil record or a calibrated coil-voltage integral
  verifies the flux amplitude.
- **Measure input and copper loss separately.** Record the synchronized voltage and
  current waveforms, coil resistance at the test temperature, and average input power.
  A temperature change in the winding alters the copper-loss subtraction.
- **Inspect insulation and clamping.** Lamination coating must block cross-layer
  current while the sheets remain magnetically well coupled. Damaged coating, a
  conducting bolt, or a conductive clamp can restore a broad loop and raise the loss.
- **Use temperature as a secondary record.** Temperature rise integrates loss over
  time and depends on cooling. It confirms a power trend only when the thermal boundary
  conditions and run duration are matched.

### Eddy currents in measurement and control

Three applications use the same local flux-rate calculation with different measured
outputs.

- **Damped indicator:** a conducting vane moves through a magnet gap. The eddy-current
  torque removes oscillation energy while the restoring and drive torques set the
  equilibrium position. At rest, the material-loop flux rate is zero and this torque
  vanishes.
- **Probe coil:** an alternating field drives currents in a nearby conducting target.
  The target response changes the coil terminal voltage-current relation. Crack size,
  thickness, conductivity, lift-off distance, and drive frequency require calibration
  against reference specimens.
- **Magnetic brake:** a disk or rail passes through a field gradient. The available
  flux rate falls with relative speed, so drag and heating fall as the vehicle or disk
  slows. A separate holding brake exerts force at zero speed.

Each apparatus needs an observable matched to the model: a decay envelope for the
indicator, complex coil impedance for the probe, or force and speed for the brake.
The current distribution remains internal to the conductor, so the measurement must
close through mechanical, thermal, or electrical power rather than a single series
ammeter.

### Sign, unit, and magnitude checks

Magnetic flux has unit weber, and its rate has unit volt. A representative eddy-loop
emf $A\d B/\d t$ therefore has unit

$$
\mathrm{m^2}\,\frac{\mathrm T}{\mathrm s}
=\frac{\mathrm{Wb}}{\mathrm s}
=\mathrm V.
$$

Dividing by loop resistance gives ampere. Multiplying a current density by an electric
field gives watts per cubic metre. Multiplying a drag force by speed gives watts. These
unit checks connect the induction calculation to the mechanical or thermal energy
calculation and catch errors such as using field magnitude where field rate is required.

A direction check proceeds locally. Increasing into-page flux calls for an out-of-page
induced contribution and counterclockwise conventional circulation as viewed. Decreasing
into-page flux calls for the reverse circulation. A moving sheet entering a field region
and a stationary sheet exposed to a strengthening field can have the same local current
direction if their material loops experience the same signed flux change.

The induced current may be strong while the net force on a whole symmetric object is
small. Opposite regions can experience forces that cancel in translation while producing
a torque, or they can carry counter-rotating currents whose external fields partly
cancel. Force, torque, heating, and current pattern must be evaluated as separate
quantities for the stated geometry.

### Material and geometry record

Record the material, geometry, and operating conditions for every eddy-current calculation or experiment.

- **Field history:** magnitude, direction, spatial extent, and time dependence of the external field.
- **Conductor geometry:** thickness, holes, slots, laminations, current-path dimensions, and motion relative to the field.
- **Electrical properties:** conductivity or resistivity, temperature range, and electrical contacts if present.
- **Mechanical state:** speed, angular speed, applied force or torque, mass or moment of inertia, and other damping forces.
- **Thermal boundary conditions:** heat capacity, cooling path, ambient temperature, and duty cycle.

Faraday's law and Lenz's law determine the induced-current direction; electrical
and mechanical power identify the energy source and destination. Use the material and
geometry record as a calculation input rather than a descriptive appendix.

- **Static-field control.** Hold the conductor at rest in a static field. Every
  material loop then has constant external flux, so induced emf and resistive
  eddy-current heating vanish after transient charge redistribution. Motion through
  a field boundary or a time-varying field restores a flux rate and identifies the
  mechanical drive or electrical source that provides the dissipated energy.
- **Scale comparison.** The response depends on available conducting-loop area,
  loop resistance, and local field rate. Larger or faster-changing configurations
  therefore need separate thermal and mechanical limits even when their current
  direction is unchanged. Record the selected material loop, field orientation, and
  motion direction before assigning current arrows.
- **Repeated operating conditions.** Retain reference field map, conductor
  temperature, motion history, sensor spacing, and system boundary for every
  comparison. A change in any one of these inputs can alter current paths or measured
  force. Compare power, force, temperature rise, and decay rate against the same
  geometry.
- **Independent closure.** Agreement among the power, force, thermal, and decay
  records supports the assigned current pattern and energy balance. A disagreement
  identifies an omitted current path, thermal loss, field-gradient error, or a
  measurement boundary that differs from the model.