---
title: Lenz's Law
module: Electromagnetic Induction
moduleNumber: 8
lessonNumber: 3
order: 803
summary: >
  The minus sign in Faraday's law is not decoration: it decides which way the induced
  current flows, and it always chooses the direction that fights the change that produced
  it. Lenz's law reads that sign off energy conservation — a current that aided the change
  would be free energy — and turns it into a repeatable procedure. We fix a surface normal
  and a positive loop direction so the sign is calculable, then work through approaching
  magnets, expanding loops, coupled coils, and rotating generators, using mechanical work
  and Joule heating as an independent check on every direction we draw.
topics: [Electromagnetic Induction]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 28 — Magnetic Induction; §28-3 Lenz's Law"
---

Faraday's law gives a signed emf around an oriented circuit,

$$
\mathcal E=-\frac{\d\Phi_B}{\d t}.
$$

The minus sign fixes the direction of the induced emf. The resulting current produces
a magnetic field whose flux opposes the change in externally produced flux. An increase of positive
external flux calls for negative induced flux. A decrease of positive external flux
calls for positive induced flux. The flux rate sets the direction; static flux
produces no induced emf.

External flux can change through relative magnet--coil motion, changing nearby
current, changing overlap area, or a changing electromagnet.
The geometry fixes the signed external-flux change. The induced field direction then
fixes the current direction through the right-hand rule. A resistance, switch, and
instrument matter only after the emf direction has been found.

## Orientation and Sign Method

Start by selecting one surface normal for the loop. The associated positive circuit
traversal follows the right-hand rule: point the right thumb along the selected normal,
and curled fingers give the positive path direction. Signed flux, emf, and current are
then measured against that same paired convention. Reversing both the normal and the
positive traversal changes every algebraic sign while leaving the physical current in
the wire unchanged.

$$
% caption: The oriented-loop convention pairs a surface normal $\hat n$ with a positive circulation by the right-hand rule. With $\hat n$ out of the page, the positive path is counterclockwise; every signed flux, emf, and induced-current statement below is read against one fixed pairing.
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$$

Write the external flux as

$$
\mathcal E=-\frac{\d\Phi_{\rm ext}}{\d t}.
$$

The subscript separates external flux from the flux created by the loop current. The current induced in the loop can create
its own flux $\Phi_{\rm ind}$. Lenz's law compares the sign of $\Phi_{\rm ind}$ with
the sign of $\d\Phi_{\rm ext}/\d t$. A positive change in external flux is met by an
induced field whose flux is negative through the chosen surface. A negative change in
external flux is met by an induced field whose flux is positive. In an ordinary
resistive loop, the induced flux only partly offsets the external change; the exact
amount depends on the resistance, geometry, and inductance of the circuit.

$$
% caption: Lenz's law reads the induced field off the sign of the external change. Left: the into-page external field grows, so the induced field points out of the page (central dot). Right: the into-page external field weakens, so the induced field points into the page (central cross). The induced field opposes the change, not the field.
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$$

The sign of $\Delta\Phi_{\rm ext}$ is a geometric statement. A flat loop in a
uniform field has

$$
\Phi_{\rm ext}=B_{\rm ext}A\cos\theta,
$$

so its change can come from the field strength, the area inside the circuit, or the
angle between field and selected normal. A loop moving sideways through a perfectly
uniform region may have $\Delta\Phi_{\rm ext}=0$ even while the wire is moving. A
loop crossing the edge of that region changes the area threaded by the field and can
therefore have an induced current. Relative motion alone is insufficient; the flux
change provides the test.

### A sign-stable direction procedure

Use this four-step sequence with one fixed orientation.

1. Draw the loop and mark a chosen normal.
2. Determine whether the external flux through that normal becomes more positive or more negative.
3. Draw an induced magnetic field whose flux change has the opposite sign.
4. Curl the right-hand fingers around that induced-field direction to obtain the conventional current direction.

The second step uses the external source only. The third step introduces the induced
source. Keeping those arrows separate matters most when the external field reverses
direction or when a loop leaves a field region. A phrase such as “the field is
decreasing” carries too little information because it omits the direction of the field
and the normal convention. “Positive into-page flux decreases” carries enough
information to finish the argument.

The same procedure works with a normal into the page. In that convention, an into-page
field gives positive flux and clockwise circulation is positive. The algebraic signs
change, but the wire current obtained from the final diagram points the same way in
space. A written solution should state the chosen normal once, then retain it without
switching midway through the calculation.

## Stationary-Loop Cases

Consider a circular wire loop in the page, with external field perpendicular to the
page. Determine each case from the signed external-flux rate shown in the diagram.
An increasing into-page field produces an out-of-page induced field and a
counterclockwise current. A decreasing into-page field produces an into-page induced
field and a clockwise current. Reversing the external field reverses both results.

$$
% caption: One out-of-page normal, two flux histories. Left: a growing into-page external field drives counterclockwise current, whose induced field points out of the page. Right: a shrinking into-page field drives clockwise current and an into-page induced field. The current sense flips with the sign of the flux change.
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  \end{scope}
\end{tikzpicture}
$$

Current directions in these figures refer to conventional current. Electron drift in a
metal runs in the opposite direction. A current arrow remains the correct tool for the
magnetic right-hand rule because magnetic-field formulas and circuit sign conventions
are written for conventional current. Adding an electron arrow to every induction
diagram can obscure the central flux reasoning, so use it only when a carrier-level
question explicitly requests electron motion.

An external field with a constant nonzero magnitude produces a constant flux through a
stationary fixed loop. The induced emf is then zero. A galvanometer connected to the
loop deflects while the source current changes or while a magnet moves, then returns to
zero when the geometry and source become steady. The brief deflection is evidence of a
flux rate, rather than a measurement of static flux.

### Area change at a field boundary

A rectangular loop moving upward into a field region gives a direct area-change case. The region has a uniform magnetic field into the page. The part of
the loop inside the region grows, so the into-page external flux grows in magnitude.
The induced current must produce an out-of-page field. Viewed from the observer, that
requires a counterclockwise conventional current. The force on the current-carrying
segment inside the region points downward, opposing the imposed upward motion.

$$
% caption: A rectangular loop enters an into-page field region from below. The exposed area grows at rate $wv$, so into-page flux grows; the counterclockwise induced current makes an out-of-page loop field, and the magnetic force on the upper segment points down, opposing the upward motion.
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  \node[black, below] at (0,-1.46) {entering};
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$$

Let the loop width be $w$, let its leading edge cross the boundary at speed $v$, and
keep the loop fully within the vertical extent of the field region. During entry,

$$
\frac{\d A_{\rm in}}{\d t}=wv,
\qquad
\left|\mathcal E\right|=Bwv,
\qquad
|I|=\frac{Bwv}{R}
$$

for a one-turn resistive loop. The current exists only while the exposed area changes.
Once the entire rectangle lies in a uniform field, the flux becomes constant and the
induced current vanishes. During exit, exposed area decreases and every current and
force direction reverses. The loop does not receive a persistent push from a static
uniform field; the opposing forces occur at the boundaries where the flux changes.

The mechanical work required to pull the loop through the boundary becomes electrical
energy. With resistance $R$, the current dissipates power $I^2R$. The external puller
does matching mechanical work at rate $F_{\rm ext}v$ in the idealized steady-speed
case. Substitution of $I=Bwv/R$ gives

$$
F_{\rm ext}v
=\left(\frac{Bwv}{R}\right)^2R
=I^2R.
$$

The equality fixes the force direction. A proposed current direction that produces a
magnetic force in the direction of motion would turn the moving loop into an energy
source without a matching loss of mechanical energy. Lenz's-law currents therefore
appear as a magnetic drag force whenever a person or motor drives the flux-changing
motion.

### Bar magnets and the energy check

A bar magnet approaching a conducting loop gives a vivid version of the same sign
calculation. Place the magnet's north pole on the left of the loop and choose the loop
normal along the common axis, pointing from the magnet toward the loop. Magnetic field
lines emerge from a north pole, so the external field through the loop points along the
chosen normal. As the north pole approaches, the positive external flux increases.
The induced current must produce a field directed back toward the magnet. The near
face of the loop becomes a north face, and the two north faces repel.

$$
% caption: A north pole approaches a loop along its axis. The magnet's field $B$ through the loop points right and its positive flux grows, so the induced field points left, back toward the magnet. The loop's near face is then a north face and the force $F$ resists the approach.
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  \node[black, below] at (.90,-1.34) {loop};
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$$

Looking from the magnet toward the loop, the induced current is counterclockwise: the
right-hand thumb points from the loop back toward the magnet, and curled fingers point
counterclockwise in that view. Looking from the opposite side reverses the apparent
clockwise sense. A current direction must therefore name the viewing side or be paired
with a drawn induced-field arrow. “Counterclockwise” by itself has no three-dimensional
meaning.

When the north pole recedes, the positive axial flux decreases. The induced loop field
points toward the loop's far side, preserving part of the positive flux. The near loop
face becomes south and attracts the retreating north pole. The force still opposes the
relative motion. Reversing the bar magnet changes the source-field direction, yet the
same flux procedure gives a resisting force in each approach and recession case.

$$
% caption: Four axial magnet--loop cases sorted by flux change. An approaching pole increases the external axial flux magnitude; a receding pole decreases it. The loop's near face repels an approaching pole and attracts a receding one, so the force opposes the relative velocity in every panel.
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  \node[black] at (4.90,2.55) {receding};
  % top-left: N approaching -> repel
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$$

The induced force opposes the imposed magnet motion. Pushing the magnet toward the
loop requires external mechanical work because the induced current produces a
repulsive force. The induced current dissipates electrical energy in the loop
resistance and may also store magnetic energy in the circuit. Pulling a magnet away
also requires work because the induced loop becomes an attracting magnet. The
mechanical agent transfers energy to the circuit.

Suppose a magnet is held fixed beside a loop after the motion stops. The loop may then
have a substantial static flux from the magnet, but $\d\Phi_{\rm ext}/\d t=0$ and the
induced emf vanishes. Persistent current requires some other source, such as a battery
or a superconducting-loop preparation. Lenz's law describes the transient response to
the changing flux during the motion into or out of the final position.

Time-dependent magnet--loop separation determines the external flux. A stationary
magnet and a loop pulled away have the same separation history as a magnet pulled away
from a stationary loop. The external flux through the loop changes in the same sense,
and the induced force opposes the change in separation. Detailed frame analysis can
use electric fields, magnetic forces on moving charges, or both; the measured current
and energy balance agree.

## Changing Geometry and Sources

A changing current in one circuit can supply the external flux change for a second
circuit. Consider two nearby coils. A battery, resistor, and switch in the primary
circuit establish a current $I_1$. Part of the primary magnetic field threads the
secondary circuit, so the secondary flux is proportional to $I_1$ when the geometry
is fixed. Closing the primary switch produces a finite interval of increasing $I_1$;
the secondary current then takes a direction that produces flux opposite the increase.

$$
% caption: Coupled circuits during a primary-current rise. The battery and switch drive a growing current $I_1$ in coil 1; its changing field $B_1$ threads coil 2, whose induced current $I_2$ makes a secondary field of the opposite flux-change sign through the shared core.
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  % primary circuit (left)
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  % secondary circuit (right)
  \draw[black, thick] (3.20,-1.10) -- (3.20,1.10) -- (1.10,1.10);
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  % field arrow between coils
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  % currents
  \draw[->, acc, very thick] (-2.86,0.66) -- (-1.30,0.66) node[midway,above] {$I_1$};
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  \node[black, below] at (-2.15,-1.30) {primary};
  \node[black, below] at (2.15,-1.30) {secondary};
\end{tikzpicture}
$$

Once $I_1$ reaches a steady value, the flux through the secondary stops changing and
the secondary induced emf becomes zero. Opening the primary switch makes $I_1$ fall,
reverses the sign of $\d\Phi_{\rm ext}/\d t$, and reverses the secondary current. A
galvanometer in the secondary therefore gives one deflection at switch closure, returns
to zero while the primary current is steady, and deflects the opposite way at opening.
The secondary emf is proportional to $\d I_1/\d t$; steady $I_1$ gives zero induced
emf.

With fixed geometry, mutual-inductance notation summarizes the magnitude relation as

$$
\Phi_{2\leftarrow 1}=M I_1,
\qquad
\mathcal E_2=-M\frac{\d I_1}{\d t},
$$

after compatible coil orientations have been chosen. The sign may instead be carried
by the orientation convention for $M$; a diagram of the windings removes ambiguity.
The direction procedure remains unchanged: determine how the primary contribution to
secondary flux changes, draw the opposing secondary contribution, and use the
right-hand rule for the secondary current.

During a primary-current rise, the primary source transfers energy to the induced
secondary response. If the secondary is connected to a resistor, energy dissipated in
that resistor comes from the source driving the primary circuit. If the secondary is
open, charge redistribution and electric fields can still establish an induced voltage;
little energy is delivered to a high-resistance meter. The size of the secondary
current depends on its circuit resistance and any self-inductance, while its direction
follows Lenz's law before those quantitative details are calculated.

### Changing area, changing angle, and changing source strength

The external flux through a flat one-turn loop has the form

$$
\Phi_{\rm ext}=B_{\rm ext}A\cos\theta.
$$

Its time derivative separates the three geometric routes to induction,

$$
\frac{\d\Phi_{\rm ext}}{\d t}
=A\cos\theta\,\frac{\d B_{\rm ext}}{\d t}
+B_{\rm ext}\cos\theta\,\frac{\d A}{\d t}
-B_{\rm ext}A\sin\theta\,\frac{\d\theta}{\d t}.
$$

Each term enters the Lenz procedure with its own sign. A field source can strengthen
or weaken at a fixed loop. A conducting boundary can expand or contract inside a
fixed field. A rigid loop can rotate, changing the projection of its area onto the
field direction. Several terms can act at once, and the induced current responds to
their signed sum rather than to any one visible motion.

Imagine a loop whose normal initially points along a uniform upward field. Rotating it
away from that alignment decreases positive flux even though neither the field
magnitude nor the geometric area changes. Its induced current produces an upward
magnetic contribution through the loop, tending to preserve the aligned-flux state.
The external agent turning the loop feels a resisting magnetic torque. Reversing the
rotation reverses the induced current and the torque.

Choose the upward normal for the initially horizontal loop. During a rotation that makes $\theta$
increase from zero, $\cos\theta$ decreases, so $\d\Phi_{\rm ext}/\d t<0$. Faraday's
law gives positive emf with respect to the positive traversal. The current selected by
that emf makes a positive induced flux, which points upward through the selected
normal. The diagram and algebra encode the same decision.

At $\theta=90^\circ$, the instantaneous external flux is zero. A rotating loop can
still have its largest emf at that orientation because

$$
\left|\frac{\d\Phi_{\rm ext}}{\d t}\right|
=BA\left|\frac{\d\theta}{\d t}\right|
$$

there. A zero flux value and a zero flux rate are different conditions. The first says
the field lies tangent to the loop surface at that instant. The second says the signed
projected field-threaded area is momentarily unchanging. Confusing those two conditions
is responsible for many incorrect generator-direction answers.

$$
% caption: Flux and emf during steady loop rotation. The signed flux follows a cosine, peaking when the normal aligns with the field and crossing zero when the loop plane contains the field. The induced emf follows the negative slope of the flux, so its magnitude peaks at each flux zero crossing.
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  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-1.55) -- (0,1.68) node[above] {f\/lux};
  \draw[->, black] (-0.15,0) -- (5.15,0) node[right] {$t$};
  \draw[acc, very thick, domain=0:4.8, samples=150] plot (\x,{1.20*cos(75*\x)});
  \draw[black, dashed] (6.15,-1.68) -- (6.15,1.78);
  \draw[->, black] (7.00,-1.55) -- (7.00,1.68) node[above] {emf};
  \draw[->, black] (6.85,0) -- (12.15,0) node[right] {$t$};
  \draw[acc, very thick, domain=0:4.8, samples=150] plot ({\x+7.00},{1.20*sin(75*\x)});
\end{tikzpicture}
$$

A variable field strength can combine with rotation. Suppose $B_{\rm ext}$ points
upward and grows while a loop turns away from alignment. The first derivative term is
positive, while the angle term is negative. Their magnitudes may cancel at a particular
instant, giving zero induced emf even though both the source current and the loop angle
are changing. A complete solution evaluates the full derivative before applying Lenz's
law. The drawing should show the source field, normal, and rotation sense; prose alone
often conceals one of those signs.

### Multiple external sources and flux bookkeeping

Magnetic fields obey superposition, so the external flux may contain several signed
contributions:

$$
\Phi_{\rm ext}=\Phi_1+\Phi_2+\cdots,
\qquad
\frac{\d\Phi_{\rm ext}}{\d t}
=\frac{\d\Phi_1}{\d t}+\frac{\d\Phi_2}{\d t}+\cdots.
$$

One source can increase positive flux while another increases negative flux. The loop
reacts to the total. The larger visible field can give the wrong direction when
the smaller field changes faster. State the selected normal, give every contribution a
sign, then add rates before choosing the induced field.

> **Worked example (signed flux rates from two sources).** With an out-of-page normal,
> let source 1 produce $+3.0\times10^{-4}\ \mathrm{Wb}$ increasing at
> $+6.0\times10^{-4}\ \mathrm{Wb\,s^{-1}}$, and source 2 produce
> $-5.0\times10^{-4}\ \mathrm{Wb}$ whose magnitude decreases at
> $2.0\times10^{-4}\ \mathrm{Wb\,s^{-1}}$. Source 2's signed rate is positive, because a
> negative flux becomes less negative, so the total rate is
>
> $$
> \frac{\d\Phi_{\rm ext}}{\d t}
> =\left(+6.0+2.0\right)\times10^{-4}\ \mathrm{Wb\,s^{-1}}
> =+8.0\times10^{-4}\ \mathrm{Wb\,s^{-1}}.
> $$
>
> The induced flux must be into the page under this normal. The static flux values, whose
> sum is negative here, do not reverse that direction; only the positive total rate
> matters. A right-hand-rule sketch converts the required into-page induced field into a
> clockwise conventional current, viewed from the out-of-page side.

Equal and opposite flux rates give zero net induced emf, even if each source changes
rapidly. This cancellation concerns the flux integral through the chosen loop. A
changing magnetic field may still exist in different parts of space, and another loop
with a different area or placement can have a nonzero emf. Flux cancellation is
geometrical rather than a claim that all local magnetic effects have disappeared.

## Coupled Circuits and Fields

The emf around a loop is a closed-path integral,

$$
\mathcal E=\oint_C\vec E_{\rm ind}\cdot \d\vec\ell
=-\frac{\d\Phi_B}{\d t}.
$$

Its direction exists even when no wire occupies $C$. A changing magnetic
field produces a circulating electric field in the surrounding space. A conducting
loop placed along the same path gives charges a route in which that electric field can
drive a current. The wire changes the charge response; it does not create the
circulating electric field from nothing.

$$
% caption: A growing into-page magnetic region produces a circulating induced electric field, drawn as dashed loops both inside and outside the region. A conducting ring on either path carries current in the tangential field direction; the field exists even with no wire present.
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An electrostatic field from stationary charges has zero circulation around a closed
path. The induced electric field from changing magnetic flux has nonzero circulation,
so it cannot be represented globally as the negative gradient of a single-valued scalar
potential. A voltmeter reading between two points on an induction apparatus can depend
on the meter leads and their path through the changing-field region. Circuit language
remains adequate when the apparatus is small compared with the field-variation scale, but
the closed-loop Faraday integral is the more general statement.

The Lenz direction for the circulating electric field follows the same flux sign rule as
the current direction. If into-page magnetic flux increases, the induced electric field
circulates counterclockwise as seen from the page. A positive test charge would be
pushed counterclockwise along a circular conducting ring. Electrons drift clockwise,
while conventional current follows the positive-charge direction. The induced magnetic
field of that conventional current points out of the page and offsets the increasing
into-page external contribution.

### Self-induced emf as a current-change response

A circuit's own current creates magnetic flux through the circuit. When that current
changes, the circuit experiences a self-induced emf whose direction opposes the change
in current. A coil with many turns makes the effect conspicuous because each turn links
the magnetic field created by the other turns. Define the inductance $L$ for a fixed
geometry through the flux linkage $\Lambda=LI$; the self-induced emf is

$$
\mathcal E_L=-L\frac{\d I}{\d t}.
$$

The minus sign has the same Lenz meaning as before. When a source attempts to increase
the conventional current, the induced emf has a polarity that resists the increase. When
the source current falls, the induced emf has a polarity that drives conventional
current in its previous direction. Inductance therefore moderates changes in current;
it does not establish a preferred current direction after all sources and transients
have ended.

The opening-switch event gives a practical energy check. Before the switch opens, a
current in the coil stores magnetic energy. Immediately after the conducting path is
interrupted, the induced emf maintains the existing current direction and can make
the voltage across the opening contacts large enough to ionize air and form a spark.
The spark is hazardous around fuel vapors and is also a visible sign that the magnetic
energy must go somewhere. A diode placed across a relay coil provides a safer current
path and limits voltage by allowing current to decay over a longer interval.

The magnitude and time constant of the decay are the province of RL-circuit analysis;
the direction is Lenz's law: increasing current produces a back emf, decreasing current
produces an emf that supports the old current direction. That statement holds whether the
changing current arose from a battery switch, a generator, or a coupled coil.

## Rotation and Force Checks

A rotating coil passes through a periodic sequence of flux changes. Let a coil with
$N$ turns and area $A$ rotate at constant angular speed $\omega$ in a uniform external
field. Choose the normal fixed in the coil and set $\theta=\omega t$ from an aligned
position. The external flux linkage and induced emf are

$$
\Lambda_{\rm ext}=NBA\cos(\omega t),
\qquad
\mathcal E=NBA\omega\sin(\omega t).
$$

The current reversal follows the flux slope rather than the flux value. For the chosen
orientation,

$$
\frac{\d\Lambda_{\rm ext}}{\d t}<0
\ \Longrightarrow\quad
\mathcal E>0,
\qquad
\frac{\d\Lambda_{\rm ext}}{\d t}>0
\ \Longrightarrow\quad
\mathcal E<0.
$$

The first and second quarter turns both have a falling signed flux, so they have the
same emf sign even though the flux crosses zero between them. The third and fourth
quarter turns have the opposite emf sign. A slip-ring generator transfers this
alternating emf to an external circuit. A commutator reverses the external connections
at the half-turn boundaries and therefore gives a unidirectional terminal current for
a suitable load.

Mechanical torque is required to maintain the rotation whenever the generator delivers
current to a load. The induced current has a magnetic dipole moment, and the field
exerts a torque that resists the imposed angular motion. The instantaneous mechanical
input is related to the electrical output by

$$
P_{\rm mech}=\tau_{\rm ext}\omega,
\qquad
P_{\rm elec}=\mathcal E I.
$$

A rotor with kinetic energy $K$ has power record

$$
\tau_{\rm ext}\omega
=\mathcal E I+P_{\rm mech\,loss}+\frac{\d K}{\d t}.
$$

At steady speed, $\d K/\d t=0$. The drive torque then delivers electrical output and
mechanical-loss power. A magnetic torque aligned with the imposed rotation while a resistor
receives positive power would give the wrong sign in this balance.

- **Open circuit:** $I\approx0$, so reaction torque is small even when the induced emf
  has its full kinematic value $NBA\omega$.
- **Resistive load:** $I$ follows the terminal emf and total circuit resistance; the
  reaction torque and mechanical input rise with delivered electrical power.
- **Reactive or time-varying load:** instantaneous $\mathcal E I$ can change sign as
  energy moves between the generator, field storage, and load. The cycle-average
  mechanical input follows the average real power plus losses.

The flux-rate calculation sets the open-circuit emf. The external circuit sets the
current and therefore the reaction torque.

### Force directions as an independent check

The magnetic force on a current-carrying straight segment is

$$
\vec F=I\,\vec\ell\times\vec B_{\rm ext}.
$$

The power equation gives a second route to the mechanical opposition predicted by Lenz's
law. It should be applied only after the current direction has been established from
flux. The force calculation then checks the result and locates where the external agent
must supply work. In a loop partly entering a uniform region, segments outside the
region feel no magnetic force from that region. Opposite forces on the two vertical
sides often cancel. The horizontal segment inside the field experiences the net force
opposite the translational motion.

Choose an entering rectangular loop with velocity upward and external $\vec B$ into
the page. The Lenz result gives counterclockwise conventional current. On the upper
horizontal segment, conventional current runs left. With leftward $\vec\ell$ and
into-page $\vec B_{\rm ext}$, the cross product points downward. The lower segment
lies outside the field during the entry interval, so it has no balancing upward force.
The net force is downward, opposing the upward velocity.

An exiting loop reverses the current. The horizontal segment still inside the field is
now the lower one, and its force points opposite the outward velocity. The force check
remains valid when the field is nonuniform, but the net force may be distributed among
several segments. A calculation using $I\,\vec\ell\times\vec B$ must use
the local field along each segment rather than a uniform-field shortcut.

Rotational induction has an analogous force check. Each side of a rotating coil carries
current in the external field and experiences a force. The pair of forces creates a
torque opposite the rotation. When the coil reaches an orientation where induced current
is zero, the magnetic torque also vanishes instantaneously. The externally imposed
rotation can continue through that orientation by inertia, then encounters an opposing
torque again as the flux slope changes sign.

## Worked Direction Analysis

> **Worked example (a falling out-of-page field).** Choose the loop normal out of the
> page. The external field points out of the page and its magnitude falls from
> $0.80\ \mathrm T$ to $0.20\ \mathrm T$ through a fixed area, so both external fluxes
> are positive and
>
> $$
> \Delta\Phi_{\rm ext}=A\left(0.20-0.80\right)\ \mathrm T<0.
> $$
>
> The induced flux must be out of the page to oppose this negative change. With the
> right thumb out of the page, the conventional current is counterclockwise as viewed.
> The current persists only during the ramp; a constant $0.20\ \mathrm T$ field through
> the same loop has nonzero positive flux and zero induced emf.

> **Worked example (a shrinking overlap area).** A loop moves to the right out of a
> region with $\vec B$ into the page. Choose a normal into the page, so the external flux
> is positive. The overlapping area decreases, making $\d\Phi_{\rm ext}/\d t<0$. The
> induced field must point into the page, and the current must be clockwise as viewed
> from the out-of-page side. That current creates a leftward force on the segment still
> in the region, so an external hand must pull to the right to keep the loop speed
> constant.
>
> Choosing the normal into the page turns the visible area shrinkage into a positive flux
> decrease. A different normal reverses the algebraic signs and keeps the same clockwise
> wire current in space.

> **Worked example (opposing source rates).** A small loop sits between two coils. The
> left coil's rising current produces out-of-page flux at rate
> $+4.0\times10^{-5}\ \mathrm{Wb\,s^{-1}}$; the right coil's rising current produces
> into-page flux at rate $-7.0\times10^{-5}\ \mathrm{Wb\,s^{-1}}$. With an out-of-page
> normal, the total rate is
>
> $$
> \frac{\d\Phi_{\rm ext}}{\d t}
> =\left(+4.0-7.0\right)\times10^{-5}=-3.0\times10^{-5}\ \mathrm{Wb\,s^{-1}}.
> $$
>
> The loop produces out-of-page induced flux and carries counterclockwise conventional
> current: the larger-magnitude negative rate controls the direction. Raising the left
> source rate to $7.0\times10^{-5}\ \mathrm{Wb\,s^{-1}}$ would make the total rate zero
> and induce no emf, even though both source currents are changing.

### A written solution that can be audited

Record each step in an induction-direction solution.

1. **Normal and positive path.** Draw the normal; state the viewing side whenever a clockwise direction is used.
2. **External contribution.** Draw or calculate the external flux through that normal.
3. **Signed change.** Mark whether the external flux becomes more positive or more negative.
4. **Induced contribution.** Draw the induced magnetic direction that gives the opposite signed flux change.
5. **Wire current and force check.** Use the right-hand rule for conventional current; when motion is involved, verify that the resulting force or torque opposes the imposed motion.

The record distinguishes three directions that are frequently conflated: the external
magnetic field, the induced magnetic field, and the conventional current. Each has a
different physical role. The external field sets the flux change. The induced field
opposes that change according to Lenz's law. The current is the charge-transport response of a particular
conducting circuit. A broken loop can retain an induced emf and induced electric-field
circulation while lacking a steady current around the boundary.

The energy audit is especially decisive in ambiguous diagrams. A current that makes a
moving magnet speed up, a moving loop accelerate farther into a field, or a generator
turn itself while delivering power has been assigned the wrong direction. The corrected
direction must make the external agent supply energy during the flux-changing motion.
That result follows from Faraday's minus sign and remains true across the magnet,
boundary-crossing, coupled-coil, and rotating-coil examples above.

### How much opposing flux appears in a real circuit

Lenz's law fixes the sign of the induced response. Its magnitude comes from the whole
circuit. A single resistive loop with external flux rate $\d\Phi_{\rm ext}/\d t$ has an
emf $-\d\Phi_{\rm ext}/\d t$, and a low-frequency circuit model gives

$$
I\approx-\frac{1}{R}\frac{\d\Phi_{\rm ext}}{\d t}.
$$

The magnetic field produced by that current creates an induced flux with the required
opposing sign. Large resistance makes the current and its magnetic response small;
small resistance permits a larger response. The direction is identical in both cases.
Changing the resistance cannot reverse the Lenz direction unless a separate active
source drives current through the loop.

An ideal superconducting loop illustrates the limiting behavior. Once a changing
external flux attempts to alter the flux through the loop, a persistent current can
develop whose induced flux keeps the total flux linkage fixed under appropriate
conditions. The response then approaches full cancellation of the applied flux change.
An ordinary copper loop has resistance, so its induced current decays after the external
change ends and its induced flux decays with it. The “opposes the change” wording holds
in both materials; the amount and duration of the response differ.

Self-inductance also affects magnitude. A loop with appreciable $L$ obeys a circuit
equation of the form

$$
L\frac{\d I}{\d t}+RI=-\frac{\d\Phi_{\rm ext}}{\d t}.
$$

The first term describes the loop's response to changing its own current, while the
right side describes the externally imposed flux change. A sudden external-flux step
does not generally produce an instantaneous jump to the resistive value
$I=-\left(\d\Phi_{\rm ext}/\d t\right)/R$. Inductance moderates the current rise or fall.
The sign of the initial current response still gives an induced magnetic contribution
opposite the external change.

Magnetic diffusion in a thick conductor gives another physical description. A rapidly
changing field drives circulating currents in the material. Those currents create fields
that oppose the changing external flux in the conductor's interior. The resulting eddy
currents can screen high-frequency field changes more strongly near the surface than
deep inside the material. Detailed current patterns and skin depth belong to the eddy-
current treatment, but their direction is another direct consequence of Lenz's law.

$$
% caption: Eddy-current braking. A conducting sheet leaves an into-page field region to the right; the falling flux in the still-immersed part drives a circulating eddy current whose magnetic force opposes the motion, slowing the plate without contact.
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## Three-Dimensional Conventions

Page symbols are convenient only when the relevant field is perpendicular to the page.
Tilted loops and three-dimensional apparatus require vectors and a named normal.
Let the selected normal be $\hat n$. The signed external flux is positive
when the average field component along $\hat n$ is positive. A positive rate
of that flux requires an induced magnetic moment with a component along
$-\hat n$; a negative rate requires a component along
$+\hat n$. The right-hand rule then maps that magnetic-moment direction to
conventional current around the actual three-dimensional wire.

An observer's view can reverse clockwise labels without reversing any vector. A loop
lying in the $yz$ plane has a normal along $+x$. Viewed from the $+x$ side, a current
that produces $+x$ magnetic field is counterclockwise. Viewed from the $-x$ side, the
same moving charges appear clockwise. Vector arrows avoid the ambiguity. When the
wording requests “clockwise,” identify the specified side before answering.

The selected surface for a loop can also be curved. Faraday's law uses any surface
bounded by the same circuit path, provided the same boundary orientation is retained.
Different spanning surfaces give the same magnetic flux change in ordinary classical
electromagnetism because magnetic fields have zero net flux through a closed surface.
For calculation, select the surface that makes $\vec B\cdot \d\vec A$ easiest to
evaluate. Lenz's rule then uses the sign associated with that surface normal and its
boundary traversal.

## Limits and Common Errors

### A sign and unit audit

Carry the signed external linkage through the calculation.

$$
\Lambda_{\rm ext}=N\Phi_{\rm ext},
\qquad
\mathcal E=-\frac{\d\Lambda_{\rm ext}}{\d t},
\qquad
\frac{\d\Lambda_{\rm ext}}{\d t}=0\ \Longrightarrow\ \mathcal E=0.
$$

Apply these checks before numerical substitution.

| check | required record | diagnostic result |
| --- | --- | --- |
| flux rate | selected normal and signed $\d\Phi_{\rm ext}/\d t$ | its sign determines induced-field direction |
| current arrow | induced field followed by the right-hand rule | conventional current follows the induced field |
| force or torque | $I\vec\ell\times\vec B_{\rm ext}$ or magnetic torque | interaction resists imposed relative motion |
| units | $\mathrm{Wb\,s^{-1}}=\mathrm V$, then $\mathrm V/\Omega=\mathrm A$ | circuit current and mechanical force have compatible units |

- **Differentiate the external contribution.** An increasing and a decreasing
  out-of-page field produce opposite currents. Magnet motion alone sets neither
  sign nor magnitude; field direction, overlap, and the chosen normal are
  required.
- **Draw fields before current.** First draw the signed external change, then the
  opposing induced field, then the conventional current. This order remains valid for
  field reversal, changing area, and multiple winding directions.
- **Use the external field in the force calculation.** The induced current interacts
  with the applied field. The magnet experiences the field of the induced current.
  These reciprocal descriptions yield equal and opposite interaction forces and the
  same mechanical-work requirement.
- **Separate direction from magnitude.** Lenz's law sets the emf polarity.
  Resistance, inductance, capacitance, motional-emf geometry, and field distribution
  determine the current magnitude and time record.
- **Sum turn linkages with winding signs.** Identical turns multiply the external
  flux rate by $N$. Oppositely wound turns contribute with the opposite traversal
  convention and can reduce the net linkage.
- **Reverse the controlled rate.** Reverse magnet velocity, angular velocity, or the
  source-current ramp while holding the geometry fixed. The emf, induced current, and
  reaction force or torque must reverse. Setting that rate to zero removes the induced
  emf while retaining any static external flux and any stored current already present in
  a separate inductive circuit.

### State variables and reversible checks

An induction calculation should preserve the state variable that causes the flux
change. A translating magnet requires position and speed; a rotating loop requires
orientation and angular speed; a current-driven solenoid requires current and its
time derivative. Reversing the sign of that state rate must reverse the induced emf.
Setting the rate to zero must remove the induced emf even when the instantaneous
flux remains large. These two substitutions check the time derivative without
relying on a remembered clockwise rule.

Mechanical response provides a second check for systems that move. When a load
current flows, the induced magnetic interaction resists the imposed change in flux.
An external drive maintains the motion and transfers the energy converted to circuit
heat or stored magnetic energy. Reversing the motion reverses the current and the
reaction force. A sign assignment that predicts the reaction force assists the
imposed motion while positive resistor power is generated violates this energy
balance.