---
title: Faraday's Law
module: Electromagnetic Induction
moduleNumber: 8
lessonNumber: 2
order: 802
summary: >
  Move a magnet toward a coil, or ramp the current in a nearby circuit, and a voltage
  appears with no battery in sight. Faraday's law names the cause: the emf around a loop
  equals minus the rate of change of the magnetic flux through it, so any change of
  field, area, orientation, or position that alters the flux drives an emf. We separate
  the emf, which lives around the boundary whether or not current can flow, from the
  current that follows only when the path is closed; fix the single sign convention that
  ties flux to loop orientation; and read the emf off rotating coils and off flux
  sampled at discrete times.
topics: [Electromagnetic Induction]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 28 — Magnetic Induction; §28-2 Induced EMF and Faraday's Law"
---

An induced emf is associated with a changing magnetic flux through a surface bounded
by a circuit. The emf exists around the boundary even when the conducting path is
incomplete and no current can flow. Completing the path gives charge carriers a
continuous route; the induced emf can then drive a current whose magnitude also depends
on the circuit resistance and other circuit elements.

For one oriented loop, Faraday's law is

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

A tightly wound coil of $N$ identical turns has flux linkage
$\Lambda_B=N\Phi_{B,1}$ and obeys

$$
\mathcal E=-\frac{\d\Lambda_B}{\d t}
=-N\frac{\d\Phi_{B,1}}{\d t}.
$$

The sign refers to an oriented loop and its associated surface normal. The physical
current direction that this sign selects is fixed by Lenz's law. The magnitude of the
induced emf is the rate at which signed flux changes.

## EMF Before Circuit Current

An emf is energy transferred per unit charge around a circuit path. It differs from a
current. A closed resistive loop can carry current $I=\mathcal E/R$ in a simple
low-frequency model. An open loop has no continuous charge path, so steady circuit
current is zero even though a changing flux produces an emf around the broken path.

$$
% caption: A changing flux induces an emf around both loops. The closed loop provides a complete conducting path and can carry current; the gap in the right loop prevents a continuous current, while the emf associated with the same changing flux remains defined around the boundary.
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\end{tikzpicture}
$$

The distinction is visible in a charge measurement. In a closed wire, charges drift
around the entire loop while the flux changes. In an open wire, charge accumulates near
the ends until an electrostatic field balances the induced driving effect in the
material. A voltmeter connected across the gap can register a potential difference
related to the induced emf, while an ammeter inserted into a complete loop registers
current. A voltmeter measures the gap voltage; an ammeter measures loop current.

Faraday's law gives the unit relation

$$
[\mathcal E]=\frac{\mathrm{Wb}}{\mathrm s}=\mathrm V.
$$

One volt is one weber per second.

> **Worked example (emf from a flux change).** A flux that changes by
> $2.0\times10^{-3}\ \mathrm{Wb}$ in $0.50\ \mathrm s$ produces a one-turn emf of
> magnitude
>
> $$
> |\mathcal E|
> =\frac{2.0\times10^{-3}}{0.50}
> =4.0\times10^{-3}\ \mathrm V,
> $$
>
> provided the rate is approximately constant over that interval.

## Flux-Rate Laws

A flat loop of fixed area $A$ in a uniform field with fixed orientation has

$$
\Phi_B=BA\cos\theta,
$$

so a changing field magnitude gives

$$
\mathcal E=-NA\cos\theta\,\frac{\d B}{\d t}.
$$

The same law applies when flux changes because the loop area changes or because the
loop rotates. The full rate for a uniform field through a flat loop is

$$
\frac{\d\Phi_B}{\d t}
=A\cos\theta\,\frac{\d B}{\d t}
+B\cos\theta\,\frac{\d A}{\d t}
-BA\sin\theta\,\frac{\d\theta}{\d t}.
$$

Flux changes can arise from variation in field magnitude, enclosed area, or loop
orientation. Faraday's law responds to the total flux rate, including any combination
of these changes.

$$
% caption: A changing field magnitude at fixed loop geometry changes the flux. The loop area and normal stay fixed while the field arrows grow from the earlier to the later panel, producing a nonzero induced emf through Faraday's law.
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  }
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    \draw[->, acc, thick] (\x,-1.35) -- (\x,{-1.35+\h});
  }
  \node[black, anchor=north] at (5.95,-1.55) {later};
  \node[acc, anchor=west] at (7.20,1.20) {larger B};
\end{tikzpicture}
$$

If flux varies linearly in time, the induced emf is constant. A graph with positive
flux slope has a constant negative emf under the same normal convention; a negative
flux slope has a constant positive emf. A stationary nonzero flux gives zero slope and
therefore zero induced emf. Field magnitude alone is not enough; its time dependence
and the loop geometry determine the result.

$$
% caption: Flux and induced-emf records for a constant flux rate. A straight increasing flux trace has a constant slope, so the induced emf is a constant horizontal line with the opposite sign under the chosen loop orientation.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.20,0) node[right] {time};
  \draw[->, black] (0,0) -- (0,2.45) node[above] {Flux};
  \draw[acc, very thick] (0.38,0.30) -- (4.65,2.05);
  \node[acc, anchor=south] at (2.35,1.55) {constant slope};
  \draw[black, dashed] (6.15,-1.20) -- (6.15,2.60);
  \draw[->, black] (7.00,0) -- (11.60,0) node[right] {time};
  \draw[->, black] (7.00,-1.20) -- (7.00,2.10) node[above] {emf};
  \draw[acc, very thick] (7.40,-0.75) -- (11.25,-0.75);
  \node[acc, anchor=south west] at (7.55,-0.68) {constant emf};
\end{tikzpicture}
$$

Use the finite-interval form when a problem gives two flux values rather than a
function:

$$
\mathcal E_{\rm avg}
=-N\frac{\Phi_{B,f}-\Phi_{B,i}}{\Delta t}.
$$

It gives the average emf over the stated interval. A rapidly changing flux can have an
instantaneous emf that differs from this average; the derivative form gives the
instantaneous value when the flux function is known.

## Circuit Response

Faraday's law determines an emf around an oriented circuit boundary. The resulting
current requires a circuit model. A closed loop whose resistance $R$ dominates its
other electrical properties over the time interval of interest has

$$
I=\frac{\mathcal E}{R}
=-\frac1R\frac{\d\Phi_B}{\d t}.
$$

The same flux-rate change produces a larger current in a lower-resistance loop. An
open switch makes the conducting path incomplete, so the steady loop current is zero
even though the emf around the boundary remains associated with the changing flux. A
high-resistance voltmeter can measure the potential difference between points on an
open loop without creating the large current that a low-resistance wire would permit.

$$
% caption: Induced emf and circuit current have separate roles. A changing field through the loop produces an emf around the boundary; the resistor $R$ limits the resulting current, and breaking the loop removes the continuous current path without removing the emf.
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    -- (4.80,1.66) -- (4.40,1.46) -- (4.80,1.26) -- (4.40,1.06) -- (4.60,0.94)
    -- (4.60,0);
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    \draw[->, acc, thick] (\x,0.40) -- (\x,2.20);
  }
  \draw[->, black, thick] (1.40,2.60) -- (3.20,2.60);
  \node[black, anchor=south] at (2.30,2.70) {current I};
  \node[acc, anchor=north] at (2.30,-0.18) {$B$ changes};
\end{tikzpicture}
$$

At constant resistance, current has the same time dependence as emf. A flux that
changes at a constant rate gives a constant current. A flux that changes
only during a short interval produces a current pulse. Once the flux becomes constant,
$\d\Phi_B/\d t$ is zero and the induced emf from that flux change vanishes. A permanent
magnet held stationary near a stationary loop can produce a substantial static flux
with no induced current.

> **Worked example (current in a resistive loop).** A one-turn loop of resistance
> $4.0\ \Omega$ has flux that decreases uniformly from $8.0\times10^{-4}\ \mathrm{Wb}$
> to zero in $0.20\ \mathrm s$. The emf magnitude is
>
> $$
> |\mathcal E|
> =\frac{8.0\times10^{-4}}{0.20}
> =4.0\times10^{-3}\ \mathrm V,
> $$
>
> and the current magnitude in the resistive model is
>
> $$
> |I|=\frac{|\mathcal E|}{R}=\frac{4.0\times10^{-3}}{4.0}
> =1.0\times10^{-3}\ \mathrm A.
> $$
>
> The flux endpoints and interval fix the average emf, not the detailed current waveform
> if the flux changes unevenly or if inductance and capacitance cannot be neglected.

Resistance is only one possible circuit response. A coil's self-inductance opposes a
changing current, and a capacitor accumulates charge; both need an inductance or
alternating-current model. Faraday's law remains the source relation for the emf
produced by the specified changing magnetic flux.

## Coil and Field Geometry

Each closely spaced turn in a coil encloses nearly the same flux when the field is
uniform over the winding. The flux linkage is $\Lambda_B=N\Phi_{B,1}$, so the emf
scales with the number of turns:

$$
\mathcal E=-N\frac{\d\Phi_{B,1}}{\d t}.
$$

Doubling $N$ doubles the emf for the same single-turn flux rate. The result does not
mean that one turn produces a larger flux; it means that the emf contributions around
the consistently wound turns add in series. Reversing one turn's winding gives that
turn an opposite oriented flux contribution and reduces the total linkage rate.

$$
% caption: A multiturn pickup coil in a changing uniform magnetic field. Every turn shares the same normal and single-turn flux rate, so the emf contributions add in series and the total magnitude scales with $N$.
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  }
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  \node[acc, anchor=west] at (1.55,0.55) {$B$ changes};
  \node[black, anchor=west] at (1.55,-0.05) {same area};
  \node[black, anchor=west] at (1.55,-0.65) {same normal};
\end{tikzpicture}
$$

> **Worked example (emf scaling with turns).** A $200$-turn coil whose one-turn flux
> changes at $3.0\times10^{-5}\ \mathrm{Wb/s}$ has emf magnitude
>
> $$
> |\mathcal E|
> =N\left|\frac{\d\Phi_{B,1}}{\d t}\right|=(200)(3.0\times10^{-5})
> =6.0\times10^{-3}\ \mathrm V.
> $$

The turn count is dimensionless, so the unit remains volts. In a practical coil,
increasing the number of turns can also change resistance, inductance, area, and the
field sampled by outer turns. The simple linear scaling holds when the one-turn flux
rate is held fixed as $N$ changes.

### Nonuniform fields and the surface-integral form

The compact product $BA\cos\theta$ applies only to a flat surface in a field whose
normal component is uniform across that surface. Faraday's law itself uses the full
flux integral:

$$
\mathcal E
=-\frac{\d}{\d t}\int_S\vec B(\vec r,t)\cdot \d\vec A.
$$

The time derivative acts on every source of change in the integrand. The field may
change at fixed points, the loop may move through a spatial field gradient, the surface
may deform, or its local normal may rotate. Use the flux integral until a justified
symmetry or approximation reduces it.

> **Worked example (emf from a nonuniform field).** A rectangular loop of height $h$
> is fixed with normal along $+z$, and the normal field over its area has the profile
>
> $$
> B_z(x,t)=B_0(t)\left(1+\frac{x}{L}\right),
> \qquad 0\leq x\leq L.
> $$
>
> Its flux is
>
> $$
> \Phi_B(t)
> =\int_0^L B_0(t)\left(1+\frac{x}{L}\right)h\,\d x
> =\frac32B_0(t)Lh,
> $$
>
> and Faraday's law gives
>
> $$
> \mathcal E
> =-\frac32Lh\,\frac{\d B_0}{\d t}.
> $$
>
> The factor $3/2$ is the area average of the profile. The left-edge value $B_0$ would
> omit the stronger right side; the right-edge value $2B_0$ would overestimate the flux
> rate. The surface integral supplies the correct area weighting.

Field nonuniformity also changes the interpretation of motion through a magnetic
region. A loop translated from one location to another can have changing flux even if
the magnetic source is static. The field at the loop's leading edge and trailing edge
need not be equal, so the change cannot always be described by one field value times
one moving area. Divide the loop surface into regions or evaluate the spatial integral
as the loop position changes.

A loop entering a rectangular field region has flux proportional to overlap area. If
the loop moves at constant speed and the overlap
width grows linearly with time, the flux changes linearly and the induced emf magnitude
is constant during entry. Once the loop is fully inside a uniform field region, its flux
is constant and the induced emf from translation is zero. The exit interval has the
opposite flux slope. Motional emf reaches the same geometry from the magnetic force on
the moving charges.

$$
% caption: Overlap area controls flux while a loop crosses the edge of a uniform-field region. During entry (left) the overlap grows and the flux changes; when the loop is fully inside (right) the flux is constant and the induced emf from translation is zero.
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  \node[black, anchor=north] at (1.55,-1.00) {region};
  \node[black, anchor=south] at (3.60,1.20) {loop};
  \draw[black, dashed] (6.35,-1.55) -- (6.35,1.55);
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  }
  \draw[black, very thick] (7.55,-1.12) rectangle (9.45,1.12);
  \node[black, anchor=north] at (8.50,-1.00) {full overlap};
\end{tikzpicture}
$$

Refined meshes of the same oriented surface and field must converge to the same flux
rate. A disagreement that persists under refinement signals a mismatched surface
normal, an omitted area, or a field map assigned to the wrong coordinates.

### Rotation of a coil in a uniform field

Consider a rigid coil of $N$ turns and area $A$ rotating at constant angular speed
in a uniform field of magnitude $B$. If the
angle from the selected coil normal to the field is

$$
\theta(t)=\omega t+\theta_0,
$$

then its flux linkage is

$$
\Lambda_B(t)=NBA\cos(\omega t+\theta_0).
$$

Differentiation gives

$$
\mathcal E(t)
=NBA\omega\sin(\omega t+\theta_0).
$$

The sign convention in this expression follows the selected normal and loop direction.
The peak emf magnitude is

$$
\mathcal E_{\rm peak}=NBA\omega.
$$

Larger field magnitude, coil area, turn count, or angular speed raises the emf peak in
direct proportion. The dependence on $\omega$ comes from a faster rate of change of
orientation, not from a larger instantaneous flux magnitude.

$$
% caption: Rotating-coil geometry at two orientations. When the normal is aligned with $B$ (left) the flux magnitude is maximal and its instantaneous rate is zero; at the quarter-turn orientation (right) the flux is zero and its magnitude changes most rapidly.
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  \draw[->, black, thick] (5.65,0) -- (5.65,0.92) node[above] {$n$};
  \node[black, anchor=north] at (5.65,-0.20) {quarter turn};
\end{tikzpicture}
$$

At a flux maximum or minimum, the flux curve has zero slope, so the induced emf is
zero. At a flux zero crossing, the slope magnitude is largest, so the emf magnitude is
maximal. The emf waveform is shifted by one quarter period from the flux waveform.
Changing the time origin changes whether the formulas use sine or cosine; the physical
quarter-cycle shift remains.

$$
% caption: Phase relation for a uniformly rotating coil. Flux linkage follows a cosine curve, while induced emf follows its negative time derivative and reaches extrema at the linkage zero crossings.
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  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.25,0) node[right] {$t$};
  \draw[->, black] (0,-1.60) -- (0,1.70) node[above] {linkage};
  \draw[acc, very thick, domain=0.22:5.48, samples=140]
    plot (\x,{1.22*cos(59.60*(\x-0.22))});
  \draw[black, dashed] (6.95,-1.90) -- (6.95,1.95);
  \draw[->, black] (7.75,0) -- (13.95,0) node[right] {$t$};
  \draw[->, black] (7.75,-1.60) -- (7.75,1.70) node[above] {emf};
  \draw[acc, very thick, domain=7.97:13.29, samples=140]
    plot (\x,{1.22*sin(60*(\x-7.97))});
\end{tikzpicture}
$$

> **Worked example (peak emf of a rotating coil).** A coil has $N=120$ turns,
> $A=2.5\times10^{-3}\ \mathrm{m^2}$, $B=0.40\ \mathrm T$, and constant angular speed
> $\omega=180\ \mathrm{rad/s}$. Its peak emf is
>
> $$
> \mathcal E_{\rm peak}=NBA\omega
> =(120)(2.5\times10^{-3})(0.40)(180)
> =21.6\ \mathrm V.
> $$

The result assumes that the field is uniform across the coil throughout its rotation
and that the coil angular speed is constant. A nonuniform source field makes the flux
waveform depart from a pure sinusoid. A varying angular speed changes the phase rate
and hence the instantaneous emf. The flux integral remains the starting point in both
cases.

The rotating-coil calculation specifies induced emf before an external load is chosen.
A resistive load converts the emf to a current and dissipates power. An inductive or
capacitive load changes the current phase and amplitude. Those circuit-response effects
are separate from the flux-rate derivation.

## Induced Electric Fields

The circuit form of Faraday's law can be written as a line integral of the induced
electric field around the oriented boundary:

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

The integral gives emf per unit charge around the closed curve $C$. A conducting wire
samples this induced field and provides mobile charges, while the field relation extends
through the surrounding space. A broken loop therefore retains an induced emf, and a
complete loop is selected by its boundary rather than by a localized battery-like source.

An electrostatic field from stationary charges has zero circulation around a closed
path in a simply connected region. The induced electric field associated with changing
magnetic flux has nonzero circulation. A scalar potential difference between two fixed
points cannot capture the entire closed-loop emf without specifying a path and the
time-dependent magnetic configuration.

$$
% caption: A changing magnetic field through the central disk is associated with a circulating induced electric field. The neutral circular arrows represent the electric-field circulation in space, not a current that requires a wire; a wire loop placed on one circle samples the same emf integral.
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  \draw[black, dashed] (0,0) circle (2.15);
  \node[acc, anchor=west] at (1.15,1.32) {$B$ changes};
  \node[black, anchor=west] at (2.55,-0.55) {circulating E};
  \node[black, anchor=north] at (0,-2.35) {chosen loop};
\end{tikzpicture}
$$

> **Worked example (circulating field around a changing-flux region).** A uniform
> magnetic field fills a circular region of radius $a$ normal to the page, its magnitude
> changing at rate $\d B/\d t$. On a circular path of radius $r$ centered on the region,
> symmetry makes the induced field tangent and of one magnitude $E(r)$, so
>
> $$
> \oint_C\vec E_{\rm ind}\cdot \d\vec\ell
> =E(r)(2\pi r)=-\frac{\d\Phi_B}{\d t}.
> $$
>
> Inside the region, $r<a$, the enclosed flux is $B\pi r^2$, giving
>
> $$
> E(r)=-\frac r2\frac{\d B}{\d t},
> \qquad r<a.
> $$
>
> Outside, $r>a$, the path encloses only the flux of the radius-$a$ disk, giving
>
> $$
> E(r)=-\frac{a^2}{2r}\frac{\d B}{\d t},
> \qquad r>a.
> $$
>
> The magnitude rises linearly with radius inside the changing-field region and falls as
> $1/r$ outside it. The field persists outside the region where $\vec B$ changes because
> the circulation depends on the total changing flux the path encloses.

The calculation separates electric-field circulation from circuit current. A wire loop
of resistance $R$ placed on a chosen circular path has current determined by the emf
and circuit properties. A different circular path encloses the same changing flux once
it lies outside the field region, so it has the same total emf but a smaller induced
electric-field magnitude spread around a longer circumference. The line integral, not
the local field magnitude alone, gives the emf.

The sign of the tangential induced field follows the oriented flux rate. Reversing the
loop traversal reverses both the line-integral sign and the associated surface normal,
leaving the physical circulation unchanged. Lenz's law sets the corresponding current
direction when a conducting loop is specified.

## Measurements and Time Variations

Experimental data often give flux linkage at discrete times rather than an analytic
function. Over an interval from $t_i$ to $t_{i+1}$, the average induced emf is

$$
\mathcal E_{\rm avg,i}
=-\frac{\Lambda_B(t_{i+1})-\Lambda_B(t_i)}{t_{i+1}-t_i}.
$$

The result belongs to that interval. Assigning it to the interval midpoint produces a
consistent time record when the samples are evenly spaced. A centered difference gives
an estimate of instantaneous emf at an interior sample:

$$
\mathcal E(t_i)
\approx-\frac{\Lambda_B(t_{i+1})-\Lambda_B(t_{i-1})}
{t_{i+1}-t_{i-1}}.
$$

Shorter sample intervals improve time resolution but amplify the effect of flux
measurement noise when the linkage difference is small. The interval should resolve
the physical flux variation while retaining a difference large enough to exceed sensor
uncertainty.

Suppose a $50$-turn coil has measured one-turn flux values shown below.

| $t$ (s) | $\Phi_{B,1}$ ($10^{-4}\ \mathrm{Wb}$) |
|---:|---:|
| $0.00$ | $0.0$ |
| $0.10$ | $1.0$ |
| $0.20$ | $2.0$ |
| $0.30$ | $2.0$ |
| $0.40$ | $1.0$ |

> **Worked example (emf from sampled flux).** From $0.00$ to $0.20\ \mathrm s$ the
> one-turn flux above rises at $1.0\times10^{-3}\ \mathrm{Wb/s}$, so the coil emf is
>
> $$
> \mathcal E
> =-N\frac{\d\Phi_{B,1}}{\d t}=-(50)(1.0\times10^{-3})
> =-5.0\times10^{-2}\ \mathrm V.
> $$
>
> From $0.20$ to $0.30\ \mathrm s$ the flux is constant and the emf is zero. From
> $0.30$ to $0.40\ \mathrm s$ the flux slope is negative, so the emf has the opposite
> sign and the same magnitude. The sign flips at the linkage maximum, where the
> derivative changes sign.

$$
% caption: Discrete flux-linkage samples and interval emf estimates. Equal positive linkage increments give equal negative average emf values; the flat linkage interval gives zero emf; decreasing linkage reverses the emf sign under the same orientation convention.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
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  \draw[->, black] (0,0) -- (5.35,0) node[right] {$t$};
  \draw[->, black] (0,-0.30) -- (0,2.70) node[above] {linkage};
  \draw[black, thick] (0.42,0.22) -- (1.48,1.16) -- (2.54,2.10) -- (3.60,2.10) -- (4.66,1.16);
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  \draw[->, black] (7.00,-1.20) -- (7.00,2.35) node[above] {emf};
  \draw[acc, very thick] (7.40,-0.72) -- (9.44,-0.72);
  \draw[acc, very thick] (9.80,0) -- (10.72,0);
  \draw[acc, very thick] (11.08,0.72) -- (11.78,0.72);
  \node[black, anchor=north] at (8.42,-0.85) {negative};
  \node[black, anchor=south] at (11.43,0.80) {positive};
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$$

Flux uncertainty propagates directly into an interval emf estimate. If two independent
flux-linkage readings have equal uncertainty $\delta\Lambda$, the difference has an
uncertainty of approximately $\sqrt2\,\delta\Lambda$. A known interval $\Delta t$
gives

$$
\delta\mathcal E_{\rm avg}
\approx\frac{\sqrt2\,\delta\Lambda}{\Delta t}.
$$

Halving the interval doubles this uncertainty contribution unless the flux measurement
precision improves. A time-series analysis therefore balances temporal detail against
derivative noise.

Calibration errors also enter through the flux model. A probe that measures magnetic
field at one point must be combined with an area map to obtain total flux; a single
point measurement represents a uniform field only when field variation over the loop
is known to be negligible. Coil area, orientation, and turn count carry their own
uncertainties. Treating a measured field value as the entire flux without the surface
geometry can produce a precise but incorrect emf estimate.

The numerical derivative should be checked against the physical configuration. A
measured sign reversal in the emf requires a flux-rate reversal under a fixed normal
convention. A large apparent emf with nearly unchanged field and loop geometry often
indicates a timing mismatch, an offset subtraction error, or an overlooked moving
portion of the circuit.

### Time-varying source fields at a fixed pickup coil

Faraday's law applies when the circuit geometry is fixed and the magnetic source field
changes in time. A current in a long primary solenoid gives an approximately uniform
internal field

$$
B(t)=\mu_0n_pI_p(t),
$$

where $n_p$ is the primary turn density. A coaxial pickup coil of $N_s$ turns and area
$A_s$ placed in the uniform central region has flux linkage

$$
\Lambda_B(t)=N_sA_s\mu_0n_pI_p(t).
$$

Its induced emf is

$$
\mathcal E_s
=-N_sA_s\mu_0n_p\frac{\d I_p}{\d t}.
$$

The primary current itself is not the quantity that determines pickup emf. A large
steady current produces a large static field and static flux but zero induced emf after
the current has settled. A small current changing rapidly can produce a larger emf
because its time derivative is larger.

$$
% caption: Fixed pickup coil inside a current-driven primary solenoid. The primary current changes the approximately uniform central field; the pickup coil remains stationary, so its induced emf follows the time derivative of the primary current rather than the current value itself.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
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  \node[black, anchor=west] at (6.05,0.55) {primary};
  \node[black, anchor=west] at (6.05,0.00) {pickup};
  \node[acc, anchor=west] at (6.05,-0.55) {$B$ changes};
\end{tikzpicture}
$$

A linear primary-current ramp, $I_p(t)=I_0+\alpha t$, has constant derivative

$$
\frac{\d I_p}{\d t}=\alpha,
\qquad
\mathcal E_s=-N_sA_s\mu_0n_p\alpha.
$$

The pickup emf is constant during the ramp and vanishes before and after it if the
primary current is constant outside the ramp. A current pulse that rises and falls
produces emf pulses of opposite signs. The sign relation follows the flux derivative
and selected normals; the resulting current direction is addressed by Lenz's law.

A sinusoidal source current $I_p=I_0\cos\omega t$ gives pickup emf

$$
\mathcal E_s
=N_sA_s\mu_0n_pI_0\omega\sin\omega t.
$$

The emf is shifted by one quarter cycle from the source current and has peak magnitude
$N_sA_s\mu_0n_pI_0\omega$. Raising the driving frequency increases the emf peak only
while the quasistatic field approximation remains valid over the apparatus size.

> **Worked example (pickup-coil emf).** A pickup coil with $N_s=80$ and area
> $A_s=1.5\times10^{-3}\ \mathrm{m^2}$ sits in a primary field rising at
> $0.20\ \mathrm{T/s}$. Its emf magnitude is
>
> $$
> |\mathcal E_s|
> =N_sA_s\left|\frac{\d B}{\d t}\right|=(80)(1.5\times10^{-3})(0.20)
> =2.4\times10^{-2}\ \mathrm V.
> $$

The calculation uses the field rate at the pickup coil, so it remains valid without
knowing the primary solenoid's individual current and turn-density values. Use the
solenoid expression when the source current is measured. Use the field-rate expression
when a probe has characterized the pickup region.

## Signs and Superposition

Faraday's law uses an oriented boundary. Select a traversal direction around the loop;
the associated surface normal follows from the right-hand rule. Evaluate both the
signed flux and the emf with that convention:

$$
\Phi_B=\int_S\vec B\cdot \d\vec A,
\qquad
\mathcal E=-\frac{\d\Phi_B}{\d t}.
$$

Reversing the traversal reverses $\d\vec A$, hence reverses the reported flux and
the reported emf. It also reverses the positive direction used for the line integral
of induced electric field. The physical electric-field circulation has not changed;
only the signed description has been reversed consistently.

An edge-on rotation requires one continuous orientation convention. Let a fixed field point along the selected normal at $t=0$, and let the
normal rotate away so that $\theta$ increases continuously. The flux

$$
\Phi_B(t)=NBA\cos\theta(t)
$$

decreases from a positive maximum, reaches zero at $\theta=90^\circ$, and becomes
negative beyond that angle. The flux is continuous through the zero crossing. Its
sign changes because the normal component of the field has reversed, not because the
field magnitude has become negative.

The corresponding emf depends on the slope,

$$
\mathcal E
=NBA\sin\theta\,\frac{\d\theta}{\d t}
$$

for a fixed positive $B$, $N$, and $A$. A positive angular speed $\d\theta/\d t$ gives an
emf sign set by $\sin\theta$. At the aligned and anti-aligned orientations, the flux
has extrema and the emf is zero. At the edge-on orientation, the flux is zero but its
rate of change and emf magnitude are largest. Flux value and flux rate must therefore
be kept distinct in a sign analysis.

An angle specified relative to the loop plane must be converted to the angle relative
to the normal before a cosine is used. If $\alpha$ is measured from the plane, then

$$
\Phi_B=NBA\sin\alpha.
$$

Differentiating the wrong trigonometric form reverses or shifts the predicted emf
phase. A geometric sketch with $\vec B$, the surface normal, and the stated angle
prevents that error before any derivative is taken.

The sign of a measured terminal voltage requires an additional circuit convention.
Faraday's law gives the positive circulation around an oriented loop. A voltmeter
measures a potential difference between its marked terminals along a particular lead
configuration. In a time-varying magnetic field, different lead paths can enclose
different changing flux and therefore contribute different induced emf. State the loop
path and terminal polarity when comparing a calculated emf with a voltage reading.

A small loop in a slowly varying field can keep one orientation convention through a
sequence of measurements. Reassigning the normal
mid-calculation to keep flux positive obscures the derivative sign. A negative flux is
ordinary directional information. It should remain in the record until the physical
current direction is interpreted using Lenz's law.

The sign convention also handles multiple coils. A secondary coil wound in the same
sense as the selected primary reference has a positive linkage relation under the same
normal choice. Reversing the secondary winding reverses its signed emf for the same
source-field change. Transformer dot conventions encode this winding orientation
compactly.

Flux rates add linearly. If two independently controlled fields contribute fluxes
$\Phi_{B,1}$ and $\Phi_{B,2}$ through the same oriented surface, then

$$
\mathcal E
=-\frac{\d}{\d t}(\Phi_{B,1}+\Phi_{B,2})
=\mathcal E_1+\mathcal E_2.
$$

One source can increase the positive flux while another decreases it. Their emf
contributions can cancel even though both magnetic fields are changing. The sign of
each contribution is set by its normal component and time derivative, not by its field
magnitude alone.

### Model conditions and quantitative checks

Faraday's law is exact in its integral electromagnetic form. A particular calculation
introduces additional approximations that should be stated explicitly. The expression

$$
\mathcal E=-N\frac{\d}{\d t}(BA\cos\theta)
$$

assumes that every turn encloses the same flat area and that the normal component of
the field is uniform across that area. A large coil near a small magnet, a loop partly
inside a field region, or a winding with substantial radial thickness requires a
surface integral or a turn-by-turn linkage calculation.

The quasistatic circuit model adds another condition. Writing $I=\mathcal E/R$ assumes
that resistance dominates the electrical response over the relevant frequency range.
A coil's self-inductance, stray capacitance, radiation, and propagation delay can
matter when the flux changes rapidly. Those effects alter the current response, while
the flux-rate relation remains the emf source relation.

Several limits check a result before numerical substitution.

- **Constant flux:** $\d\Phi_B/\d t=0$ gives $\mathcal E=0$, even when $\Phi_B$ is
  nonzero.
- **Zero normal component:** a field tangent to a fixed flat loop gives zero flux; a
  rotation can still give nonzero emf if that tangential condition changes with time.
- **Turn scaling:** multiplying the number of identical turns by a factor multiplies
  emf by the same factor when each turn samples the same flux rate.
- **Time scaling:** the same flux change completed in half the time doubles the average
  emf magnitude.
- **Orientation reversal:** reversing the selected normal reverses calculated flux and
  emf signs while preserving their magnitudes.

> **Worked example (average emf from a field ramp).** A $75$-turn coil of area
> $4.0\times10^{-3}\ \mathrm{m^2}$ has its fixed normal parallel to a field that rises
> from $0.10\ \mathrm T$ to $0.34\ \mathrm T$ in $0.060\ \mathrm s$. The linkage change is
>
> $$
> \Delta\Lambda_B
> =NA\,\Delta B=(75)(4.0\times10^{-3})(0.34-0.10)
> =7.2\times10^{-2}\ \mathrm{Wb},
> $$
>
> and the average emf is
>
> $$
> \mathcal E_{\rm avg}
> =-\frac{\Delta\Lambda_B}{\Delta t}=-\frac{7.2\times10^{-2}}{0.060}
> =-1.2\ \mathrm V.
> $$
>
> The sign follows the chosen normal and the positive field increase; reversing the
> normal reports $+1.2\ \mathrm V$ for the same event. A linear ramp makes this average
> equal to the instantaneous emf throughout; a curved ramp varies the instantaneous emf
> while the endpoint average stays $-1.2\ \mathrm V$.

The dimensions offer a compact error check. Flux linkage has units of weber because
turn count is dimensionless. Dividing by seconds gives volts. A result in
$\mathrm{T/s}$ is a field-rate value; it becomes an emf only after multiplication by
an area and turn count. A result in webers is a linkage or flux change; it becomes an
emf only after division by a time interval or differentiation.

Sampling data requires the same distinction between interval and instantaneous
quantities. A flux meter that reports values at times $t_i$ and $t_{i+1}$ gives the
interval average
$\mathcal E_{\rm avg}=-(\Lambda_B(t_{i+1})-\Lambda_B(t_i))/(t_{i+1}-t_i)$.
Assign that value to the interval or its midpoint; assigning it to an endpoint
silently changes the timing convention. A rapid flux change needs a sample interval
shorter than the time scale on which the slope varies. Otherwise the calculated
average can be accurate for a long interval while concealing a much larger
instantaneous emf pulse inside it.

Faraday-law problems should preserve the chosen normal, source geometry, and time
interval from the first sketch through the final sign. A current calculation begins
only after the emf has been found and a complete circuit model has been stated. This
separation keeps the flux calculation, the induced-electric-field relation, and the
circuit response from being mixed into one unsupported equation.

### Superposed flux sources and cancellation

Magnetic fields superpose as vectors, so the flux through one oriented loop also
superposes:

$$
\Phi_B
=\int_S(\vec B_1+\vec B_2+\cdots)\cdot \d\vec A
=\Phi_{B,1}+\Phi_{B,2}+\cdots.
$$

Faraday's law then gives the sum of emf contributions,

$$
\mathcal E
=\mathcal E_1+\mathcal E_2+\cdots.
$$

The source fields must be projected onto the same selected loop normal before their
rates are added. Two field magnitudes that both increase can produce opposite flux
rates when one field points with the normal and the other points against it. Field
magnitude trends alone do not determine the net emf sign.

> **Worked example (net emf from two sources).** A $100$-turn coil of area
> $2.0\times10^{-3}\ \mathrm{m^2}$ has fixed normal along $+z$. One source drives a
> positive normal field increasing at $0.80\ \mathrm{T/s}$; a second drives a negative
> normal field whose magnitude increases at $0.30\ \mathrm{T/s}$. The signed total field
> rate is
>
> $$
> \frac{\d B_n}{\d t}=0.80-0.30=0.50\ \mathrm{T/s},
> $$
>
> so the induced emf is
>
> $$
> \mathcal E
> =-NA\frac{\d B_n}{\d t}=-(100)(2.0\times10^{-3})(0.50)
> =-0.10\ \mathrm V.
> $$
>
> Had the second field magnitude increased at $0.80\ \mathrm{T/s}$, the normal field
> rates would cancel and the net emf would be zero. Each source still contributes a
> nonzero flux rate; cancellation occurs only in their sum.

Spatial superposition may require separate integrals. If source 1 is uniform over the
whole coil and source 2 is localized near one edge, write

$$
\Phi_B
=\int_S\vec B_1\cdot \d\vec A
+\int_S\vec B_2\cdot \d\vec A.
$$

Replacing the localized source by its field value at the coil center is justified only
when its normal component varies negligibly across the coil. The same area-weighting
rules used for static flux apply to every time-dependent source term.

## Piecewise Histories and Checks

The emf follows the local slope of the flux history. Use a piecewise linear flux
function with the linkage

$$
\Lambda_B(t)=
\begin{cases}
kt, & 0\leq t<T,\\
kT, & T\leq t<2T,\\
k(3T-t), & 2T\leq t<3T.
\end{cases}
$$

Its derivative is

$$
\frac{\d\Lambda_B}{\d t}=
\begin{cases}
k, & 0<t<T,\\
0, & T<t<2T,\\
-k, & 2T<t<3T.
\end{cases}
$$

Faraday's law gives

$$
\mathcal E(t)=
\begin{cases}
-k, & 0<t<T,\\
0, & T<t<2T,\\
+k, & 2T<t<3T.
\end{cases}
$$

The endpoint corners represent idealized abrupt changes in flux rate. A real apparatus
rounds those corners over a finite time, so the emf changes continuously rather than
jumping infinitely fast. The piecewise model applies when the transition times
are short compared with intervals of nearly constant flux rate.

The average emf over the full interval from $0$ to $3T$ is determined only by the
initial and final linkage values. In this example both are zero, so

$$
\mathcal E_{\rm avg,full}=0.
$$

The circuit experiences a negative and a positive induction interval despite the zero
average. Each interval can drive measurable current or transfer energy to a resistive
load. Averaging across opposite pulses merges the shorter-time behavior.

An exponential field change gives a different time profile. If the flux linkage is

$$
\Lambda_B(t)=\Lambda_0\left(1-e^{-t/\tau}\right),
$$

then

$$
\mathcal E(t)=-\frac{\Lambda_0}{\tau}e^{-t/\tau}.
$$

The emf has its largest magnitude immediately after the change begins and decays on
the time scale $\tau$. The linkage approaches a constant value, so the emf approaches
zero. This behavior appears whenever a magnetic source changes rapidly at first and
then approaches a steady state.

Flux histories with the same endpoint change can have very different peak emfs. A
change of $\Delta\Lambda_B$ spread uniformly over time $T$ has constant magnitude
$|\Delta\Lambda_B|/T$. The same total change compressed into a short subinterval
$T/10$ has a peak or interval magnitude ten times larger. Faraday's law depends on the
rate, so time shape matters whenever circuit breakdown, sensor saturation, or current
limits are relevant.

The flux function should be differentiated before a resistor law is applied. First
obtain the signed emf from the linkage rate. Then introduce resistance, inductance, or
capacitance according to the circuit model. Reversing this order can replace a
time-dependent induced source with an unsupported constant-voltage assumption.

### Complete fixed-coil calculation

> **Worked example (complete fixed-coil calculation).** A stationary coil has $N=240$
> turns, area $A=3.5\times10^{-3}\ \mathrm{m^2}$, and selected normal fixed at $30^\circ$
> to a spatially uniform field of magnitude
>
> $$
> B(t)=0.20+0.60t^2\ \mathrm T
> $$
>
> with $t$ in seconds. The geometry is constant, so
>
> $$
> \Lambda_B(t)
> =NA\cos30^\circ\,B(t),
> \qquad
> NA\cos30^\circ
> =(240)(3.5\times10^{-3})(0.866)
> =0.727\ \mathrm{m^2}.
> $$
>
> Thus $\Lambda_B(t)=(0.727)(0.20+0.60t^2)\ \mathrm{Wb}$, and differentiating,
>
> $$
> \frac{\d\Lambda_B}{\d t}
> =(0.727)(1.20t)\ \mathrm{Wb/s},
> \qquad
> \mathcal E(t)=-\frac{\d\Lambda_B}{\d t}=-0.873t\ \mathrm V.
> $$
>
> At $t=0.50\ \mathrm s$, $\mathcal E=-0.436\ \mathrm V$. The negative sign follows from
> an increasing positive flux under the selected normal. The field is already nonzero at
> $t=0$, but its derivative is zero there, so the emf is zero at that instant. Field
> value, flux linkage, flux rate, and emf have different time dependence and different
> units.
>
> If the coil resistance is $6.0\ \Omega$ and inductive effects are negligible over the
> interval, the instantaneous current is
>
> $$
> I(0.50\ \mathrm s)=\frac{\mathcal E}{R}=\frac{-0.436}{6.0}
> =-7.3\times10^{-2}\ \mathrm A.
> $$
>
> The current sign is relative to the same chosen loop direction; reversing the coil
> leads reverses it at the same physical state. The current magnitude depends on
> resistance, the emf magnitude does not.

Several changes provide direct checks on the expression. Doubling the turn count
doubles the geometry factor and the emf. Replacing the $30^\circ$ angle with
$90^\circ$ makes the normal field component zero at every time and gives zero flux and
zero emf for this fixed orientation. Reversing the field direction changes the signs of
linkage and emf. Replacing $0.60t^2$ with a linear term $ct$ makes the emf constant
because the field rate becomes the constant $c$.

The calculation would change if the field were nonuniform across the coil. In that
case, replace $BA\cos30^\circ$ by the surface integral of the local normal component
before differentiating. It would also change if the coil rotated or its area changed:
the geometry factor would then be time dependent and product-rule terms would enter.
Those modifications follow directly from the integral form of Faraday's law rather than
from a separate rule for each apparatus.

The final emf should be reported with a sign convention, a numerical magnitude, and
the conditions under which the coil resistance was used to infer current. This record
keeps field geometry, induction, and circuit response distinguishable when the model is
extended to Lenz's law, motional emf, inductance, and alternating-current circuits.

## Scope of Faraday's Law

Faraday's law gives an oriented loop emf from the signed flux rate,

$$
\mathcal E_{\partial S}
=-\frac{\d}{\d t}\int_S\vec B\mathbin{\cdot}\d\vec A.
$$

The input specification consists of the boundary $\partial S$, its selected normal,
a spanning surface, the magnetic field over that surface, and the time dependence of
the field-surface configuration. The result is a line-integral quantity around the
boundary, whether the boundary is a closed conductor, an open conducting path, or a
geometrical curve.

| quantity | additional model required | later treatment |
|---|---|---|
| induced emf | signed flux history and orientation | present lesson |
| circuit current | topology and constitutive relations | resistance, inductance, capacitance |
| induced-current direction | closed-path current and response field | Lenz's law |
| sliding-conductor emf | carrier force and moving geometry | motional emf |
| self-induced emf | current-dependent flux linkage | self-inductance |

- **Orientation first.** The minus sign relates the emf sign to the selected
  flux-rate sign. Lenz's law then determines the induced-current direction in a
  closed circuit. Retain signed flux until after differentiating; select the normal
  and loop direction before applying any right-hand rule.
- **Fixed and moving configurations.** A fixed coil in a prescribed field isolates
  the relation between flux history and emf. A nonuniform field requires the surface
  integral of its local normal component. Rotation or changing area makes the
  geometry time dependent and introduces product-rule terms. A moving conductor
  also requires the velocity-dependent magnetic force on its carriers and a stated
  mechanical energy boundary.
- **Circuit response.** A resistor relates a specified emf to current only under its
  stated model conditions. An inductor changes current through its stored magnetic
  energy; a capacitor can accumulate charge at an open gap. Terminal voltage along
  arbitrary external leads therefore requires circuit topology after the induced
  source term has been established.
- **Instantaneous and interval results.** A derivative gives the instantaneous emf.
  With endpoint flux data alone, report the interval average
  $\overline{\mathcal E}=-\Delta\Lambda_B/\Delta t$. Equal endpoint fluxes do not
  determine the peak emf: the same endpoints can arise from steady change, a late
  rapid change, or a reversal inside the interval. Record the interval endpoints and
  the selected normal with every average value.
- **Units and report.** Convert the complete configuration to SI units before
  differentiating or forming a finite difference: area in square metres, flux in
  webers, and time in seconds. State the sign convention, numerical emf, field
  geometry, turn count, and the resistance conditions used when inferring current.
  A millisecond interval or square-centimetre area changes an emf by powers of ten
  even when the symbolic flux relation is correct.

A sampled-flux reduction benefits from four separate columns: time, signed flux
linkage, derivative or finite difference, and induced emf. Do not label an endpoint
difference as an instantaneous derivative. With samples at $t_1$ and $t_2$, the
reported quantity is the average $-[\Lambda_B(t_2)-\Lambda_B(t_1)]/(t_2-t_1)$
unless a model for the intervening flux history has been supplied. A central
difference can estimate a local derivative only when samples on both sides are close
enough for the stated time resolution. Retain the normal convention beside every
column so that a later polarity reversal, lead swap, or Lenz-law construction can be
checked without inferring signs from a graph.

When a measured terminal trace is used to test the calculation, align its time axis
with the flux record and state the lead polarity used by the voltmeter. A lead reversal
changes the displayed terminal sign without changing the physical flux history.
Comparing the signed terminal trace with the computed emf therefore tests the combined
orientation, lead, and instrument conventions. Comparing magnitudes alone cannot
separate a correct flux derivative from a reversed connection.