---
title: Current and Resistance
module: Direct-Current Circuits
moduleNumber: 5
lessonNumber: 1
order: 501
summary: >
  What does it mean, physically, for charge to flow, and what sets how hard a wire
  resists that flow? Current counts charge crossing a surface,
  $I=\int\vec J\cdot\d\vec A$, and traces back to a slow drift of many carriers,
  $\vec J=nq\vec v_d$. We establish when the linear law $V=IR$ actually holds, how
  resistivity and geometry combine into bulk resistance, why real sources sag under
  load through their internal resistance, and how the three power forms
  $P=IV=I^2R=V^2/R$ tie electrical work to heating and component ratings.
topics: [Direct-Current Circuits]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 25 — Electric Current and Direct-Current Circuits; §§25-1–25-3"
---

Electric current is charge crossing a surface per unit time:

$$
I=\frac{\d Q}{\d t}.
$$

Conventional current has the direction of positive-charge flow. In a metal, the
mobile carriers are electrons and their drift velocity is opposite conventional
current. For carrier density $n$, carrier charge magnitude $q$, cross-sectional
area $A$, and drift-speed magnitude $v_d$,

$$
I=nqAv_d,\qquad \vec J=nq\vec v_d.
$$

The drift speed is typically very small even when a circuit responds rapidly; the
electric field becomes established throughout the conducting path at a substantial
fraction of the speed of light.

$$
% caption: Electron drift in a metal runs opposite the conventional current $I$. Each carrier's drift velocity $\vec v_d$ (short blue arrows) is a small net bias on top of large random thermal motion; the conventional current $I$ points the way positive charge would move.
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\node[black,below] at (3.2,-1.42) {conventional current $I$};
\node[acc,above] at (3.2,0.95) {drift velocity};
\end{tikzpicture}
$$

## Temperature coefficients and resistance thermometry

Metal resistance commonly changes with temperature because carrier scattering changes
as the lattice vibrates. A calibration therefore specifies a reference temperature
and a reference resistance. Over a limited interval, resistance may be approximated
by a linear temperature coefficient. That coefficient is local to the material,
composition, and calibration interval; it should not be extrapolated casually to a
wide temperature range or to a differently fabricated sensor.

Resistance thermometry uses a measured sensor resistance to infer temperature from
that calibration. The sensing current must be small enough that electrical power in
the sensor does not raise its temperature appreciably. Self-heating produces a
reading above the ambient temperature for a positive-coefficient metal sensor, and
the error depends on thermal contact, airflow, mounting, and current. Reducing the
current, using pulsed measurements, or applying a measured self-heating correction
can control this effect.

Lead and contact resistance can distort a low-resistance measurement. A four-wire
connection separates current-carrying leads from voltage-sensing leads, so the sensed
voltage is taken close to the element rather than across the full current path. This
arrangement is particularly important for precision resistance thermometers and long
lead runs. Poor thermal coupling and a sensor response altered by age or strain remain
separate error sources.

Calibration points should span the intended operating range and include repeated
measurements at stable temperatures. A straight-line fit is suitable only while its
residuals are consistent with the required accuracy. Curvature, hysteresis, or drift
requires a higher-order or tabulated calibration. A reported temperature should state
the reference point, measurement current, lead arrangement, calibration model, and
valid temperature range.


### Shunt current measurements and calibration

A shunt resistor converts current into a small voltage drop. In a four-terminal
connection, the force leads carry the load current while separate sense leads measure
the potential directly across the calibrated shunt element. Sense-lead current is
small, so lead and contact resistance contribute far less error than in a two-wire
measurement. The inferred current is the measured shunt voltage divided by the
calibrated resistance at its stated temperature.

A shunt adds series resistance, causing burden voltage and $I^2R$ heating.
Self-heating
changes shunt resistance through its temperature coefficient, shifting a calibration
made at low current. A smaller resistance reduces burden voltage and heating but also
reduces signal amplitude relative to amplifier noise, offset, and digitizer
resolution. Shunt choice is therefore a compromise between circuit disturbance and
measurement sensitivity.

Calibration should state the reference resistance, temperature range, sense method,
and uncertainty in measured voltage. Dynamic current measurements add bandwidth
limits: shunt inductance, lead geometry, amplifier response, and filtering can make a
fast transient voltage differ from the dc resistance prediction. Include measurement
bandwidth and uncertainty with every reported current value.

An ohmic conductor at fixed temperature obeys $\Delta V=IR$. Uniform material of
length $L$, cross section $A$, and resistivity $\rho$ has

$$
R=\rho\frac{L}{A}.
$$

Resistivity is a material property; resistance also depends on geometry. Many
metals have approximately $\rho=\rho_0[1+\alpha(T-T_0)]$ over a limited range.

An ideal source of emf $\mathcal E$ raises energy per unit charge by $\mathcal E$.
A real source with internal resistance $r$ has terminal voltage $V=\mathcal E-Ir$
while delivering current. Electrical power is

$$
P=I\Delta V=I^2R=\frac{(\Delta V)^2}{R}.
$$

Joule heating is the local conversion of electrical energy into internal energy.

$$
% caption: A real source is an ideal emf $\mathcal E$ in series with an internal resistance $r$. While it drives current $I$ through an external load $R$, the terminal voltage between $a$ and $b$ sags to $V=\mathcal E-Ir$, below the open-circuit emf.
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\draw[black,thick] (1.5,2.6)--(5.3,2.6);
\draw[black,thick] (1.5,0)--(5.3,0);
% left branch: emf plates then internal resistance r
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\draw[black,thick] (1.5,2.15)--(1.5,2.6);
% right branch: load R
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\draw[black,thick] (5.00,0.95) rectangle (5.60,1.65);
\node at (5.3,1.30) {$R$};
\draw[black,thick] (5.3,1.65)--(5.3,2.6);
% terminals a,b
\filldraw[draw=acc,fill=acc!12] (1.5,2.6) circle (2.2pt);
\filldraw[draw=acc,fill=acc!12] (1.5,0) circle (2.2pt);
\node[acc,left] at (1.5,2.6) {$a$};
\node[acc,left] at (1.5,0) {$b$};
% current arrow
\draw[->,very thick] (3.0,2.6)--(3.9,2.6);
\node[above] at (3.45,2.62) {$I$};
\end{tikzpicture}
$$

### Resistor combinations

Series resistors carry the same current, so $R_{\rm eq}=\sum_iR_i$. Parallel
resistors share the same voltage, so $1/R_{\rm eq}=\sum_i1/R_i$. Kirchhoff's
junction rule, $\sum I_{\rm in}=\sum I_{\rm out}$, is charge conservation.
The loop rule, $\sum\Delta V=0$, is energy conservation for a closed traversal.

## Microscopic current and drift

Current is the rate at which charge crosses a chosen surface,

$$
I=\frac{\d Q}{\d t}.
$$

Conventional current points in the direction positive charge would move. In a metal,
electrons drift oppositely. For carrier density $n$, charge magnitude $q$, wire area
$A$, and drift speed $v_d$,

$$
I=nqAv_d,\qquad\vec J=nq\vec v_d.
$$

The drift speed is usually small because conductors contain many carriers. Circuit
response follows rapid establishment of the electric field configuration along the
conducting path after a switch closes.

Current density is a local vector. In a uniform wire its magnitude is $I/A$. At a
junction, charge conservation requires the total current entering to equal the total
leaving. This condition is the basis of Kirchhoff's junction rule.

## Resistance, resistivity, and Ohm's law

Ohm's law for an ohmic element at fixed temperature is $\Delta V=IR$. A uniform
wire of length $L$, cross-sectional area $A$, and material resistivity $\rho$ has

$$
R=\rho\frac LA.
$$

$$
% caption: Bulk resistance of a uniform conductor. Resistance grows with length $L$ and falls with cross-sectional area $A$: $R=\rho L/A$, so doubling $L$ doubles $R$ while doubling $A$ halves it. The current $I$ enters one face and leaves the other along the length.
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\draw[->,very thick] (-1.05,0.55)--(-0.15,0.55) node[midway,above] {$I$};
\draw[->,very thick] (4.55,0.55)--(5.45,0.55);
\end{tikzpicture}
$$

Resistivity is a material property; resistance includes geometry. Doubling wire
length doubles resistance, while doubling cross-sectional area halves it. Metals
often have approximately $\rho=\rho_0[1+\alpha(T-T_0)]$ over a moderate temperature
range. Semiconductors and filament lamps need not have constant resistance because
their carrier populations or temperatures vary with current.

The microscopic form for a simple conductor is $\vec J=\sigma\vec E$, where
$\sigma=1/\rho$ is conductivity. This local relation becomes $V=IR$ only after a
uniform field and geometry have been imposed. A device can have a nonlinear
current-voltage characteristic and still obey charge conservation.

## Emf and terminal voltage

An ideal source supplies energy $\mathcal E$ per unit charge. A real source with
internal resistance $r$ has terminal voltage

$$
V_{\rm term}=\mathcal E-Ir
$$

while delivering current. The difference $Ir$ is energy converted to internal heat.
When a source is charged, the current direction reverses and terminal voltage can
exceed its emf. Electromotive force is a voltage-like energy-per-charge quantity,
not a mechanical force.

## Power and energy conversion

Electrical power entering an element is

$$
P=I\Delta V=I^2R=\frac{(\Delta V)^2}{R}
$$

for an ohmic resistor. The positive result is Joule heating. A source supplies power
$I\mathcal E$, of which $I^2r$ may be lost internally. Power calculations require
consistent current direction and voltage polarity. The expression $I^2R$ applies to
a resistor, not automatically to a battery or capacitor.

> **Worked example (resistive heating).** A $12\ \mathrm V$ source drives a
> $6.0\ \Omega$ resistor. Find the current, the dissipated power, and the heat
> delivered in $30\ \mathrm s$.
>
> The current is $I=V/R=12/6.0=2.0\ \mathrm A$, so the power is
>
> $$
> P=I\Delta V=(2.0)(12)=24\ \mathrm W.
> $$
>
> Over $30\ \mathrm s$ the resistor converts $24\times30=720\ \mathrm J$ to heat.
> Units check the chain: $\mathrm{A\cdot V=W}$ and $\mathrm{W\cdot s=J}$.

## Series and parallel resistance

Series resistors carry identical current and their voltage drops add,

$$
R_{\rm eq}=R_1+R_2+\cdots.
$$

Parallel resistors share identical voltage and their currents add,

$$
\frac1{R_{\rm eq}}=\frac1{R_1}+\frac1{R_2}+\cdots.
$$

The equivalent parallel resistance is smaller than the smallest branch resistance.
A network must be labelled by nodes before reduction; resistors merely drawn near
one another are not necessarily parallel or series elements.

$$
% caption: Two ways to combine resistors. In series (left) one current $I$ passes through both and the drops add, so $R_{\rm eq}=R_1+R_2$. In parallel (right) both share the same terminal voltage and the currents add, so $1/R_{\rm eq}=1/R_1+1/R_2$; the parallel value is smaller than either branch.
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% --- series panel ---
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\node at (0.95,1.25) {$R_1$};
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\draw[black,thick] (1.85,1.00) rectangle (2.65,1.50);
\node at (2.25,1.25) {$R_2$};
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\draw[->,acc,very thick] (0.25,1.95)--(2.95,1.95);
\node[acc,above] at (1.60,1.97) {$I$};
\node[black,below] at (1.60,0.80) {series};
% --- parallel panel ---
\draw[black,thick] (4.10,1.90)--(6.50,1.90);
\draw[black,thick] (4.10,0.60)--(6.50,0.60);
\draw[black,thick] (3.60,1.90)--(4.10,1.90);
\draw[black,thick] (3.60,0.60)--(4.10,0.60);
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\end{tikzpicture}
$$

## Safety and model limits

Real wires have finite resistance, sources have internal resistance, and temperature
changes can alter circuit behaviour. Large currents can overheat conductors because
heating scales as $I^2$. The ideal circuit model assumes charge does not accumulate
at ordinary nodes after the short electromagnetic transient. At high frequency,
distributed capacitance and inductance make a simple lumped resistance model
inadequate.

### Potential changes around circuit elements

An ideal wire has negligible resistance and is one equipotential node in the lumped
circuit approximation. Across a resistor, potential decreases in the direction of
conventional current by $IR$. Across an ideal source, potential increases from its
negative terminal to its positive terminal by $\mathcal E$. These sign statements
allow a circuit equation to be constructed without guessing a direction after the
fact.

Charge carriers in a resistor transfer energy to the lattice through collisions.
The electric field does work on carriers between collisions; the resulting increase
in microscopic random motion is Joule heating. A resistor does not consume charge.
The same current enters and leaves it in steady state, while energy per unit charge
falls across it.

### Resistor combinations by conservation

For resistors $R_1$ and $R_2$ in series, current $I$ is common and terminal voltage
is $I R_1+I R_2=I(R_1+R_2)$. For parallel resistors, voltage $V$ is common and total
current is $V/R_1+V/R_2=V(1/R_1+1/R_2)$. Series reduction follows the common
current; parallel reduction follows the common voltage. A voltage divider has

$$
V_1=V_{\rm source}\frac{R_1}{R_1+R_2}
$$

only when the output node is not significantly loaded by another branch. Adding a
load creates a parallel resistance and alters the division.

### Temperature and non-ohmic elements

Ohm's law is an empirical linear relation over a specified operating range. A metal
wire heated by its own current may have rising resistance, so its $I$--$V$ graph is
curved. A diode strongly favours one current direction. A battery has an emf and
internal resistance rather than a fixed terminal voltage at all currents. In each
case, the local conservation laws still hold, but the relation between voltage and
current varies with operating point rather than remaining a constant resistance.

Superconductors are idealized as zero-resistance conductors below critical conditions.
Their existence does not imply an ideal source can drive infinite current: circuit
inductance, source limits, critical current, and other effects constrain real loops.

> **Worked example (source with internal resistance).** An emf
> $\mathcal E=12\ \mathrm V$ and internal resistance $r=1.0\ \Omega$ drive an
> external $R=5.0\ \Omega$ resistor. Find the current, terminal voltage, and the
> split of source power.
>
> The single loop carries
>
> $$
> I=\frac{\mathcal E}{R+r}=\frac{12}{6.0}=2.0\ \mathrm A.
> $$
>
> Terminal voltage is $\mathcal E-Ir=12-2.0=10\ \mathrm V$, equal to $IR$ across
> the load. The source delivers $I\mathcal E=24\ \mathrm W$; of this,
> $I^2R=20\ \mathrm W$ reaches the external resistor and $I^2r=4\ \mathrm W$ heats
> the internal resistance, and $20+4=24\ \mathrm W$ closes the energy balance.
> Maximum power to a variable external resistance occurs at $R=r$, where the
> efficiency is only one-half.

### Dimensional and directional checks

Resistivity has $\mathrm{\Omega\,m}$, so $\rho L/A$ has ohms. Current density has
$\mathrm{A/m^2}$. The sign of an assumed current can be selected arbitrarily; a
negative solved value reverses its actual direction. Voltage drop labels must be
consistent with that assumed direction. In a steady resistor network, a node cannot
continuously accumulate charge, so the algebraic current sum at every junction is
zero. Check units, current orientation, voltage labels, and junction balance before
calculation.

## Drude transport picture

In the Drude model, conduction electrons accelerate between collisions with lattice
ions and defects. Their random thermal speeds are large, but the electric field
produces a small average drift velocity. If the mean time between momentum-randomizing
collisions is $\tau$, the average drift response is proportional to $q\vec E\tau/m$.
The resulting conductivity has the form

$$
\sigma=\frac{nq^2\tau}{m}.
$$

The expression is a simplified microscopic model, but it explains why greater
carrier density or a longer collision time produces higher conductivity. It also
explains the usual increase of metal resistivity with temperature: lattice vibrations
increase collision frequency and reduce the effective $\tau$. In semiconductors,
heating can instead increase carrier density enough to reduce resistivity.

The drift equation $I=nqAv_d$ is a count of carriers passing a cross section. It
does not state that individual electrons retain a constant speed along a wire.
Scattering repeatedly changes their velocities while the net charge flow remains
steady. Current density can vary with position in a nonuniform conductor, but for a
steady series wire the same total current crosses each cross section.

### Geometry and material selection

Resistance scales with $L/A$. A long thin lead can have appreciable voltage drop even
when its material is a good conductor. Power transmission uses high voltage partly
because delivering fixed power $P=IV$ at larger $V$ requires smaller $I$, reducing
line loss $I^2R$. The reduction is quadratic in current. Insulation thickness and
electrical safety then limit how high a practical voltage can be used.

Resistivity should not be confused with resistance per unit length unless cross
section is specified. Two wires of the same material may have very different
resistances. A four-terminal resistance measurement separates the voltage-sensing
leads from current-carrying leads, reducing errors from contact resistance when a
very small sample resistance is being measured.

## Power transfer and efficiency

With a source of emf $\mathcal E$ and internal resistance $r$ connected to load
$R$, load power is

$$
P_R=\frac{\mathcal E^2R}{(R+r)^2}.
$$

Differentiating with respect to $R$ gives maximum load power at $R=r$. At that point,
half the source power is lost in $r$, so efficiency is $R/(R+r)=1/2$. Power systems
usually prioritize high efficiency and use $R\gg r$. The maximum-power condition
serves signal matching and finite-source problems with deliberately specified loads.

### Measurement conventions and error cases

An ammeter is placed in series and should have small internal resistance. A voltmeter
is placed in parallel and should have large internal resistance. An ideal voltmeter
draws no current; a real one can load a high-resistance circuit. An ideal ammeter has
zero voltage drop; a real one changes the series resistance slightly. These effects
must be included when instrument resistance is comparable with circuit resistance.

Power signs must identify whether an element absorbs or supplies energy. Under the
passive sign convention, an element absorbs power when current enters its labelled
positive-voltage terminal. A source can have negative absorbed power, indicating
delivery. Reporting every $IV$ product as positive heat removes the distinction
between absorbed and delivered power.

Circuit diagrams suppress spatial detail, but every reported current, voltage, and
power value refers to a stated pair of terminals, a chosen direction, and an
operating condition. Record terminal pairs, current orientations, and operating
conditions with each result.

Record the time at which a reading was taken after a source change. A resistor,
connector, or sensor can still be warming while a meter display appears steady.
Temperature, source mode, lead routing, and instrument range define the operating
condition alongside the schematic connections.

### Current density in tapered and composite conductors

The same steady current crosses every complete transverse section of a simple
series conductor. Its current density need not be uniform along the conductor. A
tapered metal strip with local area $A(x)$ has the one-dimensional approximation

$$
J(x)=\frac{I}{A(x)},\qquad E(x)=\rho\frac{I}{A(x)}.
$$

The narrower region therefore has larger current density, larger electric field,
and larger power converted per unit length. For a short segment $\d x$, the
resistance and dissipation are

$$
\d R=\rho\frac{\d x}{A(x)},\qquad \d P=I^2\rho\frac{\d x}{A(x)}.
$$

Thin regions of a fuse exploit this concentration. Their resistance is a small
part of the total circuit resistance at ordinary current, yet their temperature
rises rapidly when the current exceeds the rated value. The result depends on heat
transfer to the surroundings as well as on $I^2R$; a fuse rating cannot be derived
from geometry alone.

$$
% caption: The same steady current $I$ crosses every section of a tapered conductor. Where the area shrinks from $A_1$ to $A_2$, the current density rises ($J=I/A$) and so does the local field $E=\rho J$; the narrow region therefore dissipates more power per unit length.
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\node[black,above] at (4.55,0.44) {area $A_2$};
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\node[black,below] at (4.55,-0.44) {$J_2>J_1$};
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\end{tikzpicture}
$$

At a change in material, continuity applies to the normal current density when
there is no charge accumulation at the interface. Ohmic materials obey
$\vec J=\sigma\vec E$, so the electric field changes when conductivity
changes. A high-resistivity film between two metal pieces can produce a substantial
voltage drop and local heating even when its physical thickness is small. Contact
quality matters in connectors, switches, and battery terminals for this reason.

The relation $I=nqAv_d$ also needs local interpretation. A reducing wire area
raises $J$ for fixed $I$; it does not require the carrier density $n$ to change.
In an inhomogeneous material, $n$, mobility, and carrier type can vary together.
The macroscopic quantities $I$, $V$, and $R$ remain reliable when their terminals
and operating temperature are specified.

### Contacts, leads, and four-terminal measurements

A two-terminal resistance measurement includes every series contribution between
the instrument terminals: leads, clips, oxide layers, solder joints, and the test
specimen. That is appropriate when the total installed resistance is required. It
is unsuitable for a milliohm-scale sample when each contact resistance is comparable
to or larger than the sample resistance. The measured value then has the form

$$
R_{\rm two}=R_{\rm lead,1}+R_{\rm c,1}+R_{\rm sample}+R_{\rm c,2}+R_{\rm lead,2}.
$$

A four-terminal, or Kelvin, measurement separates the current path from the
voltage-sensing path. A known current passes through the outer contacts. A
high-input-resistance voltmeter connects to two inner contacts, drawing negligible
current from them. The voltage leads then develop negligible voltage drop, and the
sample resistance follows from $R_{\rm sample}=V_{\rm sense}/I$. This arrangement
does not erase contact resistance; it keeps the sensed voltage between contacts
placed on the desired portion of the specimen.

$$
% caption: Four-terminal (Kelvin) measurement. The outer force leads carry the test current $I$; the inner sense leads feed a high-resistance voltmeter and draw negligible current, so the measured drop is that across the marked sample length alone. Lead and contact resistances in the force path drop out of the result.
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\draw[thick] (2.15,0)--(2.15,0.75);
\draw[thick] (4.25,0)--(4.25,0.75);
% force leads
\draw[black,thick] (0.15,0.375)--(1.0,0.375);
\draw[black,thick] (5.4,0.375)--(6.25,0.375);
\draw[->,black,very thick] (0.30,0.375)--(0.85,0.375);
\node[black,above] at (0.55,0.42) {$I$};
\draw[->,black,very thick] (5.55,0.375)--(6.10,0.375);
\node[black,above] at (5.85,0.42) {$I$};
% sense leads to voltmeter
\draw[acc,thick] (2.15,0.75)--(2.15,1.70)--(2.86,1.70);
\draw[acc,thick] (4.25,0.75)--(4.25,1.70)--(3.54,1.70);
\draw[acc,thick] (3.20,1.70) circle (0.32);
\node[acc] at (3.20,1.70) {V};
% sample bracket
\draw[black] (2.15,-0.22)--(4.25,-0.22);
\draw[black] (2.15,-0.12)--(2.15,-0.32);
\draw[black] (4.25,-0.12)--(4.25,-0.32);
\node[black,below] at (3.20,-0.30) {sample length};
\end{tikzpicture}
$$

Instrumentation introduces a further condition. A voltmeter with finite input
resistance is a parallel branch, and an ammeter with finite internal resistance is
a series branch. The fractional loading error is small only when the meter
resistance is separated by a large ratio from the resistance being measured. A
voltmeter rated at $10\ \mathrm{M\Omega}$ barely loads a $1\ \mathrm{k\Omega}$ divider
node, but it substantially alters a divider whose Thevenin resistance is several
megohms. Calibration specifications therefore list input resistance, burden voltage,
accuracy, and measurement range rather than a single universal uncertainty.

### Thermal limits, ratings, and time dependence

Near ambient temperature, a resistor reaches thermal equilibrium when electrical
heating equals heat transfer to the surroundings. A simple lumped
model is

$$
C_{\rm th}\frac{\d T}{\d t}=I^2R(T)-G_{\rm th}(T-T_{\rm amb}),
$$

where $C_{\rm th}$ is thermal capacitance and $G_{\rm th}$ is thermal conductance to
the surroundings. This equation gives a finite thermal response time. A resistor
can tolerate a brief pulse that would exceed its continuous power rating, while a
longer pulse of lower power may still raise its temperature above the allowable
limit. Datasheet pulse curves encode the component geometry and thermal path that
the simple model groups into two parameters.

A metal with positive temperature coefficient has higher $R$ after heating, thereby
raising dissipation at fixed current. At fixed applied voltage, heating raises $R$
and reduces $I=V/R$, which can limit the electrical power. The operating condition
matters: current-driven and voltage-driven circuits have different thermal feedback.
Filament lamps illustrate the effect because their cold resistance is much smaller
than their operating resistance. The initial inrush current can exceed the steady
current by a large factor.

$$
% caption: Thermal balance of a current-driven resistor. Electrical heating $I^2R(T)$ rises slowly with temperature, while heat loss to the surroundings climbs faster from ambient. Below the crossing the resistor heats up; the steady operating temperature $T_{\rm op}$ is where the two curves meet.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(5.2,0) node[right] {$T$};
\draw[->,black] (0,0)--(0,3.1) node[above] {$P$};
\draw[acc,thick] plot[smooth] coordinates {(0.30,0.95) (1.40,1.25) (2.50,1.52) (2.94,1.65) (3.60,1.85) (4.60,2.15)};
\draw[black,thick] plot[smooth] coordinates {(0.30,0.15) (1.40,0.78) (2.50,1.40) (2.94,1.65) (3.60,2.05) (4.60,2.60)};
\filldraw[draw=black,fill=black!8] (2.94,1.65) circle (1.6pt);
\draw[black,dashed] (2.94,0)--(2.94,1.65);
\node[acc,above] at (1.20,1.80) {$I^2R$};
\node[black,above] at (3.75,2.55) {heat loss};
\node[black,below] at (2.94,0) {$T_{\rm op}$};
\end{tikzpicture}
$$

Wire insulation, connector contacts, and enclosure temperature can set the safe
current before the conductor itself approaches its melting temperature. Circuit
protection therefore combines conductor ampacity, overcurrent devices, source
capacity, and the prospective fault current. A resistance calculation alone does
not establish a safe operating current.

### Voltage-current characteristics and operating points

An I--V characteristic records the current through an element as its terminal voltage
changes. An ideal ohmic resistor gives a straight line through the origin, with
constant resistance equal to the ratio of voltage to current. Many real elements are
nonlinear: a metal filament heats as current rises, a semiconductor diode changes
carrier transport with bias, and a source with internal resistance has terminal
voltage that falls as delivered current rises. A resistance value must therefore be
identified as either a static ratio at one operating point or a local differential
quantity determined by the slope of the curve.

Static resistance is the quotient of terminal voltage and current at the stated
point. Differential resistance is the small-signal ratio of a small voltage change to
the resulting current change at that same point. The two agree for a straight-line
ohmic characteristic but differ on a curved characteristic. Reporting a single
resistance for a nonlinear element without an operating voltage or temperature leaves
the static and differential quantities unspecified.

A source and load operate at the intersection of their characteristics. For a source
with emf and internal resistance, terminal voltage decreases linearly with current.
The load characteristic gives a second relation. Their intersection determines the
simultaneous current and terminal voltage from both relations. Temperature can
move the load curve, shifting the operating point and changing dissipated power.

$$
% caption: Operating point of a source driving a nonlinear load. The descending source line (terminal voltage falls as current rises) meets the rising load characteristic at one point, fixing the simultaneous current and voltage. The dashed tangent there sets the differential resistance $\d V/\d I$; the ratio from the origin to the point sets the static resistance $V/I$.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(4.9,0) node[right] {$V$};
\draw[->,black] (0,0)--(0,3.2) node[above] {$I$};
\draw[black,thick] (0.40,2.70)--(4.30,0.40);
\draw[acc,thick] plot[smooth] coordinates {(0.40,0.15) (1.30,0.45) (2.00,0.85) (2.60,1.40) (3.00,2.00) (3.30,2.60)};
\draw[acc,dashed] (2.00,0.56)--(3.20,2.24);
\filldraw[draw=black,fill=black!8] (2.60,1.40) circle (1.7pt);
\node[black,below right] at (3.95,0.55) {source};
\node[acc,left] at (2.55,2.60) {load};
\end{tikzpicture}
$$

The load-line method remains a dc graphical model. Time-dependent capacitive or
inductive elements require their own state equations, and thermal changes can make an
operating point drift rather than remain fixed.

### Line resistance and remote sensing

Two-wire delivery leads have finite resistance, so a load receives less voltage than
the source terminals provide. If each lead has resistance $R_{\rm lead}$ and current
is I, the round-trip drop is $2IR_{\rm lead}$. The load voltage is source terminal
voltage minus this drop. A measurement made only at the source cannot distinguish a
healthy source from a voltage loss in long or undersized leads.

Remote sensing uses a separate pair of high-resistance sense leads connected at the
load terminals. Their current is negligible, so their own voltage drop is small. A
regulated source can compare this sensed load voltage with its target and increase its
output to compensate for the power-lead drop. Compensation is limited by source
voltage range, stability of the feedback loop, and the changing resistance of hot
power leads. The sense pair measures load voltage while the current-carrying leads
continue to dissipate their power loss.

### Source loading and power curves

A real voltage source has an emf and an internal resistance. Its terminal voltage
falls linearly as delivered current rises because part of the energy per unit charge
is dissipated inside the source. At open circuit, current is zero and terminal voltage
equals emf. At short circuit, terminal voltage approaches zero in the ideal model
while current is limited by internal resistance. Neither endpoint is normally a safe
operating condition: open circuit delivers no load power, while short circuit can
produce damaging internal heating.

Load power depends on both current and terminal voltage. Starting from open circuit,
increasing load current initially increases power delivered to the load. At larger
current, terminal voltage falls enough that load power eventually decreases. The
maximum occurs when load resistance equals internal resistance in the ideal
Thevenin-source model. This condition maximizes load power but gives only half of the
source power to the load; the other half becomes internal heating. Efficient power
systems instead use load resistance much larger than internal resistance.

Measurements alter this curve when instrument resistance is not negligible. A
voltmeter draws a small current in parallel with a load, while an ammeter adds series
resistance. Source emf and internal resistance can be inferred from several
terminal-voltage/current measurements by fitting the linear terminal-voltage relation,
rather than treating one loaded voltage as an unloaded source rating.

$$
% caption: Loading a source that has internal resistance $r$. Terminal voltage falls linearly from the open-circuit emf (at $I=0$) to zero at the short-circuit current. Load power $P=IV$ rises, peaks, then falls; the maximum sits at half the short-circuit current, where the load resistance equals $r$ and only half the source power reaches the load.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(4.9,0) node[right] {$I$};
\draw[->,black] (0,0)--(0,2.9) node[above] {$V$, $P$};
\draw[black,thick] (0,2.40)--(4.20,0.15);
\draw[acc,thick] plot[smooth] coordinates {(0.15,0.22) (0.90,0.85) (1.50,1.25) (2.10,1.48) (2.70,1.35) (3.40,0.90) (4.10,0.22)};
\draw[black,dashed] (2.10,0)--(2.10,1.48);
\filldraw[draw=acc,fill=acc!10] (2.10,1.48) circle (1.7pt);
\node[black,above] at (1.05,2.10) {terminal $V$};
\node[acc,right] at (3.45,1.00) {load power $P$};
\node[black,below] at (2.10,0) {$R=r$};
\end{tikzpicture}
$$

### Electrical noise, bandwidth, and low-level resistance measurements

At low signal levels, a resistance measurement is limited by fluctuations as well as
by meter resolution. A resistor at absolute temperature $T$ produces Johnson
voltage noise with spectral density $\sqrt{4k_BTR}$. For an approximately flat
measurement bandwidth $\Delta f$, the rms noise voltage is
$\sqrt{4k_BTR\Delta f}$. High-resistance sources produce a larger noise voltage
and greater sensitivity to input leakage, cable contamination, and amplifier current
noise. Consider the instrument input resistance and noise model together with the
resistor under test before taking a reading.

Bandwidth determines how much of this random noise reaches the result. A voltmeter
that averages slowly has a narrower effective bandwidth than an oscilloscope taking
individual fast samples, so the same circuit can show very different rms variation on
the two instruments. Reducing bandwidth lowers uncorrelated white-noise variance but
also slows response to a changed resistance or current. If $N$ statistically
independent readings are averaged, random zero-mean noise in the mean falls roughly
as $1/\sqrt{N}$. Offset, thermal drift, contact rectification, and mains pickup are
not made harmless by this rule because successive readings of those effects are
correlated.

Use a current chosen to create a voltage well above the expected noise while keeping
Joule heating negligible. For a small resistance, four-terminal sensing separates
the current-carrying leads from the high-impedance voltage leads, so lead resistance
does not enter the measured voltage drop. For a large resistance, source current may
be limited by leakage rather than the nominal component value. A guarded connection
surrounds the sensitive high-impedance node with a conductor held near its potential.
Surface leakage then flows to the guard instead of through the measurement input.
Clean insulating supports and short, dry cable runs complete the same circuit
strategy.

Verification requires more than one displayed resistance. Reverse the test current
and compare the corresponding voltage change; this cancels many fixed thermoelectric
offsets and exposes polarity-dependent contacts. Record the effective bandwidth,
integration time, source current, temperature, and guarding arrangement. A residual
time trace helps distinguish white noise from a slow drift: random points scatter
without a persistent trend, whereas a steadily moving baseline indicates that longer
averaging will improve precision only superficially. The final uncertainty should
include both the observed spread and systematic limits from current calibration,
leakage, and self-heating.

### Distributed resistance, skin effect, and frequency limits

The dc lumped-resistance model treats a conductor as one element with
$R=\rho L/A$. It is reliable when current density is essentially uniform across
the cross-section, the voltage is nearly the same at every point of a nominal node,
and propagation delay, stray capacitance, and inductance are negligible on the time
scale of interest. A short copper lead at low frequency often meets these conditions.
Length, cross-sectional area, and resistivity then explain its resistance directly;
doubling length doubles resistance, while doubling area halves it. A nonuniform
temperature or a narrow connection can invalidate the use of one bulk geometry even
at dc, because local resistivity and current density then differ from the assumed
values.

At higher frequency, a changing magnetic field inside a conductor induces electric
fields that oppose changes in interior current. Current crowds toward the surface,
an effect called skin effect. For a good conductor, the characteristic penetration
depth is $\delta=\sqrt{2\rho/(\omega\mu)}$. When the wire radius is small compared
with $\delta$, the current remains nearly uniform and ac resistance is close to dc
resistance. When radius greatly exceeds $\delta$, only a surface layer carries most
of the current, reducing effective conducting area and increasing resistance. Nearby
conductors can further redistribute current through proximity effect, so tightly
packed windings and wide parallel traces can have more ac loss than a single isolated
wire would predict.

The same frequency range also exposes the distributed nature of a long conductor.
Its inductance and capacitance per unit length create voltage gradients and phase
delay, so assigning one series resistance does not capture the measured impedance.
The dc value can remain correct as the zero-frequency limit while the ac impedance
has both a larger real part and a reactive imaginary part. The relevant transition is
set by geometry, material, termination, and required accuracy; there is no universal
frequency at which every wire stops being a resistor.

Measure the limit by using a small-signal frequency sweep with known source and
fixture impedance. Four-terminal sensing removes lead drop at low frequency, then
record both voltage magnitude and phase across the specimen as frequency increases.
A rising in-phase voltage drop indicates increasing ac resistance; a growing phase
shift indicates inductive or capacitive effects. Repeat at low current to separate
frequency effects from self-heating. Comparing round wire, foil, and litz-wire
samples of equal dc resistance is an especially direct test: geometry changes the
current distribution and high-frequency loss even when the resistance meter reports
the same dc value.

### Power measurement, energy integration, and efficiency

Electrical power is a signed instantaneous quantity. With the passive sign
convention, $p(t)=v(t)i(t)$ is positive when the assigned current enters the
terminal marked positive for voltage; the element then absorbs power. A negative
value identifies delivery to the rest of the circuit. This convention matters for a
source, a battery under charge, or a changing load: reporting only positive current
and positive voltage magnitudes can conceal whether the measured device is supplying
or receiving energy.

In a time-varying circuit, voltage and current must be sampled at corresponding
instants before multiplication. A current shunt produces a voltage proportional
to current, while a differential voltage probe measures the load voltage; both
signals need known gain, polarity, and timing. A delay between channels can create a
large power error when waveforms change rapidly, even if each individual trace looks
accurate. Sampling rate and analog bandwidth must cover the significant waveform
content, and anti-alias filtering is needed when high-frequency noise could fold into
the recorded data.

Energy transferred over an interval is the signed integral
$W=\int p(t)\,\d t$. For sampled data, form each synchronized product and apply a
documented numerical sum using the sample interval. Inspect cumulative energy as
well as average power: a brief high-power event can contribute substantial energy
while disappearing in a long average. A zero-mean noise floor in power may average
down, but an offset in either voltage or current can accumulate into a false energy
trend. Record the integration start and stop conditions, particularly when the load
contains stored energy that changes between those times.

Efficiency compares delivered load power or energy with input power or energy under
the same boundary definition. For a dc source feeding a resistive load,
$\eta=P_{\rm load}/P_{\rm input}$ after source and lead losses have been assigned to
the input side. The quotient is meaningful only when both measurements use the same
time interval and sign convention. Calibrate the voltage channel against a stable
reference and the current channel against a known shunt or current source. Then test
the complete measurement chain with a precision resistor: calculated $VI$,
$I^2R$, and $V^2/R$ should agree within stated uncertainty. Reversing current
through the test resistor checks polarity and tests for fixed offsets before energy is
integrated over a long run.

### Traceability, contact offsets, and uncertainty

A resistance result is traceable when its voltage, current, temperature, and
connection method can be related to stated reference standards. A nominal resistor
value printed on a package does not provide that chain by itself. Precision work
uses a calibrated standard resistor or a reference current source, records the
measurement temperature, and identifies the uncertainty of each instrument range.
The calibration date matters when an amplifier gain, shunt value, or digitizer
offset can drift between checks.

The resistance inferred from a four-terminal dc measurement is

$$
R=\frac{V_{\rm sense}}{I_{\rm force}}.
$$

Its relative uncertainty contains voltage-channel gain, current-channel gain,
reference-resistor value, lead thermoelectric offsets, repeatability, and
self-heating. Independent random terms can be combined in quadrature; a common
reference-scale error remains correlated across a set of measurements and should be
reported separately. A measurement series that quotes many decimal places while
omitting its reference and temperature has high numerical resolution without a
defensible absolute accuracy.

Thermoelectric voltage becomes important when a low resistance is measured with a
small current. Junctions of dissimilar metals at different temperatures generate a
dc offset that adds to or subtracts from the resistive voltage. Reverse the current
through the specimen while keeping the sense polarity fixed. The resistive voltage
changes sign with current, while a slowly varying thermal offset retains its sign.
Combining the two readings separates the resistance term from the offset:

$$
V_+=IR+V_{\rm off},
\qquad
V_-=-IR+V_{\rm off},
\qquad
R=\frac{V_+-V_-}{2I}.
$$

The reversal interval must be long enough for current settling and short enough that
the junction temperatures do not drift appreciably between readings. A rapidly
heated contact can defeat the assumption of constant offset. Repeated positive and
negative measurements reveal this problem through a changing average offset or a
resistance estimate that depends on reversal order.

Contact resistance has a separate physical origin from bulk resistance. Surface
films, oxide layers, small real contact area, and mechanical pressure determine the
voltage drop at a connector. A two-terminal reading includes that drop; a
four-terminal reading can exclude it when the sense contacts lie inside the force
contacts. Connector heating, vibration, or corrosion can make contact resistance
time dependent. Measuring voltage at several locations along a current path locates
the dominant drop and separates a damaged joint from a uniformly resistive wire.

An uncertainty budget should remain tied to the operating point. A low-current
measurement may be limited by amplifier noise and thermoelectric offset. A
high-current measurement may be limited by self-heating, shunt power, lead drop, and
source regulation. A resistance thermometer adds thermal coupling and calibration
curve uncertainty. The same component can therefore have different quoted
uncertainties in different circuits without a contradiction; the measurement
conditions have changed.

### Documentation and cross-checks for resistance data

Record the specimen identity, geometry, material condition, connection positions,
ambient temperature, drive current, measurement bandwidth, and the time allowed for
thermal settling. These entries determine whether two resistance values can be
compared. A copper lead measured immediately after a high-current pulse and the same
lead measured after cooling are different thermal states. A thin-film resistor
measured before and after soldering can have different strain and contact conditions
even when its printed marking is unchanged.

At least one independent cross-check should accompany a precision result. Compare a
four-terminal dc reading with a calibrated bridge, repeat the measurement at two
currents to test self-heating, or measure the voltage drops across separate sections
of a uniform wire and compare them with length ratios. A disagreement localizes a
model failure when the test conditions are controlled. It does not become an
acceptable uncertainty term merely because the numerical readings are averaged.

Power data provide another check. For an ohmic specimen at stable temperature, the
three forms $VI$, $I^2R$, and $V^2/R$ agree within the stated channel uncertainties.
If the value changes with drive current, identify whether temperature, contact
resistance, nonlinear transport, or source loading has changed. Each mechanism
predicts a different dependence on current and time. Recording those dependences
turns a resistance measurement into a characterization of the actual component and
its operating range.