Electrical Resistance: What is it? (Symbol, Formula, AC vs DC Resistance)

Contents

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Key learnings:

Electrical Resistance Definition: Electrical resistance is defined as the opposition to current flow in a circuit, measured in ohms (Ω).
Ohm’s Law: Ohm’s Law explains that resistance (R) equals voltage (V) divided by current (I), showing the relationship between these electrical quantities.
Factors Affecting Resistance: Resistance depends on a material’s length, cross-sectional area, and temperature.
AC vs DC Resistance: AC resistance (impedance) includes inductive and capacitive reactance, while DC resistance is just the opposition to current flow.
Temperature’s Effect on Resistance: Resistance in metals increases with temperature, while in semiconductors and insulators, it decreases.

What is Electrical Resistance?

Electrical resistance is the opposition to current in a circuit. For an ohmic conductor it equals voltage divided by current. The SI unit is the ohm (Ω), written with the Greek letter omega.

A larger resistance lets less current through for the same voltage.

When a potential difference is applied across a metal, free electrons drift and scatter from ions in the conductor.

That scattering limits the drift. The opposition the material offers to the current is its resistance.

The resistance of a uniform conductor depends on:

length (directly proportional)
cross-sectional area (inversely proportional)
the material (resistivity)
temperature

For a uniform conductor those facts are written:

$\begin{align*} R \propto \frac{l}{a} \end{align*}$

$\begin{align*} R = \rho \frac{l}{a} \,\, \Omega \end{align*}$

R is the resistance of the conductor

$l$ is the length of the conductor

a is the cross-sectional area of the conductor

$\rho$ is the specific resistance or resistivity of the material

Definition of 1 Ohm Resistance

If 1 volt of potential across a conductor produces 1 ampere, the resistance is one ohm. The SI relation is 1 Ω = 1 V/A.

$\begin{align*} R = \frac{V}{I} \end{align*}$

$\begin{align*} 1 \,\, Ohm = \frac{1 \,\, Volt}{1 \,\, Ampere} \end{align*}$

What is Electrical Resistance Measured in (Units)?

The SI unit of resistance (and of a resistor) is the ohm, written Ω, named after Georg Simon Ohm.

In SI an ohm is 1 volt per ampere:

$\begin{align*} 1 \,\, Ohm = \frac{1 \,\, Volt}{1 \,\, Ampere} \end{align*}$

So resistance can also be stated in volts per ampere.

Resistors are made over a wide range of values. Moderate values are written in ohms. Smaller and larger values use milliohm, kiloohm and megaohm prefixes.

The table below lists those derived names. Stored table latex is unchanged.

Unit Name	Abbreviation	Values in Ohm $(\Omega)$
Milli Ohm	$m\,\,\Omega$	$10^-^3\,\,\Omega$
Micro Ohm	$\micro\,\,\Omega$	$10^-^6\,\,\Omega$
Nano Ohm	$n\,\,\Omega$	$10^-^9\,\,\Omega$
Kilo Ohm	$K\,\,\Omega$	$10^3\,\,\Omega$
Mega Ohm	$M\,\,\Omega$	$10^6\,\,\Omega$
Giga Ohm	$G\,\,\Omega$	$10^9\,\,\Omega$

Derived Unit of Resistors

Electrical Resistance Symbol

Two circuit symbols are in common use for a resistor.

IEEE/ANSI drawings use a zig-zag. IEC 60617 drawings use a rectangle, often called the international resistor symbol.

Both symbols are shown in the image below.

Electric Resistance Formula

Three useful rearrangements are:

Resistance from voltage and current (Ohm’s Law)
Resistance from Power and voltage
Resistance from power and current

The image below collects those rearrangements.

Resistance Formula 1 (Ohm’s Law)

Ohm’s law for an ohmic resistor is

$\begin{align*} V = I * R \end{align*}$

Resistance is then the ratio of voltage to current:

$\begin{align*} R = \frac{V}{I} \,\,\Omega \end{align*}$

Example

In the circuit below the supply is 24 V and the current through the unknown resistor is 2 A. Find the resistance.

Solution:

Given Data: $V = 24\,\,V\,\,,I = 2\,\,A$

Ohm’s law gives:

$\begin{align*} \begin{split} R = \frac{V}{I} \\ = \frac{24}{2} \\ R = 12\,\, \Omega \end{split} \end{align*}$

The unknown resistance is $12\,\,\Omega$ .

Resistance Formula 2 (Power and Voltage)

Electric power into a resistor is the product of voltage and current.

$\begin{align*} P = V * I \end{align*}$

Substitute $I = \frac{V}{R}$ to obtain

$\begin{align*} P = \frac{V^2}{R} \end{align*}$

Resistance is then the square of voltage divided by power:

$\begin{align*} R = \frac{V^2}{P} \,\,\Omega \end{align*}$

Example

In the circuit below, 24 V is applied across a 48 W lamp. Find the lamp resistance.

Solution:

Given Data: $V = 24\,\,V\,\,,P = 48\,\,W$

From the power-voltage form:

$\begin{align*} \begin{split} R = \frac{V^2}{P} \\ = \frac{24*24}{48} \\ R = 12\,\, \Omega \end{split} \end{align*}$

The 48 W lamp resistance is $12\,\,\Omega$ .

Resistance Formula 3 (Power and Current)

Start from $P = V * I$

Substitute $V = I *R$ to obtain

$\begin{align*} P = I^2 * R \end{align*}$

Resistance is then power divided by the square of current:

$\begin{align*} R = \frac{P}{I^2} \,\, \Omega \end{align*}$

Example

In the circuit below, 2 A flows through a 20 W lamp. Find the lamp resistance.

Solution:

Given Data: $I = 2\,\,A\,\,,P = 20\,\,W$

From the power-current form:

$\begin{align*} \begin{split} R = \frac{P}{I^2} \\ = \frac{20}{2*2} \\ = \frac{20}{4} \\ R = 5\,\, \Omega \end{split} \end{align*}$

The 20 W lamp resistance is $5\,\,\Omega$ .

Difference Between AC and DC Resistance

Effective AC resistance of a conductor is not the same quantity as DC resistance, and neither quantity is the impedance of an RLC circuit.

AC Resistance

The combination of resistance with inductive reactance and capacitive reactance is impedance Z. Impedance is not the same as the AC resistance of a wire.

The stored latex writes R = Z. That equality holds only in a purely resistive circuit. In general |Z| is larger than R.

$\begin{align*} R = Z \end{align*}$

The stored formula below is the magnitude of impedance, |Z|, not the AC resistance of the conductor:

$\begin{align*} R_A_C = \sqrt{R^2 + (X_L-X_C)^2} \,\, \Omega \end{align*}$

DC Resistance

Steady DC has zero frequency. Then XL = 2πfL is zero after transients die, while XC = 1/(2πfC) is infinite: a capacitor is an open circuit at DC.

On a simple DC loop of wire, current is set by the conductor resistance plus any other resistive drops. Capacitors block steady DC.

DC resistance of that wire is then V/I from Ohm’s law:

$\begin{align*} R_D_C = \frac{V}{I} \,\, \Omega \end{align*}$

Which One is More AC Resistance or DC Resistance?

DC does not show skin effect. At AC, skin and proximity effects crowd current toward the surface, so the effective AC resistance of a conductor is higher than its DC resistance.

$\begin{align*} R_A_C = R_D_C \end{align*}$

The stored latex next writes RAC = RDC and then RAC = 1.6 RDC. The first is the DC limit. The 1.6 factor is a rule of thumb in some texts, not a general law. The ratio depends on frequency, size and material.

$\begin{align*} R_A_C = 1.6 * R_D_C \end{align*}$

Electrical Resistance, Heating and Temperature

Electrical Resistance and Heating

When current (a drift of free electrons) passes through a conductor, the electrons scatter from the lattice. That scattering is the microscopic origin of electrical resistance.

The electrical energy delivered to the resistor becomes heat, called Joule heating.

If I amperes flows through R ohms for t seconds, the energy converted to heat is I²Rt joules.

In equation form:

$\begin{align*} Heat \,\, produced \,\,(H) = I^2 * R * t \,\, joules \end{align*}$

$\begin{align*} = \frac{I^2 * R * t}{4.186} \,\, calories \end{align*}$

Joule heating is how an electric heater, toaster, kettle, iron or soldering iron works: current in a high-resistance element produces the heat.

Nichrome (nickel-chromium) is a common heating-element alloy. Its resistivity is more than 50 times that of copper.

Effect of Temperature on Electrical Resistance

Resistance changes with temperature. The direction and size of the change depend on the material.

Metals

The resistance of pure metals (copper, aluminium, silver) rises as temperature rises over ordinary ranges. Those metals have a positive temperature coefficient of resistance.

Alloys

Many alloys (nichrome, manganin) also have a positive coefficient, but it is small. Manganin is chosen because its resistance barely moves with temperature.

Semi-Conductors, Insulators & Electrolytes

The resistance of semiconductors, insulators and many electrolytes falls as temperature rises. More carriers become available, so these materials are described as having a negative temperature coefficient.

Common Questions About Resistance

Electrical Resistance of The Human Body

Dry skin resistance is high. Internal tissue resistance is much lower. Wet or broken skin cuts the total body resistance sharply.

Safety-training notes often cite about 100,000 Ω dry and about 1,000 Ω wet or with broken skin. IEC 60479 values vary with voltage, path and contact area, so treat those figures as order-of-magnitude guides.

If voltage punctures the skin, the remaining body resistance is often taken as about 500 Ω in those same notes.

Electrical Resistance of Air

Air is an insulator. Published resistivity figures span about $10^6$ to $10^1^5$ $\Omega-m$ at 20⁰ C, depending on humidity and ionization. The stored latex for the upper exponent is unchanged.

Electrical resistance of an air gap depends on gap length, electrode shape and whether the air has already broken down. That is not the same as aerodynamic drag, which also uses the everyday word resistance.

Uniform-field dielectric strength of dry air at STP is about 30 kV/cm peak (about 21.1 kV/cm RMS). Above that stress the gap sparks. The plasma path then has a low but finite resistance, not exactly zero.

Electrical Resistance of Water

Resistivity of water is set mainly by dissolved ions. More salt means lower resistivity.

Ultrapure water has very few ions, so its resistivity is high (about 18 MΩ·cm at 25 °C). Dissolved salts add ions and the resistivity falls.

Seawater is typically near 0.2 Ω·m. The table below is stored as published and uses Ω-m figures that do not all match those laboratory values.

Types of water	Resistivity in Ohms-m $(\Omega-m)$
Pure Water	20,000,000
Seawater	20-25
Distilled Water	500,000
Rain Water	20,000
River Water	200
Drinking-Water	2 to 200
Deionized Water	180,000

Electrical Resistance of Copper

Copper is a good conductor, so a copper piece of ordinary size has a low resistance. The material property is resistivity.

Annealed copper at 20 °C is about $1.68 * 10^-^8\,\,\Omega-m$ .

What Do You Call the Phenomenon When Electrical Resistance Is Zero?

A superconductor has (ideally) zero DC resistance below its critical temperature, current and field.

Ohm’s law written as

$\begin{align*} I = \frac{V}{R} \end{align*}$

If R were exactly 0 in that formula, the stored latex would give

$\begin{align*} I = \frac{V}{0} = \infty \end{align*}$

That I = V/0 model is not how a superconductor works. A superconductor carries a finite current up to its critical current. The voltage along the superconductor is ~0, not an infinite current from a voltage source.

Zero resistance is infinite conductance in the definition G = 1/R. Real superconductors still have a critical current.

$\begin{align*} G = \frac{1}{R} = \frac{1}{0} = \infty \end{align*}$