Electrical Resistance

Electrical resistance is a property of an object or circuit that describes its opposition to electric current. In a steady-state measurement, resistance $R$ is the ratio of the voltage across an object to the current through it. Its unit is the ohm, symbol $\Omega$. Resistance depends on the material, dimensions, and physical conditions of the object; it is distinct from resistivity, which characterizes the material itself. A resistor is a component designed to provide a specified resistance. (openstax.org)

Definition and units

For a two-terminal component carrying a nonzero current,

where $V$ is the difference in electric potential between the terminals and $I$ is the current. Thus, at a given voltage, a larger positive resistance corresponds to a smaller current. (openstax.org)

In the International System of Units (SI),

An object has a resistance of one ohm if a voltage of one volt produces a current of one ampere. The ohm is an SI derived unit, rather than a base unit. (bipm.org)

The reciprocal of resistance is electrical conductance:

Conductance is measured in siemens, symbol $\mathrm{S}$, with $1\ \mathrm{S}=1\ \Omega^{-1}$. Resistance and conductance describe the same two-terminal relationship from opposite perspectives. (bipm.org)

Relationship to Ohm’s law

An ohmic component has a current proportional to its applied voltage, provided its temperature and other relevant conditions remain fixed. Its resistance is therefore constant over the operating range considered:

This proportionality is Ohm’s law. Defining resistance as $V/I$ does not establish that a component obeys the law: for a non-ohmic device, that ratio can vary with the applied voltage or current. Ohm’s law is an empirical description of many materials, not a universal law governing every electrical device. (openstax.org)

For a nonlinear current–voltage characteristic, two quantities are distinguished:

• Static resistance: $R=V/I$, evaluated at an operating point.
• Differential resistance: $r=dV/dI$, describing the local response to a small change in current.

They coincide for a linear characteristic passing through the origin but generally differ for nonlinear devices. Some devices exhibit negative differential resistance over part of their operating range, where an increase in current accompanies a decrease in voltage. This is a statement about the local slope, not necessarily about the sign of $V/I$. (tek.com)

Resistance, resistivity, and geometry

Electrical resistivity, symbol $\rho$, measures the response of a material independently of the overall dimensions of a particular specimen. For a uniform conductor of length $L$, cross-sectional area $A$, and uniform resistivity,

A longer conductor has greater resistance because the current traverses a longer path; a greater cross-sectional area lowers resistance by providing more material through which current can flow. These comparisons assume that the material and physical conditions are unchanged. Resistivity is measured in ohm metres, $\Omega\,\mathrm{m}$. (openstax.org)

For an isotropic, locally ohmic material, the corresponding field relation is

where $\mathbf{E}$ is the electric field, $\mathbf{J}$ is current density, and $\sigma=1/\rho$ is electrical conductivity. The expression $R=\rho L/A$ follows when the field and current density are uniform along the conductor. Irregular geometry or nonuniform material properties require a more detailed calculation of the current distribution. (openstax.org)

Microscopic origin

In a metal, mobile electrons carry current. An applied field produces a net drift superimposed on their microscopic motion. Interactions within the material interrupt this directed motion, so a sustained field is needed to maintain a steady current. The electron drift speed is distinct from the much faster propagation of an electrical signal through a circuit. (openstax.org)

Resistance is also associated with energy transfer. In a normal conductor, energy supplied by the electric field is transferred from the charge carriers to the material, producing thermal energy. Consequently, a resistive conductor can carry a steady current without the carriers continuously gaining kinetic energy. (openstax.org)

The microscopic explanation depends on the material. In metals, increasing lattice vibration commonly increases resistivity. In a semiconductor, temperature can also change the number of mobile charge carriers substantially, so its effect on resistance may differ from that in a metal. (openstax.org)

Temperature dependence and superconductivity

Resistance generally depends on temperature. Over a sufficiently limited range, its variation can be approximated by

where $T_0$ is a reference temperature and $\alpha$ is the temperature coefficient of resistance near that temperature. This approximation assumes that changes in the specimen’s dimensions are negligible or incorporated into the coefficient. It should not be extrapolated indiscriminately across large temperature intervals. (openstax.org)

Most ordinary metallic conductors have a positive temperature coefficient near room temperature: their resistance increases as they become warmer. Many semiconducting materials instead show decreasing resistance with increasing temperature, although the behavior depends on composition and impurities. This dependence underlies resistance thermometers and thermistors, which infer temperature from measured resistance. (openstax.org)

Superconductivity is a distinct state in which a material can carry steady current with zero electrical resistance under appropriate conditions. It occurs below a material-dependent critical temperature and is limited by magnetic field and current. Superconductivity also involves magnetic properties, including the Meissner effect, that are not explained by zero resistance alone. (openstax.org)

Resistance in circuits

For ideal resistors connected in series, the same current passes through each component and their voltage drops add:

For ideal resistors connected in parallel, each branch has the same voltage and the branch currents add:

For positive, finite resistances, a series combination has greater resistance than any individual member, while a parallel combination has less resistance than the smallest member. These formulas assume that connecting wires contribute negligible resistance or are included separately in the model. (openstax.org)

As an illustrative calculation, two $100\ \Omega$ resistors have an equivalent resistance of $200\ \Omega$ in series and $50\ \Omega$ in parallel.

Energy dissipation and applications

For a passive resistor, electrical power absorbed is

Using $V=IR$, this becomes

This conversion of electrical energy into thermal energy is called Joule heating. The two expressions emphasize different constraints: at fixed current, increasing resistance increases dissipation; at fixed voltage, increasing resistance decreases it. (openstax.org)

Resistance is useful for limiting current and establishing voltage drops in electronic circuits. Resistive heating is deliberately employed in heaters and incandescent filaments. In other contexts, such as electrical wiring and transmission lines, resistive dissipation is an unwanted loss, making low-resistance conductors desirable. Temperature-dependent resistance is also used for sensing rather than primarily for heating or current limitation. (openstax.org)

Resistance versus impedance

In alternating-current circuits, resistance alone does not describe every relationship between voltage and current. Capacitors and inductors can produce a phase difference between them. The broader quantity is electrical impedance, which includes resistance and reactance. (openstax.org)

In complex notation,

where $R$ is the real, resistive part, $X$ is reactance, and $j^2=-1$. Both are measured in ohms. An ideal resistor has zero reactance and voltage in phase with current; an ideal capacitor or inductor has reactance and exchanges stored energy with the circuit. Therefore, the ratio of sinusoidal voltage amplitude to current amplitude generally gives $|Z|$, not resistance alone. (openstax.org)

Measurement and standards

Resistance can be determined by supplying a known current and measuring the voltage across the specimen, or by supplying a known voltage and measuring current. An ohmmeter performs such a measurement internally. The result depends on measurement conditions, including temperature and the chosen operating point for a nonlinear device. (tek.com)

In a two-wire measurement, the same leads carry current and connect the measuring instrument to the specimen. Their resistance contributes to the result. A four-terminal measurement, also called a Kelvin measurement, uses separate current-carrying and voltage-sensing leads. The high-impedance sensing circuit draws very little current, substantially reducing errors from voltage drops in the current leads. This is particularly important for low resistances. (tek.com)

Other sources of error include self-heating, thermoelectric voltages, and leakage currents. Reversing the test current and comparing voltage readings can cancel stable thermoelectric offsets; high-resistance measurements require particular attention to leakage and insulation. (tek.com)

High-accuracy resistance standards use the quantum Hall effect. Quantized Hall resistance provides a reproducible reference associated with the ratio $h/e^2$, involving the Planck constant and the elementary charge. Such references support resistance calibration and metrological traceability to the SI. (bipm.org)

Historical development

Georg Simon Ohm’s experiments established the proportional relationship between current and voltage for metallic conductors under controlled conditions. His work published in 1827 became the basis of Ohm’s law, and the resistance unit was later named in his honor. (openstax.org)

International electrical units for current and resistance were introduced at the International Electrical Congress in Chicago in 1893, with definitions confirmed at an international conference in London in 1908. Resistance measurement subsequently developed from material reference standards toward quantum electrical standards. The SI revision that took effect on May 20, 2019 fixed the numerical values of $h$ and $e$, bringing quantum realizations of electrical units into direct agreement with the SI definitions. (bipm.org)

References

1. 3 Resistivity and Resistance - University Physics Volume 2openstax.org
2. SI Brochure - 9th ed./version 3.02bipm.org
3. 4 Ohm's Law - University Physics Volume 2openstax.org
4. An Improved Method for Differential Conductance Measurementstek.com
5. 3 Resistance and Resistivity - College Physics 2eopenstax.org
6. 2 Model of Conduction in Metals - University Physics Volume 2openstax.org
7. 5 Electrical Energy and Power - University Physics Volume 2openstax.org
8. 8 Superconductivity - University Physics Volume 3openstax.org
9. 2 Resistors in Series and Parallel - University Physics Volume 2openstax.org
10. 3 RLC Series Circuits with AC - University Physics Volume 2openstax.org
11. Keithley Low Level Measurements Handbook - 7th Editiontek.com
12. Ohms: How do you decide whether to use a two-probe method or a four-probe method for resistance?tek.com