Electrical resistance and conductance

The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is electrical conductance, measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with mechanical friction. The SI unit of electrical resistance is the ohm ( $Ω$ ), while electrical conductance is measured in siemens (S) (formerly called the 'mho' and then represented by $℧$ ).

This article is about specific applications of conductivity and resistivity in electrical elements. For other types of conductivity, see Conductivity. For electrical conductivity in general, see Electrical resistivity and conductivity.

Electric resistance

$R$

ohm (Ω)

kg⋅m²⋅s⁻³⋅A⁻²

${\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-3}{\mathsf {I}}^{-2}$

$G$

siemens (S)

kg⁻¹⋅m⁻²⋅s³⋅A²

${\mathsf {M}}^{-1}{\mathsf {L}}^{-2}{\mathsf {T}}^{3}{\mathsf {I}}^{2}$

The resistance of an object depends in large part on the material it is made of. Objects made of electrical insulators like rubber tend to have very high resistance and low conductance, while objects made of electrical conductors like metals tend to have very low resistance and high conductance. This relationship is quantified by resistivity or conductivity. The nature of a material is not the only factor in resistance and conductance, however; it also depends on the size and shape of an object because these properties are extensive rather than intensive. For example, a wire's resistance is higher if it is long and thin, and lower if it is short and thick. All objects resist electrical current, except for superconductors, which have a resistance of zero.

The resistance $R$ of an object is defined as the ratio of voltage $V$ across it to current $I$ through it, while the conductance $G$ is the reciprocal: $R={\frac {V}{I}},\qquad G={\frac {I}{V}}={\frac {1}{R}}.$

For a wide variety of materials and conditions, $V$ and $I$ are directly proportional to each other, and therefore $R$ and $G$ are constants (although they will depend on the size and shape of the object, the material it is made of, and other factors like temperature or strain). This proportionality is called Ohm's law, and materials that satisfy it are called ohmic materials.

In other cases, such as a transformer, diode or battery, $V$ and $I$ are not directly proportional. The ratio $V / I$ is sometimes still useful, and is referred to as a chordal resistance or static resistance,^[1]^[2] since it corresponds to the inverse slope of a chord between the origin and an $I$ – $V$ curve. In other situations, the derivative ${\textstyle {\frac {\mathrm {d} V}{\mathrm {d} I}}}$ may be most useful; this is called the differential resistance.

geometry (shape), and

material

In the hydraulic analogy, current flowing through a wire (or resistor) is like water flowing through a pipe, and the voltage drop across the wire is like the pressure drop that pushes water through the pipe. Conductance is proportional to how much flow occurs for a given pressure, and resistance is proportional to how much pressure is required to achieve a given flow.

The voltage drop (i.e., difference between voltages on one side of the resistor and the other), not the voltage itself, provides the driving force pushing current through a resistor. In hydraulics, it is similar: the pressure difference between two sides of a pipe, not the pressure itself, determines the flow through it. For example, there may be a large water pressure above the pipe, which tries to push water down through the pipe. But there may be an equally large water pressure below the pipe, which tries to push water back up through the pipe. If these pressures are equal, no water flows. (In the image at right, the water pressure below the pipe is zero.)

The resistance and conductance of a wire, resistor, or other element is mostly determined by two properties:

Geometry is important because it is more difficult to push water through a long, narrow pipe than a wide, short pipe. In the same way, a long, thin copper wire has higher resistance (lower conductance) than a short, thick copper wire.

Materials are important as well. A pipe filled with hair restricts the flow of water more than a clean pipe of the same shape and size. Similarly, electrons can flow freely and easily through a copper wire, but cannot flow as easily through a steel wire of the same shape and size, and they essentially cannot flow at all through an insulator like rubber, regardless of its shape. The difference between copper, steel, and rubber is related to their microscopic structure and electron configuration, and is quantified by a property called resistivity.

In addition to geometry and material, there are various other factors that influence resistance and conductance, such as temperature; see below.

$t$ is time;

$u (t)$ and $i (t)$ are the voltage and current as a function of time, respectively;

$U 0$ and $I 0$ indicate the amplitude of the voltage and current, respectively;

$\omega$ is the of the AC current;

angular frequency

$\varphi$ is the displacement angle;

$U$ and $I$ are the complex-valued voltage and current, respectively;

$Z$ and $Y$ are the complex and admittance, respectively;

impedance

${\mathcal {R_{e}}}$ indicates the of a complex number; and

real part

$j\equiv {\sqrt {-1\ }}$ is the .

imaginary unit

Conductance quantum

Von Klitzing constant

Electrical measurements

Contact resistance

for more information about the physical mechanisms for conduction in materials.

Electrical resistivity and conductivity

Johnson–Nyquist noise

a standard for high-accuracy resistance measurements.

Quantum Hall effect

Resistor

RKM code

Series and parallel circuits

Sheet resistance

SI electromagnetism units

Thermal resistance

Voltage divider

Voltage drop

. Vehicular Electronics Laboratory. Clemson University. Archived from the original on 11 July 2010.

"Resistance calculator"

. wolfram.com. Wolfram Demonstrantions Project.