Resistivity
Common symbols	ρ
SI unit	ohm metre (Ω⋅m)
Other units	s (Gaussian/ESU)
In SI base units	kg⋅m3⋅s−3⋅A−2
Derivations from other quantities	${\displaystyle \rho =R{\frac {A}{\ell }}}$
Dimension	${\displaystyle {\mathsf {M}}{\mathsf {L}}^{3}{\mathsf {T}}^{-3}{\mathsf {I}}^{-2}}$

Conductivity
Common symbols	σ, κ, γ
SI unit	siemens per metre (S/m)
Other units	${\displaystyle \mathrm {s} ^{-1}}$ (Gaussian/ESU)
Derivations from other quantities	${\displaystyle \kappa ={\frac {1}{\rho }}}$
Dimension	${\displaystyle {\mathsf {M}}^{-1}{\mathsf {L}}^{-3}{\mathsf {T}}^{3}{\mathsf {I}}^{2}}$

Electrical resistivity (also called volume resistivity or specific electrical resistance) is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the Greek letter ρ (rho). The SI unit of electrical resistivity is the ohm-metre (Ω⋅m).^[1][2][3] For example, if a 1 m³ solid cube of material has sheet contacts on two opposite faces, and the resistance between these contacts is 1 Ω, then the resistivity of the material is 1 Ω⋅m.

Electrical conductivity (or specific conductance) is the reciprocal of electrical resistivity. It represents a material's ability to conduct electric current. It is commonly signified by the Greek letter σ (sigma), but κ (kappa) (especially in electrical engineering) and γ (gamma) are sometimes used. The SI unit of electrical conductivity is siemens per metre (S/m). Resistivity and conductivity are intensive properties of materials, giving the opposition of a standard cube of material to current. Electrical resistance and conductance are corresponding extensive properties that give the opposition of a specific object to electric current.

Definition

Ideal case

In an ideal case, cross-section and physical composition of the examined material are uniform across the sample, and the electric field and current density are both parallel and constant everywhere. Many resistors and conductors do in fact have a uniform cross section with a uniform flow of electric current, and are made of a single material, so that this is a good model. (See the adjacent diagram.) When this is the case, the resistance of the conductor is directly proportional to its length and inversely proportional to its cross-sectional area, where the electrical resistivity ρ (Greek: rho) is the constant of proportionality. This is written as:

where

The resistivity can be expressed using the SI unit ohm metre (Ω⋅m) — i.e. ohms multiplied by square metres (for the cross-sectional area) then divided by metres (for the length).

Both resistance and resistivity describe how difficult it is to make electrical current flow through a material, but unlike resistance, resistivity is an intrinsic property and does not depend on geometric properties of a material. This means that all pure copper (Cu) wires (which have not been subjected to distortion of their crystalline structure etc.), irrespective of their shape and size, have the same resistivity, but a long, thin copper wire has a much larger resistance than a thick, short copper wire. Every material has its own characteristic resistivity. For example, rubber has a far larger resistivity than copper.

In a hydraulic analogy, passing current through a high-resistivity material is like pushing water through a pipe full of sand - while passing current through a low-resistivity material is like pushing water through an empty pipe. If the pipes are the same size and shape, the pipe full of sand has higher resistance to flow. Resistance, however, is not solely determined by the presence or absence of sand. It also depends on the length and width of the pipe: short or wide pipes have lower resistance than narrow or long pipes.

The above equation can be transposed to get Pouillet's law (named after Claude Pouillet):

The resistance of a given element is proportional to the length, but inversely proportional to the cross-sectional area. For example, if A = 1 m², ${\displaystyle \ell }$ = 1 m (forming a cube with perfectly conductive contacts on opposite faces), then the resistance of this element in ohms is numerically equal to the resistivity of the material it is made of in Ω⋅m.

Conductivity, σ, is the inverse of resistivity:

Conductivity has SI units of siemens per metre (S/m).

General scalar quantities

If the geometry is more complicated, or if the resistivity varies from point to point within the material, the current and electric field will be functions of position. Then it is necessary to use a more general expression in which the resistivity at a particular point is defined as the ratio of the electric field to the density of the current it creates at that point:

where

The current density is parallel to the electric field by necessity.

Conductivity is the inverse (reciprocal) of resistivity. Here, it is given by:

For example, rubber is a material with large ρ and small σ — because even a very large electric field in rubber makes almost no current flow through it. On the other hand, copper is a material with small ρ and large σ — because even a small electric field pulls a lot of current through it.

This expression simplifies to the formula given above under "ideal case" when the resistivity is constant in the material and the geometry has a uniform cross-section. In this case, the electric field and current density are constant and parallel.

Derivation of the constant case from the general case

We will combine three equations. Assume the geometry has a uniform cross-section and the resistivity is constant in the material and the geometry has a uniform cross-section. Then the electric field and current density are constant and parallel, and by the general definition of resistivity, we obtain $${\displaystyle \rho ={\frac {E}{J}},}$$ Since the electric field is constant, it is given by the total voltage V across the conductor divided by the length ℓ of the conductor: $${\displaystyle E={\frac {V}{\ell }}.}$$ Since the current density is constant, it is equal to the total current divided by the cross sectional area: $${\displaystyle J={\frac {I}{A}}.}$$ Plugging in the values of E and J into the first expression, we obtain: $${\displaystyle \rho ={\frac {VA}{I\ell }}.}$$ Finally, we apply Ohm's law, V/I = R: $${\displaystyle \rho =R{\frac {A}{\ell }}.}$$

Tensor resistivity

When the resistivity of a material has a directional component, the most general definition of resistivity must be used. It starts from the tensor-vector form of Ohm's law, which relates the electric field inside a material to the electric current flow. This equation is completely general, meaning it is valid in all cases, including those mentioned above. However, this definition is the most complicated, so it is only directly used in anisotropic cases, where the more simple definitions cannot be applied. If the material is not anisotropic, it is safe to ignore the tensor-vector definition, and use a simpler expression instead.

Here, anisotropic means that the material has different properties in different directions. For example, a crystal of graphite consists microscopically of a stack of sheets, and current flows very easily through each sheet, but much less easily from one sheet to the adjacent one.^[4] In such cases, the current does not flow in exactly the same direction as the electric field. Thus, the appropriate equations are generalized to the three-dimensional tensor form:^[5][6]