9.3 Resistivity and resistance

What drives current? We can think of various devices—such as batteries, generators, wall outlets, and so on—that are necessary to maintain a current. All such devices create a potential difference and are referred to as voltage sources. When a voltage source is connected to a conductor, it applies a potential difference V that creates an electrical field. The electrical field, in turn, exerts force on free charges, causing current. The amount of current depends not only on the magnitude of the voltage, but also on the characteristics of the material that the current is flowing through. The material can resist the flow of the charges, and the measure of how much a material resists the flow of charges is known as the resistivity . This resistivity is crudely analogous to the friction between two materials that resists motion.

Resistivity

When a voltage is applied to a conductor, an electrical field $\overrightarrow{E}$ is created, and charges in the conductor feel a force due to the electrical field. The current density $\overrightarrow{J}$ that results depends on the electrical field and the properties of the material. This dependence can be very complex. In some materials, including metals at a given temperature, the current density is approximately proportional to the electrical field. In these cases, the current density can be modeled as

where $\sigma$ is the electrical conductivity    . The electrical conductivity is analogous to thermal conductivity and is a measure of a material’s ability to conduct or transmit electricity. Conductors have a higher electrical conductivity than insulators. Since the electrical conductivity is $\sigma =J\text{/}E$ , the units are

Here, we define a unit named the ohm    with the Greek symbol uppercase omega, $\text{Ω}$ . The unit is named after Georg Simon Ohm, whom we will discuss later in this chapter. The $\text{Ω}$ is used to avoid confusion with the number 0. One ohm equals one volt per amp: $1\text{Ω}=1\text{V/A}$ . The units of electrical conductivity are therefore ${\left(\text{Ω}·\text{m}\right)}^{\mathrm{-1}}$ .

Conductivity is an intrinsic property of a material. Another intrinsic property of a material is the resistivity    , or electrical resistivity. The resistivity of a material is a measure of how strongly a material opposes the flow of electrical current. The symbol for resistivity is the lowercase Greek letter rho, $\rho$ , and resistivity is the reciprocal of electrical conductivity:

The unit of resistivity in SI units is the ohm-meter $\left(\text{Ω}·\text{m}\right)$ . We can define the resistivity in terms of the electrical field and the current density,

The greater the resistivity, the larger the field needed to produce a given current density. The lower the resistivity, the larger the current density produced by a given electrical field. Good conductors have a high conductivity and low resistivity. Good insulators have a low conductivity and a high resistivity. [link] lists resistivity and conductivity values for various materials.