Chapter 9. Current and Resistance
9.3 Resistivity and Resistance
Learning Objectives
By the end of this section, you will be able to:
- Differentiate between resistance and resistivity
- Define the term conductivity
- Describe the electrical component known as a resistor
- State the relationship between resistance of a resistor and its length, cross-sectional area, and resistivity
- State the relationship between resistivity and temperature
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 [latex]\stackrel{\to }{\textbf{E}}[/latex] is created, and charges in the conductor feel a force due to the electrical field. The current density [latex]\stackrel{\to }{\textbf{J}}[/latex] 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 [latex]\sigma[/latex] 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 [latex]\sigma =J\text{/}E[/latex], the units are
Here, we define a unit named the ohm with the Greek symbol uppercase omega, [latex]\text{Ω}[/latex]. The unit is named after Georg Simon Ohm, whom we will discuss later in this chapter. The [latex]\text{Ω}[/latex] is used to avoid confusion with the number 0. One ohm equals one volt per amp: [latex]1\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}}=1\phantom{\rule{0.2em}{0ex}}\text{V/A}[/latex]. The units of electrical conductivity are therefore [latex]{\left(\phantom{\rule{0.2em}{0ex}}\text{Ω}\phantom{\rule{0.2em}{0ex}}·\text{m}\right)}^{-1}[/latex].
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 fl