Resistivity — The Specific Resistance of a Material

The resistivity of a material is the measure of it’s specific resistance to the flow of electricity, a numerical quantity unique to each material which varies only with temperature. We symbolize resistivity with the Greek letter lowercase rho, ρ, a numerical constant of proportionality enabling us to determine the resistance of a material from its length and area of a cross section.

Assuming that a conductor has a consistent width and thickness down its length, or a uniform diameter across its length (a wire of a specific gauge for example) there are fundamental relationships between resistance and the objects spatial dimensions. If the conductor is uniform, then the resistance of the conductor is directly proportional to its length (l), and inversely proportional to the area of the cross-section (A).

This means that if we double the length of a wire, for example, then the resistance of the wire also doubles. If we triple the length of the wire, the resistance triples, while if we half the length of the wire, then resistance is also cut in half. Conversely, if we double the diameter of a wire the resistance is actually cut in half, because the area of a cross section has an inversely proportional relationship.

Thicker wires allow more electricity to flow, because they have more space for electrons to flow within the wire, in the same way that a larger diameter pipe allows more water to flow at a lower pressure because it has more space within the pipe for water to flow.

If the diameter of a wire is quartered then the resistance is quadrupled. As the area of a cross-section of a conductor increases, the resistance decreases in an inversely proportional fashion; and as the length of a wire increases, the resistance increases in a directly proportional fashion.

We can sum this up by saying that the longer a conductor is, the greater it’s resistance will be from end to end. In regards to the cross-sectional area, the thinner a wire is (the smaller its diameter) the greater its resistance, while the thicker a wire, the smaller its resistance will be.

The Resistivity Equation

Assuming a stable temperature (since the resistance of a material varies with temperature) the resistance of a material is given by the equation

\[ R = \rho \frac{l}{A} \]

When calculating resistance, the three important variables are length ($m$) measured in meters, area of a cross section ($m^2$) in meters squared, and resistivity ($\Omega \cdot m$) measured in ohm-meters. As you will know, $R$ is the symbol for resistance which is measured in ohms (Ω).

Resistivity is measured in SI units called ohm-meters with the unit symbol Ω∙m. You can see how the SI Units for ρ are derived by analyzing the above equation. Rearrange the expression $R=\rho \cdot l / A$ for ρ and you will arrive at the equation $\rho = R \cdot A / l$. From this form of the equation input the unit symbol in place of their respective quantity symbol, simplify, and you will see how the units for resistivity are derived

\[ \rho = \frac{R \cdot A}{l} = \frac{\Omega \cdot m^2}{m} = \Omega \cdot m \]

Thus we arrive at the ohm-meter (Ω∙m) unit for resistivity. (Remember not to say “ohms per meter” because the “per” is what we say when one quantity divides another, which is not the case for units of resistivity!)

The Relationship Between Resistance and Resistivity

The Relationship Between Resistance and Resistivity is essentially that the resistance of a material has to do with how well said material either inhibits or allows the flow of electrons through itself. That is resistance, which is more general.

The specific resistivity changes according to the dimensions of the material. You can have two copper wires — both made of copper — which have entirely different resistivity because of their shape. One can be longer or thicker than the other (or both!) which effect how they conduct electricity.

Classifying The Resistivity of Conductors and Insulators

Resistivity (ρ) depends on the type of material. As we have discussed in previous articles on resistance, a materials resistance to electricity is determined ultimately by it’s atomic and molecular structure. In light of this fact, every material will have a different value for ρ, a constant of proportionality which will give us the resistance of a specific object in conjunction with the length and the diameter of the conductor, as well as the temperature. You will see that if the length of a material is 1 m and the area of a cross section is 1 m², then the resistance is numerically equivalent to the resistivity of the material.

The most direct way to classify conductors and insulators is by the value of their resistivity. A good conductor has a resistivity close to 10^-8 Ω∙m. Silver is the best conductor on the planet. However, it is too expensive to use in the majority of common situations, which is why copper is preferred since copper is more abundant in nature, which due to supply and demand makes it less expensive. For this reason copper is the most common conductor. Aluminum is another common conductor.

Materials with a resistivity above 10¹⁰ Ω∙m are classified as insulators. To put these numbers in perspective, let us look at these numbers in a form other than scientific notation. Insulators have a resistivity on the order of 10 billion ohm-meters (10¹⁰ Ω∙m), which we could refer to as 10 gigaohm-meters (10 GΩ∙m) while good conductors have a resistivity on the order of 100 millionths of an ohm-meter which we could also call 100 micro-ohm-meters (100 μΩ∙m).

There is a full 18 orders of magnitude of difference between the resistivity of good conductors versus insulators, which should give you at least some idea of the difference between the degree to which electrons are allowed to flow within various materials.

Insulators are used to protect people and other components within an electrical device from electricity because they inhibit the flow of electric current by not permitting electrons to move due to their atomic and molecular structure. Electricians wear insulators on their hands when they work with electricity. Conducting wires are usually coated in an insulator material (such as polyvinyl chloride (PVC) or neoprene) to prevent leakage of current.

Materials with resistivity in the range of 10^-4 Ω∙m to 10^-7 Ω∙m are called semiconductors. Semiconductors are often designed by scientists to possess just the right resistivity for some purpose. Conductor material is mixed with a small portion of an insulator material which raises the very low resistivity of the good conductor materials proportionally with the amount of insulator added. This means that we can manufacture material with a precise resistivity, to further create precision electrical components, devices, and equipment. Transistors are made of semiconductor materials.

Below is a table showing the resistivity of some common materials at 20°C.

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