## Resistivity and Resistance

5.3 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

### Resistivity

When electric charges move through a material, they bump into atoms in the material and, as a result of that bumping, the electrons move at a constant speed. We describe this “bumping” as resistance and the amount of resistance the charges experience will depend on the 3 things: (1) what the material is made from, (2) the width, or cross-sectional area of the resistor, and (3) the length of the resistor. Let’s go through each of these and discuss how they affect the resistance.

**(1) What the material is made from**

There will be more resistance on the charges moving through a material if those charges bump into a lot of stuff. Some materials have more space between their atoms than other materials, which can reduce how many collisions the charges have with atoms. If a material has more space or is structured in a way to make it easy for charges to move through without too many collisions, then that material will have a lower resistance than one that results in lots of collisions. The way we measure how much resistance a material will have is with a property called **resistivity**. We define the resistivity ρ (Greek letter rho) of a substance so that the resistance of the material is directly proportional to the resistivity. Resistivity is an *intrinsic *property of a material, and is independent of the material’s shape or size.

In general, conductors have very low resistivity values and insulators have very high resistivity values.

If you look at this table, you’ll see the resistivity values are given for a specific temperature of 23°C (room temperature). This is because the resistivity can change with temperature. You know that when a material gets hot, the atoms in that material will vibrate more. If charges are trying to move through the material, there will be more collisions between the charges and the vibrating atoms when the material is hotter. So resistivity increases with temperature. We can calculate this temperature dependence of resistivity with

where is the resistivity at temperature T_{0}, and α is the temperature coefficient of resistivity (which can be looked up on a table).

**(2) The width, or cross-sectional area of the resistor**

Since resistance decreases the current that flows through a material, if you decrease the resistance, the current will increase. One way to decrease resistance is to give the charges more room to move around and this can be done by widening the wire or increasing the cross-sectional area of the wire.

That’s all there is to it, if you make the wire thicker, the charges will be able to flow easier and the current will be higher. So a wider wire has less resistance.

**(3) The length of the resistor**

This one is pretty straightforward, too. If a wire is longer, it will take more time for the current to flow through it. Since it will take more time for the charges to move through the long wire, this will decrease the current. So a longer wire has more resistance than a shorter one because the longer wire will have less current.

We can put this all together to describe resistance in one formula, in terms of resistivity (ρ), cross-sectional area (A), and length (L). We know the resistance is proportional to the resistivity and the length, and inversely proportional to the area, so we have:

**Practice!**

Practice 5.3.1 |
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Two cylindrical resistors are made of the same material (same resistivity ). Resistor 2 is twice as long and has twice the diameter of resistor 1. What is the ratio R _{2}/R_{1}? |

Practice 5.3.2 |
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A wire of resistance R is stretched uniformly until it has twice its original length. What happens to the resistance? HINT: The wire’s density will not change, which means you cannot change the volume of the wire by stretching it. (Volume = length*cross sectional area) |

Practice 5.3.3 |
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A metal wire of resistance R is cut into three equal pieces that are then placed together side by side to form a new cable with a length equal to one-third the original length. What is the resistance of this new cable? |

### Temperature dependence of resistivity

**Practice!**

Practice 5.3.4 |
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When does an incandescent light bulb carry more current? |

*Example problem: Work through this problem on your own. Afterward, you can watch my video solution to check your results.*

An aluminum wire with a diameter of 0.090 mm has a uniform electric field of 0.220 V/m imposed along its entire length. The temperature of the wire is 40.0° C. Assume one free electron per atom.

- Determine the resistivity of aluminum at this temperature.
- What is the current density in the wire?
- What is the total current in the wire?
- What is the drift speed of the conduction electrons?
- What potential difference must exist between the ends of a 1.80-m length of the wire to produce the stated electric field?