When we build an electric circuit, we want it to be as efficient as possible. This means that we want low resistance, so it's only logical that we use materials like copper and not wood or rubber in our circuits. But why? Because materials like wood and rubber have high resistivities compared to copper.
An informal definition of resistivity is ‘the characteristic obstruction materials offer to the flow of charges per unit of length and cross-section’, which is related to the concept of electrical resistance.
What is electrical resistance?
We often explore electric phenomena in circuits where we can use materials to guide electric charges for different purposes. We use three basic quantities to characterise circuits: resistance, voltage, and current.
Electrical resistance (or simply resistance) is a measure of the opposition of a medium to the movement of charges inside of it. It is measured in ohms (Ω).
The voltage or potential difference is the amount of energy per unit of charge needed to move charges between two points of a circuit. It is usually supplied by batteries, and it is measured in volts (V).
The electric current, or simply the current, is the number of charges that pass a cross-section (transversal cut) of a conductor per unit of time. It is measured in amperes (A).
The role of resistance is most easily seen in Ohm’s law, which governs the behaviour of ohmic conductors and certain ranges of non-ohmic conductors. Its equation is the following:
\[V = I \cdot R\]
Here R is the resistance, V is the voltage, and I is the electric current. If a circuit has high resistance, less current will be produced (and vice-versa). Since the current is the flow of charges, it is clear that the bigger the resistance, the bigger the opposition to the movement of charges.
The greater the resistance, the smaller the current. Check out our explanation on the Basics of Electricity and Circuits for more info.
See our explanation on Current-Voltage Characteristics. You’ll have more info on why Ohm’s law is not universal – only some conductors behave as this law predicts, and they are called ohmic conductors. The relationship between voltage, current, and resistance may be as complex as we desire (non-ohmic conductors), but if we limit ourselves to a small region of these quantities, we can always use Ohm’s law in that range.
Above, we have defined resistance, the role it fulfils in circuits, and the movement of charges. However, the definition we have given does not include information about its fundamental nature, i.e. how resistance is generated due to microscopic phenomena. To study these matters in-depth, let’s look at the concept of resistivity.
The definition of resistivity
Studying the relationship between resistivity and resistance allows us to see why resistivity is a characteristic property of materials and resistance is not. To define resistivity of a material, we measure the resistance of a conductor per unit of length and cross-section.
Resistivity is an intrinsic characteristic of matter, describing how strongly it can resist electric current compared to other materials.
It is different for each material and depends on certain physical conditions, such as temperature. It is measured in ohm-meters or Ωm and is denoted by the Greek letter ρ.
Factors affecting resistivity
Temperature
Resistivity grows with temperature because the temperature is a measure of the average kinetic energy of the particles of a material. If the particles of the conductor move faster (on average), they are more likely to interfere with the movement of charges.
Metallic nature
Another factor that determines the resistivity of a material is its metallic nature. Metals are known to favour the movement of charges, which implies that their characteristic resistivity is lower than the resistivity of other materials like wood or rubber. When we consider metals, their atomic structure and microscopic spatial disposition will determine how easy it is for charges to move, which will ultimately determine the exact value of the resistivity.
Some examples of the characteristic resistivity values of materials are shown below:
Material | Resistivity at 20ºC (Ω·m) |
Silver | 1.59 · 10-8 |
Copper | 1.68 · 10-8 |
Iron | 9.71 · 10-8 |
Carbon | 3 · 10-5 - 60 · 10-5 |
Mercury | 98 · 10-8 |
Silicones | 1 · 10-3 - 500 · 10-3 |
Glass | 1 · 109 - 1 · 1013 |
Rubber | 1 · 1013 - 1 · 1015 |
Air | 1.3 · 1016 - 3.3 · 1016 |
Resistivity is a characteristic property of materials that does not depend on their length and their cross-section.
The resistivity equation
If we know the resistivity of a material, we can calculate the resistance of a conductor made out of this material by multiplying it by the length and dividing it by the cross-section. Here is the equation that captures the relationship between resistance and resistivity:
\[R = \rho \cdot \frac{L}{A}\]
Here, R is the resistance, ρ is the resistivity, L is the length of the conductor, and A is its cross-section.
To interpret this equation, we must remember that current is the number of charges that go through a cross-section of a conductor per unit of time.
No matter the shape of a conductor, we can always find the cross-section as the surface perpendicular to the direction of the current at each point.
Now, since we know that resistance measures the opposition of the material to the current flow, why should we consider the material's length? Because the length also directly affects the resistance: the longer a medium (or object) is, the greater the resistance. This means that resistance and length are directly proportional. On the other hand, the resistance is inversely proportional to the medium’s cross-sectional area.
Length
You are in a very crowded street. The street is the conductor, and you are a charge trying to reach the other end of the street by avoiding the people standing in the street. It will be less tiring to walk just one block instead of three blocks because you avoid fewer people by walking just one block (the shorter the distance, the fewer people you encounter, which means that the shorter the length of a conductor, the less resistance there is).
Cross-section
The role of the cross-section is much easier to explain. In the end, we know that resistance measures the opposition to the flow of a current, but the current depends on the cross-section. If we double the size of the cross-section, we also double the current. That means that the opposition (resistance) is still acting, but due to the characteristics of the medium, we get more current (which means less resistance).
Imagine you are at the end of a crowded street, and you have several friends uniformly separated from each other at the other end of the street. If you were to count how many of your friends reach your end of the street per unit of time, you would count double that amount if you were in a street that was two times wider (and, consequently, where you had double the number of friends).
You have a proportional growth in friends by enlarging the street because you are considering a uniform density of charges in a material (following the analogy).
- Resistance grows with the length of conductors since the moving charges find more particles that obstruct them.
- Resistance decreases with the cross-section since the bigger the cross-section, the bigger the number of charges crossing it per unit of time.
How to calculate resistance using resistivity
Let’s use an example to help you understand the above info!
Consider two materials, silver and carbon. Silver is really expensive and difficult to get while obtaining carbon is relatively easy. We want to make a cable to connect two parts of a circuit separated by 1 meter. Since silver is hard to get, we only have a wire with a cross-section of 1cm2 (0.0001m2).
How wide should the carbon wire be to transfer current as efficiently as silver?
By using the equation of the resistance in terms of the length, the resistivity (found in the table), and the cross-section, we can calculate the resistance of the silver wire:
\[R_S = \rho _S \cdot \frac{L}{A_S} = 1.59 \cdot 10^{-8} \Omega \cdot m \cdot \frac{1 m}{0.0001 m^2} = 1.59 \cdot 10 ^{-4} \Omega\]
We now solve the same equation for the cross-section of carbon and the same resistance:
\[A_C = \rho_C \cdot \frac{L}{R} = 3 \cdot 10^{-5} \Omega \cdot m \cdot \frac{1 m}{1.59 \cdot 10^{-4} \Omega} = 0.19 m^2\]
If we were considering approximately cylindrical wires, this would imply using a cable with a diameter of roughly 0.5 m, which is big compared to the silver cable.
If we had considered a copper cable, the diameter would need to be almost the same as for silver (around 1.1cm), which explains why we use copper instead of carbon to make the cables we use.
Resistivity - key takeaways
- Resistance is a measure of the opposition of a medium to the flow of charges flowing through it.
- Resistivity is a measure of the opposition of a medium to the flow of charges inside of it per unit of length and cross-section. It is a more fundamental quantity than resistance because it does not depend on the size or width of the conductor, just on the properties of the material.
- Resistivity is characteristic for each material at certain external conditions since it is determined by the microscopic characteristics of the material. For instance, temperature affects the resistivity of a material.
- Resistance grows with the length of conductors since the moving charges find more particles that obstruct them.
- Resistance decreases with the cross-section since the bigger the cross-section, the bigger the number of charges crossing it per unit of time.
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