KAHIBARO
Table of Contents
What Resistivity Means
Resistivity is a property of a material that tells us how strongly the material opposes the flow of electric current. It belongs to the material itself, not mainly to the shape of the object. A long thin wire and a short thick wire made of the same material do not have the same resistance, but they do have the same resistivity.
If charges move easily through a material, the resistivity is low. If the material makes charge flow difficult, the resistivity is high. Metals usually have low resistivity. Rubber, glass, and many plastics have very high resistivity.
This idea is important because resistance depends on both material and geometry, while resistivity isolates the effect of the material alone.
Resistivity and Resistance
For a uniform wire or rod, the resistance depends on its length $L$, its cross sectional area $A$, and the material resistivity $\rho$. The relation is
$$
R = \rho \frac{L}{A}
$$
This equation shows the roles clearly. If the wire is longer, the resistance increases. If the wire is thicker, the resistance decreases. If the material has larger resistivity, the resistance increases.
Important formula:
$$
R = \rho \frac{L}{A}
$$
and equivalently,
$$
\rho = R \frac{A}{L}
$$
Resistivity $\rho$ is a material property. Resistance $R$ depends on both the material and the object's dimensions.
Physical Picture
Inside a conductor, electric charges move through a lattice of atoms. As they move, they collide with atoms, impurities, and other imperfections in the material. These collisions hinder the motion of charge. A material with more hindrance has higher resistivity.
You can think of resistivity as a measure of internal difficulty for current flow. A low resistivity material gives charge carriers an easier path. A high resistivity material gives them a harder path.
SI Unit of Resistivity
From the equation
$$
R = \rho \frac{L}{A}
$$
we can determine the SI unit of resistivity. Since resistance is measured in ohms, length in meters, and area in square meters,
$$
\rho = R \frac{A}{L}
$$
so the unit is
$$
\Omega \cdot \text{m}
$$
This is read as ohm meter.
SI unit of resistivity:
$$
[\rho] = \Omega \cdot \text{m}
$$
Conductors, Poor Conductors, and Insulators
Different materials have very different resistivities. Good conductors have small resistivity. Insulators have extremely large resistivity. Semiconductors lie between these extremes.
Here is a qualitative comparison.
| Material type | Typical resistivity behavior | Current flow |
|---|---|---|
| Good conductor, like copper | Very low $\rho$ | Easy |
| Poor conductor | Moderate $\rho$ | Limited |
| Insulator, like rubber | Very high $\rho$ | Very difficult |
A lower resistivity means the same object will have lower resistance if its dimensions are unchanged.
Dependence on Material
Resistivity is determined by the microscopic structure of a material. Different materials have different numbers of free charge carriers and different amounts of scattering. That is why copper is much better for wires than iron, and why glass is not used for ordinary conducting wires.
A table of common materials helps show the idea.
| Material | Approximate resistivity at room temperature, $\Omega \cdot \text{m}$ |
|---|---|
| Silver | $1.6 \times 10^{-8}$ |
| Copper | $1.7 \times 10^{-8}$ |
| Aluminum | $2.8 \times 10^{-8}$ |
| Iron | $1.0 \times 10^{-7}$ |
| Carbon, graphite | $10^{-5}$ to $10^{-3}$ |
| Glass | about $10^{10}$ to $10^{14}$ |
| Rubber | about $10^{13}$ to $10^{16}$ |
These values vary with purity and temperature, but they show the enormous range of resistivity in nature.
Dependence on Temperature
Resistivity usually changes with temperature. For most metals, resistivity increases as temperature increases. Hotter metal means more lattice vibration, which causes more collisions for moving charges.
For many materials over a limited temperature range, the change can be written as
$$
\rho = \rho_0 \left[1 + \alpha (T - T_0)\right]
$$
where $\rho_0$ is the resistivity at reference temperature $T_0$, and $\alpha$ is the temperature coefficient of resistivity.
If $\alpha$ is positive, resistivity increases with temperature. This is common for metals.
For many metals near room temperature:
$$
\rho = \rho_0 \left[1 + \alpha (T - T_0)\right]
$$
A higher temperature usually means a higher resistivity.
Some materials, especially semiconductors, can behave differently. Their resistivity may decrease when temperature rises. The detailed reason belongs to semiconductor physics, but the key point here is that resistivity is not always constant.
Why Geometry Still Matters
Although resistivity is a material property, practical resistance in circuits depends strongly on size and shape. Two copper wires can have very different resistances if one is much longer or thinner.
A simple picture is useful.
If both wires are made of the same material, they have the same $\rho$, but the long thin wire has a much larger resistance because $\frac{L}{A}$ is larger.
Rearranging the Formula
Sometimes you know the dimensions and the material, and want the resistance. Sometimes you measure resistance and want the resistivity. The same relation can be rearranged.
$$
R = \rho \frac{L}{A}
$$
$$
\rho = R \frac{A}{L}
$$
$$
A = \rho \frac{L}{R}
$$
$$
L = \frac{RA}{\rho}
$$
These forms are useful in experiments and engineering design.
Simple Example
Suppose a wire has length $L = 2.0 \, \text{m}$, cross sectional area $A = 1.0 \times 10^{-6} \, \text{m}^2$, and resistivity $\rho = 1.7 \times 10^{-8} \, \Omega \cdot \text{m}$.
Then
$$
R = \rho \frac{L}{A}
= \left(1.7 \times 10^{-8}\right)\frac{2.0}{1.0 \times 10^{-6}}
$$
$$
R = 3.4 \times 10^{-2} \, \Omega
$$
So the wire has a resistance of
$$
R = 0.034 \, \Omega
$$
This small value is typical for a short copper wire of reasonable thickness.
Microscopic Interpretation
Resistivity can also be linked to how many charge carriers are available and how easily they move. At a deeper level, current depends on microscopic motion inside the material. For beginners, the most important point is that resistivity summarizes all those complicated internal effects into one measurable material property.
So instead of tracking every collision, we use one number, $\rho$, to describe how the material behaves electrically.
Resistivity Versus Conductivity
Another useful quantity is conductivity, written $\sigma$. Conductivity tells how well a material conducts electricity. It is the inverse of resistivity:
$$
\sigma = \frac{1}{\rho}
$$
and
$$
\rho = \frac{1}{\sigma}
$$
A material with high conductivity has low resistivity, and a material with low conductivity has high resistivity.
Resistivity and conductivity are inverses:
$$
\sigma = \frac{1}{\rho}
$$
Low resistivity means high conductivity.
Experimental Determination
To find the resistivity of a sample, measure its resistance and dimensions, then use
$$
\rho = R \frac{A}{L}
$$
For a wire, the area is often found from its radius $r$:
$$
A = \pi r^2
$$
so
$$
\rho = R \frac{\pi r^2}{L}
$$
This method is common in laboratory work.
Key Ideas to Remember
Resistivity is a property of a material that measures opposition to current flow. It is different from resistance, which also depends on length and cross sectional area. The main relation is
$$
R = \rho \frac{L}{A}
$$
The SI unit of resistivity is $\Omega \cdot \text{m}$. Good conductors have low resistivity, insulators have very high resistivity, and resistivity often changes with temperature.