Imagine you're standing in a long, crowded hallway, and suddenly, everyone starts walking in the same direction. How easily could you leave the corridor, if following the crowd was the only way out? What would happen if the hallway shortened or became more narrow, or more people entered, walking at a faster pace? Depending on the circumstances, it would become easier or more difficult to move from one end to the other. This is analogous to resistance and resistivity. In a long cylindrical wire, the resistance depends on the length and cross-sectional area of the wire. So walking through a shorter and wider corridor would be much easier: we could say there is less resistance. Resistivity, on the other hand, is an inherent characteristic of the material, or the crowd in this example. A corridor full of children would have a different resistivity to that full of adults, as kids take up less space. Similarly, different materials have different atomic structures, with various arrangements and different inter-atomic spacing. In this article, we'll develop a better understanding of resistance and resistivity and check our knowledge by applying it to real-life examples!
Difference Between Resistance and Resistivity
Even though resistance and resistivity are often used interchangeably, they are definitely not the same thing. The two concepts are intertwined but resistance describes the opposition of current due to matter, while resistivity quantifies the structural properties of matter. Let's check out their respective definitions to see the differences.
Definition of Resistance and Resistivity
Now that we know that resistance and resistivity are two different concepts, let's look at their exact definitions!
Resistance is the ability of a material to oppose an electric current.
Resistance is represented by the symbol \(R\) and it describes how easily a material will allow for free electrons to flow through. The two factors impacting the resistance of an object are the type of material of which the object is made and its shape.
Resistivity is an intrinsic characteristic of matter, describing how strongly it can resist electric current compared to other materials.
Sometimes, resistivity is referred to as specific resistance, but both terms mean the same thing. This means that resistance and specific resistance are different things!
Resistivity is usually represented by the symbol \(\rho\) and depends on the atomic structure and the temperature of the material, and has nothing to do with the shape of an object. Values of resistivity for various materials have been determined in labs and tabulated, usually, at \(0\,\mathrm{^\circ C}\) or room temperature (\(20\,\mathrm{^{\circ}C}\)). The resistivity of some of the more commonly used materials can be seen in Table 1 below.
Table 1 - Resistivity of various materials at room temperature.
Material | \(\rho\) (\(\Omega \, \mathrm{m} \)) at \(20\,\mathrm{^{\circ}C}\) |
| Copper | \(1.68\times10^{-8}\) |
| Aluminum | \(2.65\times10^{-8}\) |
| Tungsten | \(5.6\times10^{-8}\) |
| Iron | \(9.71\times10^{-8}\) |
| Platinum | \(10.6\times10^{-8}\) |
| Manganin | \(48.2\times10^{-8}\) |
| Lead | \(22\times10^{-8}\) |
| Mercury | \(98\times10^{-8}\) |
| Carbon (pure) | \(3.5\times10^{-5}\) |
| Germanium (pure) | \(600\times10^{-3}\) |
| Silicon (pure) | \(2300\) |
| Amber | \(5\times10^{14}\) |
| Glass | \(10^9-10^{14}\) |
| Hard rubber | \(10^{13}-10^{16}\) |
| Quartz (fused) | \(7.5\times10^{17}\) |
Low resistivity corresponds to conductors, semiconductors have intermediate resistivity, while insulators have high resistivity.
Based on these definitions, it's clear that resistance depends on resistivity. After all, two objects with the same shape and size but with different resistivities will clearly have different resistances. But how exactly does resistance depend on resistivity? Let's look at the equation that connects the two concepts.
Equation Connecting Resistance and Resistivity
The main factors influencing an object's resistance are its shape and the material of which it is made. Mathematically, one of the easiest shapes to analyze is a cylinder. It just so happens that this is also the shape of most electrical wires! Let's look at the one in Figure 1 and use it to obtain the expression for resistance, which can then be applied to more complex shapes.
Fig. 1 - The resistance of a cylinder is directly proportional to its length and resistivity, and inversely proportional to its cross-sectional area.
The two variables can be connected through the following equation:
$$ R = \frac{\rho\ell}{ A}, $$
where \(R\) is the resistance measured in ohms (\(\mathrm{\Omega}\)), \(\ell\) is the length of the cylinder measured in meters (\(\mathrm{m}\)), \(A\) is the cross-sectional area of the cylinder in meters squared (\(\mathrm{m}^2\)), and \(\rho\) is the resistivity of the material measured in ohm-meters (\(\Omega\, \mathrm{m}\)).
Conceptually, we can imagine a flow of charges moving through this cylinder. Increasing its length will increase the overall number of collisions between the atoms in the material and the electric charges, as they will travel a longer distance. Similarly, if we increase the diameter of the cylinder, it's easier for the current to move through it, therefore experiencing less resistance. Finally, if a material's resistivity is higher, then it means that the material can more strongly resist electric current, so the resistance of the whole cylinder will be higher. These three facts nicely explain why the equation above is true, and you can use this logic to remember the equation.
Let's apply the resistance equation to some example problems.
An aluminum wire at room temperature has a resistance of \(0.150 \, \mathrm{\Omega}\). What is the diameter of this wire, if it has a length of \(16.0 \, \mathrm{m}\)?
Solution
We are given the length \(\ell\) and the resistance \(R\) of the wire. Considering it's placed at room temperature, the value for resistivity \(\rho\) of aluminum can be checked in Table 1 above (\(\rho_{\mathrm{Al}}=2.65\times10^{-8} \, \Omega \, \mathrm{m}\)).
First, let's rearrange the equation of resistance,
$$ R = \frac{\rho \ell}{A}, $$
to find the cross-sectional area:
$$ A=\frac{\rho \ell}{R}. $$
Plugging in our values gives us
$$ \begin{align} A &= \frac{(2.65\times10^{-8} \, \bcancel{\Omega} \, \mathrm{m})(16.0 \, \mathrm{m})}{(0.150 \, \bcancel{\Omega})} =2.83\times10^{-6} \, \mathrm{m}^2. \end{align}$$
Assuming that the wire is a uniform cylinder, we can calculate the radius \(r\) of its base circle using
$$ A=\pi r^2,$$
which can be rearranged to find the radius
$$ \begin{align} r&=\sqrt{\frac{A}{\pi}} \\ r&=\sqrt{\frac{2.83\times10^{-6} \, \mathrm{m}^2}{3.14}} \\ r& =9.49\times 10^{-4} \, \mathrm{m}. \end{align}$$
The diameter is simply twice the circle's radius, so the diameter of this aluminum wire is
$$ D=1.90\,\mathrm{mm}. $$
Similarly, we can calculate the resistivity of a material. A method explaining how to determine it experimentally, however, is explained later in the article.
A cube of unknown material has a resistance of \(2.80 \, \mathrm{\mu \Omega} \). If it has a side of \(2.00 \, \mathrm{cm}\) and is kept at room temperature, what material is the cube made of?
Solution
One way to determine the material is by calculating its resistivity and comparing it to the values listed in Table 1.
Once again, we can use the equation for resistance mentioned earlier to find the resistivity of the material:
$$ \rho = \frac{R A}{\ell}.$$
The shape of the object is a cube; therefore, the length will be equal to its side, and the cross-sectional area of a square can be calculated by squaring the length, so we obtain
\begin{align} \rho &= \frac{(2.80 \times 10^{-6} \, \Omega)(0.0200^2 \, \mathrm{m^{\cancel{2}}})}{(0.0200 \, \cancel{\mathrm{m}})} \\ \rho &= 5.60\times10^{-8} \, \Omega \, \mathrm{m}. \end{align}
Based on this calculation, we can conclude that the cube is made of tungsten, which is visible below.
Fig. 2 - Tungsten, more commonly known as wolfram, is a metal with low resistivity, making it a great conductor.
Resistance and Resistivity Experiment
At this point, we've concluded that resistivity is an inherent property of matter, just like the coefficient of friction or mass density, for instance. How are these values determined for different materials? That can be done experimentally, so let's take a look at an example of such an evaluation!
Develop an experimental plan to determine the resistivity of a resistive wire!
Overview
A wire of length \(\ell\) can be connected in a DC circuit. The values for current and voltage can be measured and used to calculate the resistance \(R\) for different lengths of wire. By plotting \(R\) versus \(\ell\), and finding the slope of the line, we can determine the resistivity of the wire.
Experimental equipment
For this experiment, we need
- a voltmeter,
- an ammeter,
- a ruler,
- a micrometer,
- a power supply,
- electrical wires,
- a resistive wire.
Procedure
Fig. 3 - A circuit diagram of a DC circuit used to measure the resistance of the wire.
- Connect the voltmeter, ammeter, and resistive wire to a power supply in a DC circuit as visible in Figure 3 above.
- Measure the diameter \(D\) of the wire using the micrometer, and calculate the cross-sectional area using $$ A=\pi\left ( \frac{D}{2} \right ) ^2. $$
- Measure the voltage \(V\) and current \(I\) values for various lengths of wire \(\ell\) and compile the data in a table.
- Using Ohm's law $$ R = \frac{V}{I},$$ calculate the resistance for each length.
- Plot the obtained values of \(R\) versus \(\ell\). An example of such a plot displaying a straight line is visible in Figure 4 below.
- From the resistance and resistivity relation $$ R = \ell \left (\frac{\rho}{A} \right ) $$ we can conclude that \(\ell\) is directly proportional to resistance, with the ratio of resistivity and cross-sectional area being the constant of proportionality. That is because the same type of wire is used in this experiment, meaning its diameter and material remain constant.
- Obtain the slope of the line in the \(R\) vs. \(\ell\) plot, plug in our measured value for the area, and determine the resistivity \(\rho\).
Fig. 4 - An example of resistance plotted against the length of the wire, used to determine the resistivity of the material.
Make sure to determine and calculate all the errors as well! The more times the measurements are repeated, the more accurate value for resistivity we'll get!
Resistivity and Conductivity
Just like resistivity, conductivity is an inherent property of a material.
Conductivity describes the ability of a material to conduct electricity.
Sometimes, conductivity is referred to as specific conductance, but both terms mean the same thing!
Mathematically, it can be expressed as
$$\sigma=\frac{1}{\rho}$$
where \(\sigma\) is conductivity measured in siemens per meter (\(\frac{\mathrm{S}}{\mathrm{m}}\)), and \(\rho\) is the same resistivity as defined earlier.
The unit of siemens is simply an inverse of ohm: \(1\, \mathrm{S} = 1\,\Omega^{-1}\).
Conductivity is the inverse of resistivity, so a low conductivity corresponds to a high resistivity and vice versa.
Resistance and Resistivity - Key takeaways
- Resistance is the ability of a material to oppose the flow of charge.
- Resistivity is an intrinsic characteristic of matter, describing how strongly it can resist electric current.
- Mathematically, resistance can be calculated using \(R = \frac{\rho\ell}{A}\).
- Low resistivity corresponds to conductors, semiconductors have intermediate resistivity, and insulators have high resistivity.
- Experimentally, resistance can be determined by measuring the voltage and current of a circuit and applying Ohm's law \(V=IR\).
- Conductivity describes the ability of a material to conduct electricity.
- Mathematically, conductivity can be calculated using \(\sigma=\frac{1}{\rho}\).
References
- Table 1 - Resistivities of materials, Douglas C. Giancoli, Physics, 4th Ed, Prentice Hall, 1995.
- Fig. 1 - The resistance of a cylinder, StudySmarter Originals.
- Fig. 2 - Tungsten (https://commons.wikimedia.org/wiki/File:Wolfram_1.jpg) by Tomihahndorf is licensed by Public Domain.
- Fig. 3 - A circuit diagram for experimentally determining resistivity, StudySmarter Originals.
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