Electricity is a vital part of modern society. We rely on electricity for manufacturing, heating, lighting, and many other daily necessities. Fundamentally, electricity can be explained using its three main properties: current, voltage, and resistance, which combined form one of the most important laws of physics - Ohm's law. It was announced in 1827 by German physicist Georg Simon Ohm, who was the first to describe electricity in such a manner. He performed an experiment using a metal wire and a compass, to determine the relationship between the voltage and the current and concluded that their ratio is equal to a property we now call electrical resistance. As a result of this discovery, the units for resistance were named after Ohm. In this article, we'll explain what exactly electrical resistance is and how it is calculated.
Resistance Definition
To begin, let's define what exactly electrical resistance is.
Resistance is a measure of the degree to which an object opposes the movement of electric charges.
In electrical systems and electronic devices, such as irons, electronic watches, or mobile phones, elements with fixed or variable electrical resistance are always required. Regardless of their specific construction, all of these elements are considered resistors.
A resistor is an element in an electric circuit that limits the flow of electric charges.
Resistors can convert electrical energy to thermal energy, which may change the temperature of the resistor and the resistor’s environment.
Fig. 1 - Each color band on a resistor provides us with useful information about the specific resistor.
They are very diverse both in their resistance range and in construction. Typically, resistors have from three to six bands of various colors wrapping around them, as pictured in Figure 1 above. Each color and placement indicates a different resistance value or property.
Resistor Color Code
The most commonly used type of resistor consists of four bands, so let's explain the meaning of each of these bands. To determine the properties of a resistor, we must read it from left to right. The first two bands correspond to the first two digits. The third band is the multiplier, which simply indicates how many zeros we must add to the first two digits. Finally, the fourth band is the tolerance, or the error of the resistor. Each color is paired with a number, which is compiled in Table 1 below.
Table 1 - Resistor color code.
| Color | Digits | Multiplier, \(\Omega\) | Tolerance |
| Black | \(0\) | \(1\) | |
| Brown | \(1\) | \(10\) | \(\pm \, 1\%\) |
| Red | \(2\) | \(10^2\) | \(\pm \, 2\%\) |
| Orange | \(3\) | \(10^3\) | \(\pm \, 3\%\) |
| Yellow | \(4\) | \(10^4\) | \(\pm \, 4\%\) |
| Green | \(5\) | \(10^5\) | \(\pm \, 0.5\%\) |
| Blue | \(6\) | \(10^6\) | \(\pm \, 0.25\%\) |
| Violet | \(7\) | \(10^7\) | \(\pm \, 0.1\%\) |
| Gray | \(8\) | \(10^8\) | \(\pm \, 0.05\%\) |
| White | \(9\) | \(10^9\) | |
| Gold | | \(10^{-1}\) | \(\pm \, 5\%\) |
| Silver | | \(10^{-2}\) | \(\pm \, 10\%\) |
| No color | | | \(\pm \, 20\%\) |
To understand the application of these values better, let's analyze the resistor on the outermost right side in Figure 1.
Determine the value of a four-band resistor with the following band colors: yellow, violet, orange, and gold.
Answer:
In this case, yellow and violet are the first two digits, orange corresponds to the multiplier, and gold will tell us the tolerance.
We can extract the following information from the resistor band table above:
- 1st digit is yellow: \(4\),
- 2nd digit is violet: \(7\),
- Multiplier is orange: \(10^3\,\Omega\),
- Tolerance is gold: \(\pm\,5\%\).
Putting it all together, we conclude that the resistance of this particular resistor is \(47\times10^3\,\Omega \pm5\%\), or \(47\,\mathrm{k}\Omega \pm 5\%\).
- Three-band resistors consist of two digits and a multiplier, as the tolerance band is always "no color."
- Five-band resistors are the same as four-band resistors, only with three digits rather than two.
- Six-band resistors are the same as five-band resistors, only they have an additional band denoting the temperature coefficient.
Resistance Symbol
In equations, resistance is represented by the letter "R", which is quite straightforward. It becomes a little more tricky when we have to include resistors in an electric circuit, where just like every other component, they have their own symbol. Depending on the source and the country of origin, a fixed resistor can be represented either as a zigzag line (American style) or a rectangular box (International style), as pictured in Figure 2 below. An easy way to associate the zigzag line with resistance is imagining current passing through a straight line versus such a rapidly changing path - it will obviously be much harder for the electric charge to get through it.
Fig. 2 - Resistor symbols, rectangular box is the international version, zigzag is the American version.
In all of our examples, we'll be using the zigzag line to represent the resistor; however, it's useful to be familiar with the internationally accepted version as well.
A fixed resistor simply implies that its resistance remains constant and cannot be adjusted. Sometimes, these symbols have additional lines and arrows drawn around them. That indicates a very specific type of resistor with special functionality, which we won't be dealing with in this course.
Resistance and Resistivity
Resistance and resistivity are often thought to be the same thing, which is definitely not the case. The two concepts are intertwined, but resistance describes the opposition of current due to matter, while resistivity quantifies the structural properties of matter.
Resistivity is a fundamental property of a material that quantifies how strongly the material opposes the motion of electric charge.
Resistivity is usually represented by the symbol \(\rho\). The factors impacting a material's resistivity are the atomic structure and the temperature. Resistivity 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 \(0\, ^\circ \mathrm{C}\) or room temperature (\(20\, ^\circ \mathrm{C}\)). The resistivity of some of the more commonly used materials can be seen in Table 2 below.
Table 2 - 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) | \(7.5\times10^{17}\) |
Resistance Formula
The main factors influencing an object's resistance are its shape and the material of which it is made. We can easily obtain the expression for resistance by considering a cylinder, like the one in Figure 3.
Fig. 3 - The resistance of a cylinder is directly proportional to its length and resistivity, and inversely proportional to its cross-sectional area.
Mathematically, the expression for resistance is
\[ R=\frac{\rho \ell}{A}\]
where \(R\) is the resistance measured in ohms \(\left(\Omega\right)\), \(\ell\) is the length of the cylinder measured in meters (\(\mathrm{m}\)), \(A\) is the cross-sectional area of the cylinder in meters squared \(\left(\mathrm{m^2}\right)\), and \(\rho\) is the resistivity of the material measured in ohm-meters \(\left(\Omega\,\mathrm{m}\right)\). This equation can be applied to other, more complex shapes.
Essentially, resistance explains how easily electrical charges move 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, it means that the material can more strongly resist electric current, so the resistance of the whole cylinder will be higher.
Let's apply the resistance equation to an example problem.
A copper wire at room temperature has a resistance of \(0.120 \, \mathrm{\Omega}\). What is the diameter of this wire, if it has a length of \(18.0 \, \mathrm{m}\)?
Answer:
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 2 above (\(\rho_{\mathrm{Cu}}=1.68\times10^{-8} \, \Omega \, \mathrm{m}\)).
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
$$ \begin{align} A &= \frac{(1.68\times10^{-8} \, \bcancel{\Omega} \, \mathrm{m})(18.0 \, \mathrm{m})}{(0.120 \, \bcancel{\Omega})} =2.52\times10^{-6} \, \mathrm{m}^2. \end{align}$$
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
$$ \begin{align} r&=\sqrt{\frac{A}{\pi}} \\ r&=\sqrt{\frac{2.52\times10^{-6} \, \mathrm{m}^2}{3.14}} \\ r& =8.96\times 10^{-4} \, \mathrm{m}. \end{align}$$
The diameter is simply twice the circle's radius, so the diameter of this copper wire is
$$ D=1.79\,\mathrm{mm}. $$
Electrical Resistance and Conductance
When dealing with electric circuits, electrical resistance and conductance are important characteristics that describe their behavior. Both of these properties are closely related to the electric current and ability of a component to move through it. Let's look at each of them separately.
Resistance
Resistance can be measured using an ohmmeter or calculated using Ohm's law:
\[R=\frac{\Delta V}{I},\]
where \(R\) is the resistance in ohms (\(\Omega\)), \(V\) is the voltage in volts (\(\mathrm{V}\)), and \(I\) is the current in amperes (\(\mathrm{A}\)).
If we know the type of circuit and individual resistance of each resistor, we can find the total resistance of an electrical system using the following relations:
If the resistors are in series (i.e. next to each other), you add the value of each resistor together: \[R_\mathrm{series}=\sum_{n}R_n=R_1+R_2+ \cdots,\]
If the resistors are in parallel, the rule for finding the total resistance is as follows: \[\frac{1}{R_\mathrm{parallel}}=\sum_{n}\frac{1}{R_n} =\frac{1}{R_1}+\frac{1}{R_2}+\cdots.\]
Conductance
Conductance is simply the opposite property of resistance. If resistance tells us the extent a component opposes current, conductance explains how easily current can flow through said component.
Conductance is the ability of a particular component to conduct electricity.
Considering it's the inverse of resistance, mathematically conductance \(G\) can be expressed as
\[G=\frac{1}{R}=\frac{I}{\Delta V}.\]
It's measured in the units of siemens (\(\mathrm{S}\)), which are the inverse of an ohm: \(1\,\mathrm{S}=1\,\frac{1}{\Omega}.\)
Not to be confused with conductivity \(\sigma\), which is an inherent property of material just like resistivity.
Resistance - Key takeaways
- Resistance is a measure of the degree to which an object opposes the movement of electric charges.
- A resistor is an element in an electric circuit that limits the flow of electric charges.
- Resistance is represented by the letter "R", while the resistor in an electric circuit is represented by a zigzag line.
- Resistivity is a fundamental property of a material that quantifies how strongly the material opposes the motion of electric charge.
- Resistivity is represented by the symbol \(\rho\) and depends on the atomic structure and the temperature of the material.
- Mathematically, the expression for resistance is \( R=\frac{\rho \ell}{A}.\)
- Resistance is one of the main properties of an electric circuit, where it can be calculated using Ohm's law \(R=\frac{\Delta V}{I}.\)
- Conductance is the ability of a particular component to conduct electricity.
References
- Fig. 1 - Electronic-Axial-Lead-Resistors-Array (https://commons.wikimedia.org/wiki/File:Electronic-Axial-Lead-Resistors-Array.jpg), by Evan-Amos (https://commons.wikimedia.org/wiki/User:Evan-Amos) is licensed by Public Domain.
- Fig. 2 - Resistor symbols, StudySmarter Originals.
- Table 2 - Resistivities of materials, Douglas C. Giancoli, Physics, 4th Ed, Prentice Hall, 1995.
- Fig. 3 - The resistance of a cylinder, StudySmarter Originals.
Explanations 
Exams 
Magazine 
