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Section 1.4 Resistance

Subsection 1.4.1 Defining resistance

Resistance is opposition to the flow of current — anything that slows down electrons as they travel through a material. Resistance is a fact of life: everything 1  has some electrical resistance, including wires.

Some materials exhibit a particular kind of resistance such that the current flowing through the material is always proportional to the voltage that is applied. If we plot current through the material as a function of the voltage these materials produce a straight line:

For these materials, we can easily quantify their resistance using the slope of the line:

\begin{equation*} R = \frac{V}{I} \end{equation*}

This is known as Ohm's law, after German physiscist Georg Ohm 3 . You'll see it written in various forms, most commonly as \(V = IR\text{.}\) However, since we usually are working with batteries and constant-voltage power supplies, it is often more intuitive to think of Ohm's law as determining the current based on the voltage and resistance:

\begin{equation*} I = \frac{V}{R}\text{.} \end{equation*}

Following Ahmdal's little green men theory, electrical resistance is anything that prevents the green men from getting to their destination. If the road is narrow, or long, or bumpy, fewer green men will be going to the party. The more difficult the road, the fewer green men will be on it. At the same time, an increased need to party will result in more green men on the road. Ohm's law simply says that if you double the need to party, the number of green men on their way to the party will also double. If you make the road twice as long, or half as wide, then there will be half as many partiers on the road.

Subsection 1.4.2 Resistors

A device designed to have a particular amount of electrical resistance is called a resistor. Resistors come in all sorts of shapes and sizes, as shown in the picture below. .. todo: picture

Why would you ever want a resistor? We'll discover many uses throughout the course, but here are a few examples:

  • To limit the amount of current that flows through another device. LEDs in particular will burn out if too much current flows through them, so you should always include a resistor in the path with the LED.

  • To make a path for a small trickle of current to flow when nothing else is connected. Sometimes it's a problem for a component to be completely electrically disconnected, so a large resistor is used to make the connection without substantially affecting the circuit.

  • Multiple resistors can be used together to produce specific voltages.

Subsection 1.4.3 Not everything is a resistor!

It's tempting — now that you know about voltages and currents and resistance — to start applying Ohm's law to everything. Don't! We got to Ohm's law by observing that some materials have a linear relationship between current and voltage, and there are plenty of things that do not behave that way. Consider a few examples.

  • Batteries and DC power supplies (like laptop chargers and USB ports) are designed to deliver roughly constant voltage, regardless of the current. The amount of current flowing in or out depends on what is connected to it, not on the voltage.

  • An LED (or any diode) will have increased current with increased voltage, but the relationship is not at all linear:
  • The current consumed by a motor depends on the mechanical load in addition to the voltage.

You could imagine a tollbooth that lets little green men through at a constant rate, or a collapsed bridge that stops them dead in their tracks. In both cases, the number of green men traveling does not depend on the intensity of their need to party, and Ohm's law does not apply.

In truth, very few things follow Ohm's law perfectly. Even commercially-bought resistors have an IV relationship that is slightly dependent on temperature. That said, we will often model circuits as if all of the components are linear, because it makes the analysis so much easier. The important thing is to remember that you're making an approximation and keep in mind the limits of your analysis.

Yes, superconductors 2  exist, but they are extremely expensive and even "high temperature" superconductors require operating temperatures below -150C. They're fantastically useful for things like MRI machines and high-energy physics experiements, but don't expect to find them in your blender any time soon.