Resistors
Resistors resist! Okay, it’s in the name I know, I know. But really, resistors are circuit elements that resist current and slow down electron flow within a circuit. Since they DO let charge flow they are classified as conductors albeit ones with rather poor conductivity. We use resistance (R) measured in Ohms (Ω) to determine how much a particular resistor impedes the flow of charge. Here larger Ω values mean greater resistance and as a result lower current.
Determining Resistance
Overall there are three factors that determine the resistance of a specific resistor: what material it is made of (ρ), how large it is (A), and how long it is (L).
Material
As we saw in electrostatics, not all materials are created equally some allow charges to flow freely while others impede the flow of charge greatly. The property that describes this phenomenon is resistivity (ρ) and the greater a material’s resistivity the more it reduces charge flow.
[latexpage]
\[R \propto \rho\]
Since the resistance equation is often tested through proportionalities we will look at the proportionality of each individual factor. At the end, we will put all of the proportionalities together and compare the resistance equation to the resistance equation in fluids.
Temperature
While different materials have different resistivities the same materials can also have different resistivities if they are at different temperatures. Specifically increases in temperature cause the resistivities of different materials to increase. So iron at 200K will have a lower resistance than iron at 500K.
Length
The length of a resistor also impacts its resistance and as the length increases so does the resistance. This make sense if we think of a resistor as an electron obstacle course. The longer the obstacle the longer it take for you to get through to the other side.
[latexpage]
\[R \propto L\]
Size
Size specifically the cross-sectional area of material also impacts the resistance of a material. Here we are usually talking about wires so we will be using πr2 the area of a circle when discussing how the area affects resistance and current. As with fluids the smaller the pipe the higher the resistance. The exact same holds true of resistors the smaller the area of the resistor the higher the resistance. This make sense since the smaller the area the harder time charges have passing through.
[latexpage]
\[R \propto \frac{1}{A}\]
or
\[R \propto \frac{1}{\pi r^2}\]
Resistance Equation
Putting all of the proportionalities together we get the resistance equation:
[Latexpage]
\[ R=\frac{\rho L}{A}\]
A neat mnemonic to remember this equation is replay (RρLA). In total, resistance is due to the resistivity of a material the length of that material and the cross-sectional area.
This equation is nearly identical to the resistance equation in fluid flow so if you can remember one it can help you remember the other.
[latexpage]
\[ R = \frac{8\eta L}{\pi r^4}\]
Here viscosity (η) replaces resistivity (ρ) since the two are analogous. Length (L) remains the same and the bottom of the equation isn’t quite an area but comes pretty close to being one.
Resistors
Now that we know how to calculate the resistance of an individual resistor let’s zoom out and see how multiple resistors affect a circuit. In order to do this, we have to understand the arrangement of our resistors since it affects their impact on electron flow. We will use an analogy to investigate this idea and will treat each resistor like an obstacle in an obstacle course. Some obstacles are really challenging and have a high resistance while others are fairly easy and thus they have a lower resistance.
Path of Least Resistance
Electrons don’t want to work terribly hard so they will always follow the path of least resistance and seek out the easiest obstacle presented. This means that pathways through a circuit with lower resistance will have greater current since the electrons will “choose” to go that route.
Obstacle Arrangement
Obstacles can be arranged in different ways too. We can string them together one after the other or we can give electrons a choice and branch our obstacles off into different routes that eventually all meet up again.
Resistors in Series
When we place obstacles or resistors one after the other they are said to be in series. Here the difficulty of the obstacles compounds and electrons have to slog through each obstacle one after the other. This leads to a large drop in current and we can give the route an overall resistance by adding up the resistance of each resistor. It is kinda like a difficulty rating for the course we sum up the difficulty of each obstacle into an overall difficulty rating.
[latexpage]
\[R_{Total}=R_1+r_2+R_3…\]
Resistors in Parallel
When we give electrons choice and allow them to choose between a variety of obstacles the resistors are said to be in parallel. Here electrons are only passing through one of the obstacles and they tend to choose the easiest one. This leads to a bit of a line though so impatient electrons will divert to other routes. Again we can figure out the overall difficulty rating of the obstacle course but we can’t simply add each obstacle difficulty rating up.
Instead we will add the reciprocal of each rating up to determine the total resistance of resistors in parallel.
[latexpage]
\[ \frac{1}{R_{Total}} = \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}…\]