Resistors*

claimed by Benjamin Flamm

Resistors are elements that are inserted into circuits in order to oppose the flow of current. This page gives examples of computing resistance as well as the history and applications of resistors.

The Main Idea

Resistors have many forms throughout modern technology and are applied in electronic industries ranging from basic manufacturing (lightbulbs, portable devices, etc.) to advanced biomedical instrumentation such as electrocardiogram devices. (electronicdesign.com)

The primary goal of a resistor is to limit the current that flows through a circuit. For example, a lightbulb is a very simple application of Tungsten or another material that has a high resistance. As electrons flow into the lightbulb, they begin to collide with themselves and the high number of charge carriers in the high-resistance filament. The result of these collisions is energy released as light and heat. See the Mathematical Model section for the relationship of these factors and how they determine resistance.

A Mathematical Model

Resistance can be modeled by starting at the fundamental concept [math]\displaystyle{ {I = |q|nA\bar{v}} }[/math] where [math]\displaystyle{ I }[/math] is conventional current, [math]\displaystyle{ |q| }[/math] is the magnitude of the charge being carried, [math]\displaystyle{ n }[/math] is the number of charge carriers, [math]\displaystyle{ A }[/math] is the area of the resistor, and [math]\displaystyle{ \bar{v} }[/math] is the drift speed of the charge.

[math]\displaystyle{ {I = |q|nA\bar{v} = |q|nAuE} }[/math] and [math]\displaystyle{ {J = \frac IA } }[/math] the equation for current density

Grouping the properties of the material together and utilizing the equation for conductivity [math]\displaystyle{ {\sigma = |q|nu} }[/math]:

[math]\displaystyle{ {I = (|q|nu)AE = {\sigma}AE} }[/math]

[math]\displaystyle{ {J = \frac IA = {\sigma}E} }[/math]

Substituting in the equation for electric field we get [math]\displaystyle{ {\frac IA = {\sigma}\frac {{\Delta}V}{L}} }[/math]

Finally, using algebra we attain [math]\displaystyle{ {I = \dfrac{{\Delta}V}{\dfrac{L}{{\sigma}A}} = \frac {{\Delta}V}{R}} }[/math]

Resulting in the definition of resistance being [math]\displaystyle{ {R = \dfrac {L}{{\sigma}{A}}} }[/math]

Although resistance can be easily derived and calculated, the majority of problems that contain resistance involve circuit analysis and Ohm's Law [math]\displaystyle{ {{\Delta}V = IR} }[/math] in which resistance is usually provided beforehand.

A Computational Model

Given a diagram consisting of a thinner wire leading into a thicker wire and then back through a thinner wire, it is possible to visualize how the cross-sectional area of a wire affects its resistance. The two thin sections of wire are identical and labeled with areas A\sub{1} and the thick section of wire has an area of A2.

Charge carriers (electrons in the diagram above) are flowing through the three wires with the same charge but are forced to pass through regions with varying cross-sectional areas. Analyzing the equation (R=l/sigA) it is obvious that resistance is inversely proportional to its cross-sectional area but observing the above diagram gives a more solidifying explanation of why this relationship exists.

So let's say electron flow is from left to right; traveling through the first thin wire, the thick wire, and then finally exiting through the second thin wire. The charge of these electrons are equal throughout each individual wire, as well as the number of charge carriers and the mobility of these charges. Observing the diagram, you can see that the electron flow is more dense in the thinner wires. This is due to the fact that the electrons have less room to move around and are constantly colliding with each other (makes sense, less area:more crowded). In the thick wire, the same amount of electrons are passing through but with much more space which means less collisions and ultimately less resistance.

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