The tools of linear algebra can be used to study the application of resistor networks. An example of an electrical circuit is below.
The jagged lines denote resistors and the numbers next to them give their resistance in ohms, written as \(\Omega\). The voltage source causes the current to flow in the direction from the shorter of the two lines toward the longer (as indicated by the arrow). The current for a circuit is labeled \(I_k\).
In the above figure, the current \(I_1\) has been labeled with an arrow in the counter clockwise direction. This is an entirely arbitrary decision and we could have chosen to label the current in the clockwise direction. With our choice of direction here, we define a positive current to flow in the counter clockwise direction and a negative current to flow in the clockwise direction.
The goal of this section is to use the values of resistors and voltage sources in a circuit to determine the current. An essential theorem for this application is Kirchhoff’s law.
Kirchhoff’s law allows us to set up a system of linear equations and solve for any unknown variables. When setting up this system, it is important to trace the circuit in the counter clockwise direction. If a resistor or voltage source is crossed against this direction, the related term must be given a negative sign.
We will explore this in the next example where we determine the value of the current in the initial diagram.
A more complicated example follows. Two of the circuits below may be familiar; they were examined in the examples above. However as they are now part of a larger system of circuits, the answers will differ.