20.3

Kirchhoff's Rules

Student Notes

With this simulation, you can investigate a multiloop circuit, in which a number of batteries and resistors are connected together. In this circuit there are three branches - these are color-coded on the diagram. In each branch there is an unknown current. You can solve for these three unknown currents by using Kirchoff's Rules to set up three equations.
The first equation is arrived at using Kirchoff's junction rule. A junction is a place where three or more branches meet, so there are two junctions in the circuit shown here. The junction rule says that the total current flowing in to the junction equals the total current flowing away from the junction.
You can then set up two more equations by applying Kirchoff's loop rule, which says that the sum of all the potential differences around a complete loop equals zero. There are three loops to choose from in this circuit. You can write down a loop equation for each loop, but you only need to use two of the loop equations - the third equation is actually a combination of the other two.
Here are some things to try to see if you fully understand the concepts. For a given set of battery voltages and resistances, set up the junction and loop equations needed to solve for the currents. Do you agree with the equations arrived at by the simulation? When you solve your equations do you get the same values for the currents that the simulation gets? In this simulation the loop equations are obtained by going clockwise around each loop starting and ending in the same place. What happens to a loop equation if you go counter-clockwise around a loop instead? You should find that all the signs are reversed, but if you solve you should find exactly the same values for the currents. What happens to the equations if you reverse the direction of one of the currents? What does this do to the value of this current?

Self-test

It does not matter whether you go clockwise or counter-clockwise around a loop to obtain a loop equation, but what about the spot you choose as the starting and ending point? How does an equation change if you start in a different spot? Remember that you stop when you return to the starting point. Does the choice of starting point affect the values of the currents? There are two junctions in this circuit. Why don't we write down junction equations for both junctions? An excellent way to check to see whether the currents you calculate are correct is to label the potential at different points around the circuit. You could, for instance, decide that the potential at the bottom left corner of the circuit is V = 0. The potential at the top left would therefore be equal to V1, because to get from the bottom left to the top left you cross battery 1 from negative to positive, increasing the potential by the battery voltage. Continue this process, labeling the potential everywhere. If you get consistent results for the potential at each point around the circuit, you know your calculations must be correct.

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