Interactive Lecture Demonstration 6 – Circuits with Three Resistors

In this activity, you will predict the current and voltage difference for each of three different resistors in a circuit connected to a battery. You will make your predictions using two different methods. You will then compare your predictions with the values of current and voltage difference measured directly using a digital multimeter.

The circuit you will analyze is diagrammed below. The instructor will tell you the battery voltage and the values of the three resistances.

Method I: In any circuit involving resistors of known resistance combined using series and parallel wiring and a single voltage source, it is always possible to determine the currents and voltage differences for all the resistors by first calculating the equivalent resistance of all the resistors, then using the Loop Rule and Ohm's Law to calculate the battery current, then further applying the Loop Rule, Ohm's Law and the Junction Rule to calculate everything else. Note that this method has fairly limited applicability (since not all circuits contain only resistors that are wired in only series and parallel combinations). It is nevertheless worth knowing.

1. Calculate the equivalent resistance of the circuit above. Show your work clearly, using appropriate "equivalent circuit" diagrams along with the equivalent resistance formulas to explain.

2. Now apply the Loop Rule to the equivalent circuit consisting of just the battery and the single equivalent resistor representing the total resistance of the resistors. Label on your circuit diagram the direction of the conventional current through the battery, then use the Loop Rule to calculate the magnitude of that current.

3. Having found the battery current, find a way to use Ohm's Law, the Loop Rule and the Junction Rule to calculate the voltage difference and current for all of the individual resistors in the original circuit. You may find it helpful to apply those laws to one or more of the intermediate equivalent circuits you used in #1. Document your work clearly by using appropriate subscripts for all quantities. When you're done, check your work by making sure that the Junction Rule is satisfied at all junctions in the original circuit, and the Loop Rule is satisfied for all loops in the original circuit.

Method II: This method is more powerful than Method I, because it uses only fundamental laws of physics, which (unlike the formulas for equivalent resistance of resistors in series and parallel) can be applied to any circuit, no matter how complicated, and no matter what devices the circuit contains.

1. Start by writing down the Loop Rule for every loop that goes through the battery in the circuit. Label appropriately all currents and voltage differences with subscripts, and show/label on your circuit diagram every loop that you use. In writing your Loop Rules, use Ohm's Law to express the voltage difference for every resistor through which a loop passes.

2. Now write down the junction rule for every junction in your circuit. Label every junction on your circuit diagram.

3. You now have a set of equations. The number of equations you have is automatically equal to the number of different currents in your circuit. Thus, you have N equations with N unknowns (the currents). Since your equations are linear (i.e. only have terms that are directly proportional to the current and not to the square or some other power of the current) and are independent of each other, you can always solve them to find all of the currents (there is a math theorem that proves this is always possible). You will have to do some algebra here. Depending on your equations (i.e. depending on your circuit), you might have to solve one equation to express one of the currents in terms of the other ones, then substitute into a different equation to eliminate that current, before being able to actually solve for the value of any current. Look for algebra shortcuts (for example, adding or subtracting equations) that could help you eliminate unknown variables from your equations.

Show below how you solve your Loop and Junction Rule equations to find each of the currents in your circuit. Check your calculations afterwards, by making sure your calculated currents satisfy all the junction rules for the circuit.

4. Now that you've gotten all the currents, calculating the voltage differences is easy. Just apply Ohm's Law to every resistor to find its voltage difference. Check your calculations, by making sure your calculated voltage differences satisfy all the loop rules for the circuit.

Followup:

1. Compare your predicted currents and voltage differences with those measured directly using a digital multimeter. Making a data table would help you organize this part.

2. One more check of your work: since the energy of the charges in your circuit is conserved (i.e. your circuit satisfies the Loop Rule), the power delivered to the circuit by the battery must equal the total power consumed by the resistors. Check that this is true by calculating the power of the battery and of each resistor.

3. If you have time, repeat the entire procedure we just followed for other circuits containing the same three resistors. There are a total of four possible ways of connecting three resistors to a battery – can you find them all ?