

Guide 212. Applying Conservation Laws to Series and Parallel Circuits There are two important conservation laws that are commonly applied to circuit analysis. These are the Law of Conservation of Charge and the Law of Conservation of Energy. This guide shows how these laws are applied to series and parallel combinations of resistors. Series Circuits Consider a circuit of 2 resistors of resistances R_{1} and R_{2} and a battery in a single loop as shown in Figure 1. The current in the circuit is I (that's conventional, positive current) and is the same at all points. We can say the latter, because electric charge is conserved. All the charge that passes any point of the circuit in a given time interval must pass any other point in the same time interval. If that didn't occur, charge would be lost or gained along the way. For the single loop circuit, there's nowhere else for the charge to go. If for some reason, the current changesfor example, this could be due to a charging or discharging capacitorthe change is the same in all parts of the circuit.
If we traverse the circuit counterclockwise in the direction of the current, then we encounter a potential rise across the battery and potential drops across the resistors. We've denoted these changes in Figure 2 as ΔV_{b}, ΔV_{1}, and ΔV_{2}. (The textbook uses the script uppercase symbol for the potential difference across the battery and calls this potential difference the emf. That's certainly an acceptable practice. However, any real battery has internal resistance, and the potential difference across its terminals is less than the emf and is equal to , where r is the internal resistance.) The change in electrical potential energy of any particular amount of charge Q passing through the battery is ΔU_{b} = QΔV_{b}. Note that this is a positive energy change or increase. The change in electrical potential energy of charge Q in the resistors is ΔU_{1} + ΔU_{2} = Q(ΔV_{1} + ΔV_{2}). This is a negative change or a decrease. Making our usual assumption that the energy loss in the wires is negligible, conservation of energy applied to the circuit tells us that ΔU_{b} + ΔU_{1} + ΔU_{2} = 0. Since Q divides out from each term, we obtain the loop rule. Loop rule: ΔV_{b} + ΔV_{1} + ΔV_{2} = 0. We can state the loop rule more compactly as , where the summation is taken around the complete circuit. We can take this a step further to come up with the relationship for the equivalent resistance of resistors in series. We'll use the relationship ΔV_{r} = IR. (You may at first find the negative sign confusing, since the textbook writes V = IR. However, the textbook uses the conventional practice that V is measured in such a way that it is always positive. This is what you would get if you touched the positive probe of the multimeter to the higher potential side of the resistor. It's important to realize, however, that V = ΔV_{r} for a resistor. That's because the change in potential, ΔV_{r}, across a resistor is negative.) Substituting, ΔV_{b}  IR_{1}  IR_{2} = 0. ΔV_{b}/I = R_{1}_{ }+ R_{2}. The lefthand side, ΔV_{b}/I, can be thought of as the resistance of a circuit with a single resistor whose resistance, R_{eq}, replaces that of R_{1} and R_{2} together. The equivalent circuit is shown below. Thus, R_{eq} = R_{1}_{ }+ R_{2}. Parallel Circuits Now let's apply conservation of energy and charge to a parallel circuit. The circuit is shown to the left. The current I splits at the junction point into two parts, I_{1} and I_{2}. Applying conservation of charge, Q = Q_{1} + Q_{2}. All charge going into the junction in a particular time must leave the junction. Dividing all terms by the same Δt allows us to apply the definition of current and write I = I_{1} + I_{2}. This application of conservation of charge to a circuit is called the junction rule.
Conservation of energy applies as well. We can apply it to each loop of the circuit individually. Note that there are actually 3 loops. These are: Loop 1: including the battery and R_{1} Let's apply the loop rule to loops 1 and 2 and traverse the loops counterclockwise. Loop 1: ΔV_{b} +
ΔV_{1} = 0. We see from this that ΔV_{b} = ΔV_{1} = ΔV_{2}. Thus, the potential rise across the battery is equal to the potential drop across each resistor. For completeness, let's look at loop 3. If we traverse the loop counterclockwise, we have ΔV_{1} + ΔV_{2} = 0. Note the negative sign. This is necessary, because we're traversing R_{1} in a direction opposite to the current. We then have ΔV_{1} = ΔV_{2}. This, of course, is consistent with the result found from examining loops 1 and 2. Let's apply these results to find the equivalent resistance of resistors in parallel. We start with the junction rule: I = I_{1} + I_{2}. Using ΔV_{r} = IR or I = ΔV_{r}/R, ΔV_{b}/R_{eq} = ΔV_{1}/R_{1}  ΔV_{2}/R_{2}, where the resistance we use on the lefthand side is that of the equivalent resistance that would replace the parallel combination of R_{1} and R_{2}. Now using the relationship found earlier, namely, ΔV_{b} = ΔV_{1} = ΔV_{2}, 1/R_{eq} = 1/R_{1} + 1/R_{2}. 

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