Applying Conservation Laws to Series and Parallel Circuits, Part 1

Guide 21-2. 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 R₁ and R₂ 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 changes--for example, this could be due to a charging or discharging capacitor--the change is the same in all parts of the circuit.

Figure 1	Figure 2

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, ΔV₁, and ΔV₂. (The textbook uses the script upper-case 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 ΔU₁ + ΔU₂ = Q(ΔV₁ + ΔV₂). 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 + ΔU₁ + ΔU₂ = 0. Since Q divides out from each term, we obtain the loop rule.

Loop rule: ΔV_b + ΔV₁ + ΔV₂ = 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 - IR₁ - IR₂ = 0.

ΔV_b/I = R₁+ R₂.

The left-hand 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 R₁ and R₂ together. The equivalent circuit is shown below. Thus, R_eq = R₁+ R₂.

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, I₁ and I₂. Applying conservation of charge, Q = Q₁ + Q₂. 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 = I₁ + I₂. This application of conservation of charge to a circuit is called the junction rule.

Junction rule: The sum of currents going into a circuit junction is equal to the sum of currents going out of the junction.

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 R₁
Loop 2: including the battery and R₂Loop 3: including R₁ and R₂

Let's apply the loop rule to loops 1 and 2 and traverse the loops counterclockwise.

Loop 1: ΔV_b + ΔV₁ = 0.
Loop 2: ΔV_b + ΔV₂ = 0.

We see from this that ΔV_b = -ΔV₁ = -ΔV₂. 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 -ΔV₁ + ΔV₂ = 0. Note the negative sign. This is necessary, because we're traversing R₁ in a direction opposite to the current. We then have ΔV₁ = ΔV₂. 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 = I₁ + I₂. Using ΔV_r = -IR or I = -ΔV_r/R,

ΔV_b/R_eq = -ΔV₁/R₁ - ΔV₂/R₂,

where the resistance we use on the left-hand side is that of the equivalent resistance that would replace the parallel combination of R₁ and R₂. Now using the relationship found earlier, namely, ΔV_b = -ΔV₁ = -ΔV₂,

1/R_eq = 1/R₁ + 1/R₂.

Here's a summary of the important points so far.

For all circuits, the loop rule and junction rule apply, since these are generally valid conservation laws.
For series circuits, the current is the same in all parts. (Note that this is not the same as saying the current is constant. Constant implies no change in time. The current can be changing in all parts of the circuit at the same time. This is what happens, for example, when capacitors charge and discharge.)
For series circuits, the potential rise across the battery is equal to the sum of the potential drops across the resistors.
For parallel circuits, the potential rise across the battery is equal to the potential drop across each branch of the circuit.
For parallel circuits, the current from the battery is the algebraic sum of the currents in the branches. (We specify algebraic sum in case we select current directions in such a way that one or more are negative.)
Finally, we have the formulas for calculating equivalent resistance of series and parallel circuits.