Circuit elements can also be connected so that more than one conducting path is formed, as illustrated in Figure 31.9.
Such circuits contain more than one loop and are called multiloop circuits. These circuits contain junctions—locations where more than two wires are connected together—and branches—conducting paths between two junctions that are not intercepted by another junction. The circuit in
Figure 31.9, for example, contains two junctions (represented by open circles), three branches, and three loops.
The continuity principle permits us to draw some impor- tant conclusions about the currents in multiloop circuits.
Let us begin by applying the current continuity principle to the circuit shown in Figure 31.10. Because we can select any system boundary along a branch, the current continu- ity principle requires the current to be the same throughout
Figure 31.9 Circuit diagram of two light bulbs connected to a battery.
This multiloop circuit has two junctions, c
cthree loops, c
cand three branches.
junction
junction
battery bulb
A B
Figure 31.10 The branch rule: In each branch, the current must be the same throughout that branch.
I2
I1 I1
I2
I3
I3 A
B
branch 1 branch 2 branch 3
ConCepts 31.3 JunCtions and multiple loops 835
❶ GettinG Started I begin by drawing circuit diagrams for the single-bulb circuit and for the parallel and series two-bulb circuits (Figure 31.12). To connect the bulbs in parallel to the battery, one contact of each bulb is connected directly to the positive terminal of the battery, and the other contact of each bulb is connected directly to the negative terminal of the bat- tery. For the series circuit, one contact of bulb 1 is connected directly to the positive terminal of the battery, the other contact of bulb 1 is connected to one contact of bulb 2, and the other contact of bulb 2 is connected directly to the negative terminal of the battery.
equal I1. The general statement of this principle is known as the junction rule:
The sum of the currents directed into a junction equals the sum of the currents directed out of the same junction.
Figure 31.11b illustrates this principle for balls flowing through a junction of tubes. If we push four balls in at one end of branch 1, four balls have to come out from branches 2 and 3 combined. In the junction shown in the figure, branch 2 and branch 3 are equivalent and two balls are likely to come out from each. In general, however, the branches do not need to be equivalent, and in that case the number of balls going into branch 1 must be equal to the sum of the balls coming out of branches 2 and 3. Because the pushing in and the coming out occur during the same time interval, the “current” of balls through branch 1 equals the sum of the currents in branches 2 and 3, which is what we concluded for the currents in Figure 31.11a.
Let us return now to the circuit in Figure 31.9. Because the two bulbs are each connected to the same two junc- tions, they are said to be connected in parallel across the battery. Because each junction is at a given potential, we can conclude that for parallel circuit elements:
The potential differences across circuit elements connected in parallel are always equal.
More specifically, for the circuit shown in Figure 31.9, the potential difference across the two light bulbs is equal to the emf of the battery.
In the next example we study the resistance of parallel combinations of circuit elements.
example 31.3 series versus parallel
Two identical light bulbs can be connected in parallel or in se- ries to a battery to form a closed circuit. How do the magnitudes of the potential differences across each bulb and the current in the battery compare with those in a single-bulb circuit when the bulbs in the two-bulb circuit are connected in parallel? When they are connected in series?
Figure 31.11 The continuity principle at a junction is illustrated by the flow of balls through a branched tube.
(a) (b)
I3 I2
branch 3 branch 1
branch 2
I1
In steady state:
cflow of charge into system
equals flow out: I1 = I2 + I3. cfor every two balls pushed into tube, two come out.
Charge does not accumulate
in system, so c Balls do not accumulate in tube, so c
❷ deviSe plan To determine the magnitude of the potential difference across each bulb, I need to analyze how the bulbs are connected to the battery. To determine the current in the bat- tery, I must first determine the current through each bulb. To determine the current in each bulb, I use the fact that the resis- tance and potential difference determine the amount of current in each bulb. I can use the junction rule to determine the cur- rents in the parallel circuit, and so I label the junctions 1 and 2 in Figure 31.12b. Because there is only one branch in the series circuit, I know that the current in the battery is the same as the current in the bulbs. Because the bulbs are identical, their resis- tances are identical too. I assume the wires have no resistance.
❸ execute plan Because in the parallel circuit one contact of each bulb is connected to the battery’s positive terminal and the other contact of each bulb is connected to the negative terminal, the potential difference across each bulb is equal to the potential difference across the battery. By the same argument, the poten- tial difference across the bulb in the single-bulb circuit is also equal to the potential difference across the battery. Therefore the potential difference across each bulb in the parallel circuit is the same as that across the bulb in the single-bulb circuit. ✔
Because the potential difference is the same across these three bulbs, the current must also be the same in all three bulbs.
I’ll denote this current by Ibulb. In the parallel circuit, the fact that the current in each bulb is Ibulb means that the current in the battery must be 2Ibulb. (The current pathway at junction 1 of Figure 31.12b is just like the pathway shown in Figure 31.11, with I2=I3=Ibulb.) In the single-bulb circuit, the current must be the same at all locations in the circuit, so the current in the battery must be Ibulb. ✔
In the series circuit, the potential difference across the two-bulb combination is equal to the potential difference across the battery, which means the magnitude of the potential differ- ence across each bulb must be half the magnitude of the poten- tial difference across the battery. The potential difference across each bulb is thus equal to half the potential difference across the bulb in the single-bulb circuit. ✔
Figure 31.12
ConCepts
That doesn’t mean there is no current in the circuit. As we’ll discuss in more detail in Section 31.4, because of the wire’s very small resistance, there is a very large current in the wire. Therefore, if the wire is left connected to the battery as in Figure 31.13, the battery quickly discharges through the wire and the wire heats up. A circuit branch with negligible resistance in parallel with an element is commonly called either a short or a short circuit.
The circuits in Figures 31.8, 31.10, and 31.12 are all drawn neatly with the circuit elements on a rectangular grid. Real circuits are rarely so neatly laid out, of course, and typically look more like the one shown in Figure 31.14. It takes time and concentration to look at the tangle of wires in Figure 31.14 and figure out whether or not the bulbs light up (they do) and identify the path taken by the charge carriers through the circuit.
To analyze such a circuit, therefore, it is helpful to draw a circuit diagram, with elements and wires arranged hori- zontally and vertically. The circuit diagram must accurately show the connections between elements that are present in the actual circuit, but wires that are not connected to each other should, as far as possible, be drawn so that they do not cross.
To draw a circuit diagram for the circuit in Figure 31.14, we begin by identifying the junctions and the branches connecting the junctions. Note that the two terminals of bulb A are connected to two wires each. Because the ter- minals of the bulb are also connected to each other via the filament inside the bulb, each of these terminals is a junc- tion. There are three branches. One branch consists of the battery and the wires that connect the left and right con- tacts of light bulb A. The second branch connects the left contact of light bulb A through the filament to the right contact of light bulb A. The third branch connects the right contact of light bulb A through light bulbs B and C Because the potential difference across each bulb in series
is half the potential difference across the battery, the current in each bulb must be half the current in the single-bulb circuit
Ibulb>2. Because the series circuit in Figure 31.12c is a single-
loop circuit, the current is the same at all locations. Therefore the current in the battery must also be Ibulb>2. The current in the battery in the single-bulb circuit is therefore twice that in the series two-bulb circuit. ✔
In tabular form my result are
❹ evaluate reSult The current in the battery in the paral- lel circuit is therefore four times that in the series circuit. That makes sense because in the parallel circuit each bulb glows identically to the bulb in the single-bulb circuit, while in the se- ries circuit, the battery has to “push” the charge carriers through twice as much resistance. Therefore it makes sense that in the series circuit, both the potential difference across each bulb and the current in the battery are much smaller than they are in the parallel circuit.
31.6 In Figure 31.9, treat the parallel combination of two light bulbs as a single circuit element. Is the resistance of this element greater than, equal to, or smaller than the resistance of either bulb?
Checkpoint 31.6 highlights an important point about electric circuits: Adding circuit elements in parallel lowers the combined resistance and increases the current. How can adding elements with a certain resistance lower the com- bined resistance? The resolution to this apparent contradic- tion is that adding elements in parallel really amounts to adding paths through which charge carriers can flow, rather than adding resistance to existing paths in the circuit. If you are emptying the water out of a swimming pool, it empties faster if you have multiple hoses draining the water than if you drain the entire pool through a single hose.
Instead of connecting two light bulbs in parallel as in Figure 31.9, what if we connected a wire in parallel with a bulb as in Figure 31.13? Experimentally, we find that replac- ing bulb B of Figure 31.9 with a wire causes bulb A to stop glowing. Why does this happen? The potential difference across a branch is determined by the potential difference between the two junctions on either end of the branch, and therefore the potential difference across all branches be- tween two junctions must be the same. Because the wire is made from a conducting material, the potential difference between its two ends is zero, and therefore the two junctions in Figure 31.13 are at the same potential. Consequently there is no current in the light bulb.
Figure 31.13 A wire that is connected to a battery in parallel to a light bulb constitutes a short circuit.
short circuit A
not glowing
A B C
Figure 31.14 Real circuits don’t always look like circuit diagrams.