20.15

Concepts & Calculations

Series and parallel wiring are two common ways in which devices, such as light bulbs, can be connected to a circuit. The next example reviews the concepts of voltage, current, resistance, and power in the context of these two types of circuits.

| Concepts & Calculations Example 16 | | | The Brightness of Light Bulbs Wired in Series and in Parallel

A circuit contains a 48-V battery and a single light bulb whose resistance is . A second, identical, light bulb can be wired either in series or in parallel with the first one (see Figure 20-44). Determine the power delivered to a single bulb when the circuit contains (a) only one bulb, (b) two bulbs in series, and (c) two bulbs in parallel. Assume that the battery has no internal resistance.

Figure 20-44 When a second identical light bulb is connected to the circuit on the left, it can be connected either in series or in parallel with the original bulb.

Concept Questions and Answers How is the power P delivered to a light bulb related to the bulb’s resistance R and the voltage V across it?
Answer Equation 20.6c gives the answer as .
When there is only one light bulb in the circuit, what is the voltage V?
Answer Since there is only one bulb in the circuit, V is the voltage of the battery, V = 48 V.
The more power delivered to a light bulb, the brighter it is. When two bulbs are wired in series, does the brightness of each bulb increase, decrease, or remain the same relative to the brightness when only a single bulb is in the circuit?
Answer Since the bulbs are identical and are wired in series, each receives one-half the battery voltage V. The power delivered to each bulb, then, is

This result shows that the power delivered to each bulb in the series circuit is only one-fourth the power delivered to the single-bulb circuit. Thus, the brightness of each bulb decreases.
When two bulbs are wired in parallel, does the brightness of each bulb increase, decrease, or remain the same relative to the brightness when only a single bulb is in the circuit?
Answer Since the bulbs are wired in parallel, each receives the full battery voltage V. Thus, the power delivered to each bulb remains the same as when only one bulb is in the circuit, so the brightness of each bulb does not change.
Solution

a. | When only one bulb is in the circuit, the power that the bulb receives is

| (20.6c)

b. | When the two (identical) light bulbs are wired in series, each receives one-half the battery voltage V. According to Equation 20.6c, the power P delivered to each bulb of resistance R is

As expected, the power delivered to each bulb is only one-fourth the power delivered when there is only one bulb in the circuit.
c. | When the two bulbs are wired in parallel, the voltage across each is the same as the voltage of the battery. Therefore, the power delivered to each bulb is given by

| (20.6c)

As expected, the power delivered, and hence the brightness, does not change relative to that in the single-bulb circuit.

Kirchhoff’s junction rule and loop rule are important tools for analyzing the currents and voltages in complex circuits. The rules are easy to use, once some of the subtleties are understood. The next example explores these subtleties in a two-loop circuit.

| Concepts & Calculations Example 17 | | | Using Kirchhoff’s Rules

For the circuit in Figure 20-45, use Kirchhoff’s junction rule and loop rule to find the currents through the three resistors.

Figure 20-45 The three currents in this circuit are evaluated using Kirchhoff’s loop and junction rules.

Concept Questions and Answers Notice that there are two loops, labeled 1 and 2, in this circuit. Does it matter that there is no battery in loop 1, but only two resistors?
Answer No, it doesn’t matter. A loop can have any number of batteries, including none at all.
The currents through the three resistors are labeled as I1, I2 and I3. Does it matter which direction, left-to-right or right-to-left, has been chosen for each current?
Answer No. If we initially select the wrong direction for a current, it does not matter. The value obtained for that particular current will turn out to be a negative number, indicating that the actual current is in the opposite direction.
When we place the + and − signs at the ends of each resistor, does it matter which end is + and which is −?
Answer Yes, it does. Once the direction of the current has been selected, the + and − signs must be chosen so that the current goes from the + end toward the − end of the resistor. Notice that this is the case for each of the three resistors in Figure 20-45.
When we evaluate the potential drops and rises around a closed loop, does it matter which direction, clockwise or counterclockwise, is chosen for the evaluation?
Answer No, the direction is arbitrary. If we choose a clockwise direction, for example, we will have a certain number of potential drops and rises. If we choose a counterclockwise direction, all the drops become rises, and vice versa. Since we always set the potential drops equal to the potential rises, it does not matter which direction is picked for evaluating them. In the Solution that follows, we will use a counterclockwise direction for loop 1 and a clockwise direction for loop 2.
Solution Let’s begin by expressing the current I3 in terms of I1 and I2. Notice that I1 and I2 flow into junction C, and I3 flows out of it. Kirchhoff’s junction rule states that

| (20.23)

We now apply the loop rule to loop 1, starting at point A and proceeding counterclockwise around the closed circuit (ABCDEA):

| (20.24)

The two equations above contain three unknown variables, so we now apply the loop rule to loop 2 to obtain another equation that contains these variables. Starting at point A, we will move clockwise around loop 2 (ABCFGHA):

| (20.25)

Substituting Equation 20.23 for I3 into Equation 20.25 gives

| (20.26)

Solving Equation 20.24 for I2 in terms of I1 reveals that , and substituting this result into Equation 20.26, we find that

Solving this result for I1 yields . Substituting this result back into Equation 20.24 and solving for I2 gives . Equation 20.23 indicates, then, that .

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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