20.6

Series Wiring

Thus far, we have dealt with circuits that include only a single device, such as a light bulb or a loudspeaker. There are, however, many circuits in which more than one device is connected to a voltage source. This section introduces one method by which such connections may be made—namely, series wiring. Series wiring means that the devices are connected in such a way that there is the same electric current through each device. Figure 20-15 shows a circuit in which two different devices, represented by resistors R1 and R2, are connected in series with a battery. Note that if the current in one resistor is interrupted, the current in the other is too. This could occur, for example, if two light bulbs were connected in series and the filament of one bulb broke. Because of the series wiring, the voltage V supplied by the battery is divided between the two resistors. The drawing indicates that the portion of the voltage across R1 is V1, while the portion across R2 is V2, so V = V1 + V2. For the individual resistances, the definition of resistance indicates that and , so that and . Therefore, we have

where RS is called the equivalent resistance of the series circuit. Thus, two resistors in series are equivalent to a single resistor whose resistance is , in the sense that there is the same current through RS as there is through the series combination of R1 and R2. This line of reasoning can be extended to any number of resistors in series if we note that, in general, the voltage across all the resistors in series is the sum of the individual voltages across each resistor. The result for the equivalent resistance is

| (20.16)

Examples 8 and 9 illustrate the concept of equivalent resistance in series circuits.

Figure 20-15 When two resistors are connected in series, the same current I is in both of them.

Problem solving insight

| Example 8 | | | A Series Circuit

Suppose the resistances in Figure 20-15 are and , and the battery voltage is 24 V. Determine the equivalent resistance of the two resistors and the current in the circuit. Reasoning The two resistors are wired in series, since there is the same current through each one. The equivalent resistance RS of a series circuit is the sum of the individual resistances, so . The current I can be obtained from Ohm’s law as the voltage V divided by the equivalent resistance: I = V/RS.
Solution The equivalent resistance is

| (20.16)

The current in the circuit is

| (20.2)

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 9 | | | Power Delivered to a Series Circuit

A resistor and a resistor are connected in series with a 12.0-V battery, as Figure 20-16 indicates. Assuming that the battery contributes no resistance to the circuit, find the power delivered to each of the resistors.

Figure 20-16 A and a resistor connected in series are equivalent to a single resistor.

Reasoning The power P delivered to each resistor is the product of the current squared (I2) and the corresponding resistance R, or P = I2R. The resistances are known, and Ohm’s law can be used to find the current. Ohm’s law states that the current in the circuit (which is also the current through each of the resistors) equals the voltage V of the battery divided by the equivalent resistance RS of the two resistors: I = V/RS. Since the resistors are connected in series, we can obtain the equivalent resistance by adding the two resistances (see Figure 20-16).
Knowns and Unknowns The data for this problem are:

Description | Symbol | Value
Resistance of resistor | R1 |
Resistance of 3.00-Ω resistor | R2 |
Battery voltage | V | 12.0 V
Unknown Variables |  |
Power delivered to 6.00-Ω resistor | P1 | ?
Power delivered to 3.00-Ω resistor | P2 | ?

Modeling the Problem

| Step 1 Power
| The power P1 delivered to the resistor is given by (Equation 20.6b), where I is the current through the resistor and R1 is the resistance. In this expression R1 is a known quantity, and the current I will be determined in Step 2.

| (20.6b)

| Step 2 Ohm’s Law
| The current I in the circuit depends on the voltage V of the battery and the equivalent resistance RS of the two resistors in series (see Figure 20-16). This dependence is given by Ohm’s law (Equation 20.2) as

This result for the current can be substituted into Equation 20.6b, as indicated at the right. Note from the data table that the voltage is given. In Step 3 we will evaluate the equivalent resistance from the individual resistances R1 and R2.

| Step 3 Equivalent Resistance
| Since the two resistors are wired in series, the equivalent resistance RS is the sum of the two resistances (Equation 20.16):

The resistances R1 and R2 are known. We substitute this expression for RS into Equation 1, as shown in the right column.

Solution Algebraically combining the results of the three steps, we have

The power delivered to the resistor is

In a similar fashion, it can be shown that the power delivered to the resistor is

Related Homework: Problems 44 and 108

In Example 9 the total power sent to the two resistors is . Alternatively, the total power could have been obtained by using the voltage across the two resistors (the battery voltage) and the equivalent resistance RS:

| (20.6c)

In general, the total power delivered to any number of resistors in series is equal to the power delivered to the equivalent resistance.

Problem solving insight

Pressure-sensitive pads form the heart of computer input devices that function as personal digital assistants, or PDAs, and they offer an interesting application of series resistors. These devices are simple to use. You write directly on the pad with a plastic stylus that itself contains no electronics (see Figure 20-17). The writing appears as the stylus is moved, and recognition software interprets it as input for the built-in computer. The pad utilizes two transparent conductive layers that are separated by a small distance, except where pressure from the stylus brings them into contact (see point P in the drawing). Current I enters the positive side of the top layer, flows into the bottom layer through point P, and exits that layer through its negative side. Each layer provides resistance to the current, the amounts depending on where the point P is located. As the right side of the drawing indicates, the resistances from the layers are in series, since the same current flows through both of them. The voltage across the top-layer resistance is VT, and the voltage across the bottom-layer resistance is VB. In each case, the voltage is the current times the resistance. These two voltages are used to locate the point P and to activate (darken) one of the elements or pixels in a liquid crystal display matrix that lies beneath the transparent layers (see Section 24.6 for a discussion of liquid crystal displays). As the stylus is moved, the writing becomes visible as one element after another in the display matrix is activated.

Figure 20-17 The pressure pad on which the user writes in a personal digital assistant is based on the use of resistances that are connected in series.

The physics of personal digital assistants.

The physics of a joystick. The joystick, found in many computer games, takes advantage of resistors connected in series. A joystick contains two straight coils of resistance wire that are oriented at 90° to each other (see Figure 20-18a). When the joystick is moved, it repositions the metallic slider on each of the coils. As part b of the drawing illustrates, each coil is connected across a 1.5-V battery.* Because one end of a coil is at 1.5 V and the other at 0 V, the voltage at the location of the slider is somewhere between these values; the voltage of the left slider in the drawing is labeled V1 while that of the right slider is V2. The slider voltages are then sent via wires to a computer, which translates them into positional data. In effect, the slider divides each resistance coil into two smaller resistance coils wired in series, and allows the voltage at the point where they are joined together to be detected.

Figure 20-18 (a) A joystick uses two movable sliders that are perpendicular. Each slider makes contact with a coil resistor. (b) The sliders allow the voltages V1 and V2 to be detected, which are then sent to a computer and translated into positional data.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.