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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

PowerPoint® Lectures forUniversity Physics, Twelfth Edition

– Hugh D. Young and Roger A. Freedman

Lectures by James Pazun

Chapter 26

Direct-Current Circuits


Goals for Chapter 26

• To study resistors in series and parallel

• To consider Kirchoff’s Rules

• To see the design and learn about the use of electronic measuring instruments

• To mentally assemble R-C circuits

• To study the applications of circuits in household wiring


Introduction

• In the last chapter, we gained insight about how current flows through a resistor in simple examples like a light bulb attached to a battery.

• Now, imagine many thousands of circuits wired onto flat wafers with structure so tiny that microscopy would be necessary to view their patterns. Understanding the next step and mastering more complex circuit patterns is the goal forChapter 26.


Resistors in series and parallel I• If we took three resistors

and considered the different ways they could be connected, we arrive at the four possibilities illustrated in Figure 26.1.

• Some of the combinations will be sequential (like the line at a phone booth), some will be en masse (like a marching band moving in rows). The former are analogous to resistors in series, the latter to resistors in parallel.


Resistors in series and parallel II• If you have ever wired a Christmas tree with a series of lights

(resistors) in series, you know what happens if just one burns out. The lights have become an open circuit and will not function.

• Car headlights are a good example of resistors wired in parallel. If one light burns out, the circuit changes but still functions to allow the driver a safe trip to repair. See Figure 26.2 below.


Resistors in series and parallel III—combinations

• Consider Problem-Solving Strategy 26.1.

• Follow Example 26.1 guided by Figure 26.3 below.

• Follow Example 26.2.


Kirchoff’s Rules I—junctions• The algebraic sum of the currents into any junction is zero.

Figures 26.6 and 26.7 illustrate this rule and are shown below.


Kirchoff’s Rules II—loops

• The algebraic sum of the potential differences in any loop, including those associated with emfs and those of resistive elements, must equal zero.


Kirchoff’s Rules III—examples and strategy• Read through Problem-Solving Strategy 26.2. Figure 26.9

illustrates this strategy.

• Refer to Example 26.3, illustrated by Figure 26.10.


Kirchoff’s Rules IV—examples• Refer to Example 26.4, illustrated by Figure 26.11.

• Consider Example 26.5.

• Refer to Example 26.6, illustrated by Figure 26.12.

• Review Example 26.7.


D’Arsonval’s galvanometer• We’ll call it simply “meter” henceforth.

• The meter is a coil of wire mounted next to a permanent magnet. Any current passing through the coil will induce magnetism in the coil. The interaction of the new electromagnetism and the permanent magnet will move the meter indicator mounted to the coil.


The ammeterMaxes out at full scale deflection current, Ifs. The ammeter can only measure from zero to Ifs, unless it is wired in parallel to a shunt resistor of resistance Rsh. This shunt resistor decreases the current through the coil (with resistance Rc) because current is selective in parallel. Since voltage throughcoil and shunt is the same, Vc = Vsh or IfsRc = (I – Ifs)Rsh


The voltmeterMaxes out at full scale reading Vv. We can increase the range by adding a resistor such that the current is forced through theresistor with resistance R. Most of the voltage drop will be across the resistor, leaving a fraction of voltage drop across the coil.

The maximum voltage can be calculated by:

VV = Ifs (Rs + Rc)


Ammeters and voltmeters in combination• An ammeter and a voltmeter may be used together to measure

voltage and power.

• Figure 26.16 below illustrates meters set to measure resistance.

• Refer to Example 26.10.



Ohmmeters and digital multimeters• An ohmmeter is designed specifically to measure resistance.

• Refer to Figure 26.17 and the figure below to see an ohmmeter wiring diagram and a photograph of a digital multimeter. The multimeter can measure current, voltage, or resistance over a wide range.


R-C circuitsSo far we have assumed that all emfsand resistances are constant, time-independent quantities. This assumption fails when we consider a circuit with a capacitor.Lets assume an ideal source with emf Εconnected to a resistor of resistance Rand capacitor with capacitance C.When the switch is open i is zero, as is the charge on the capacitor q.When the switch is initially closed the current is maximized, and q is zero. As the capacitor charges i decreases as q increased until q is maximized and i is zero.


Charging a CapacitorThe capacitor in figure 26.20 is initially uncharged. That means that vbcis zero at time t = 0. According to the loop rule the voltage across the resistor, vab, is equal to the emf of the source E. The initial current through the resistor: Io = vab/R = E/RAs the capacitor charges, its voltage vbc increases and the potential difference across the resistor, vab, decreases. The sum of these two voltages must always be equal to E. When the capacitor is fully charged, vbc = E.Why does the current decrease as the capacitor charges?Let q represent the charge on the capacitor and i the current in the circuit as some time t after the switch has been closed. The current will flow counter clockwise in figure 26.20. The instantaneous voltages for the resistor and capacitor are:

iRvab = Cqvbc =


Charging a CapacitorUsing these in Kirchhoff’s loop rule:

The potential drops by iR from a to b and by q/C from b to c. Solving the above equation for i we get:

When the switch is closed at t = 0 the capacitor is not yet charged, and the initial current is, as we have stated before, E/R. Without the capacitor, this would be the constant value of the current.As q increases the voltage drop across the capacitor increases. The drop equals E when the charge reaches its final value Qf. The current eventually reaches zero. When this happens Qf = EC.

RCq

REi −=

0=−−CqiRE


Charging a CapacitorThe current and the capacitor charge are given as functions of time to the right.Current jumps from 0 to Io = E/R at t = 0, then gradually approaches zero. The charge starts at zero and gradually approaches Qf.We can derive general expressions for the charge and current as functions of time. Due to our choice of the direction of current, i equals the rate at which positive charge arrives at the left-hand plate of the capacitor, so i = dq/dt. Putting this into equation 26.10 on page 998:

( )CEqRCRC

qRE

dtdq

−−=−=1


Charging a CapacitorThe previous equation rearranged:

And then integrate both sides. Lets change the integration variables to q’and t’ so we can use q and t as the upper limits:

When we carry out the integration we get:

Exponentiating both sides, taking the inverse logarithm, and solving for q:

The instantaneous current i is just the time derivative:

RCdt

CEqdq

−=−

∫∫′

−=−′′ tq

RCtd

CEqqd

00

RCt

CECEq

−=⎟⎠⎞

⎜⎝⎛−−ln

)1()1( RCt

RCt eQeCEq f

−−

−=−=

RCt

RCt eIe

RE

dtdqi o

−−

===


Time ConstantAfter a time equal to RC, the current in the R-C circuit has decreased to 1/e of its initial value and the charge has reached (1 – 1/e) of its final value. The product RC is called the time constant or relaxation time of the circuit, τ = RC. It is a measure of how quickly the capacitor charges or discharges.

When tau is small, the capacitor charges quickly. When it is large the capacitor takes a longer time to charge. See figure 26.21 and notice the factor RC.


Discharging a CapacitorSuppose that after the capacitor in figure 26.20 has acquired a charge Qfwe remove the battery and connect points a and c to an open switch, like in figure 26.22.When we close the switch, t = 0 and q = Qo, the capacitor discharges through the resistor and the charge again returns to zero.Let i and q represent the time varying current and charge at some instant after the switch is closed. We choose our current direction to be the same as we did before so we can use equation 26.10 for Kirchhoff’s loop rule, but this time E = 0. This gives:The initial current, when t =0, Io = -Qo/RC.To find q as a function of time we take the same steps as charging:

RCq

dtdqi −==

∫∫ ′−=′′ tq

tdRCq

qd

00

1RCt

Qq

o

−=lnRC

teQq o−

=

RCt

RCt eIe

RCQ

dtdqi o

o −−

=−==


Charging, discharging and timing …• Consider Figures 26.23 and 26.24.

• Follow Examples 26.12 and 26.13.


26.4 Summary and HomeworkWhen a capacitor is charged by a battery in series with a resistor, the current and capacitor charge are not constant. Thecharge approaches its final value asymptotically and the current approaches zero asymptotically. The charge and current in the circuit are given by Eqs. (26.12) and (26.13). After a time τ = RC the charge has approached within 1/e of its final value. This time is called the time constant or relaxation time of the circuit. When the capacitor discharges, the charge and current are given as functions of time by Eqs.(26.16) and (26.17). The time constant is the same for charging and discharging.Read 1002 to 1008:On page 1012: 35, 37, 41, 43, 45, 47


Power distribution systems—a home

• Potential, resistors, outlets, input from the power company … no wonder electricians are integral contractors in home construction!


Fuses, circuit breakers, and GFI• A fuse will melt and a breaker will

open the circuit if maximum current is reached. See Figure 26.26.

• GFI stops further current flow when a sudden drop in resistance indicates that someone has offered a new path to ground. I don’t know if it will save this worker we see in Figure 26.27 who didn’t use a grounded drill.


The wiring diagram for a typical kitchen• Consider Figure 26.28 below.



26.5 SummaryIn household wiring systems, the various electrical devices are connected in parallel across the power line, which consists of a pair of conductors, one “hot”and the other “neutral.” An additional “ground” wire is included for safety. The maximum permissible current in a circuit is determined by the size of the wires and the maximum temperature they can tolerate. Protection against excessive current and the resulting fire hazard is provided by fuses or circuit breakers.


Chapter Homework11, 19, 21, 29, 31, 33, 35, 37, 41, 43, 45, 47

Read pages 1019 to 1029

ch 26 notes - fcps 26 notes.pdf · title: microsoft powerpoint - ch 26 notes.ppt author:...

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