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Chapter 28

Direct Current Circuits

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Direct CurrentWhen the current in a circuit has aconstant magnitude and direction, the

current is called direct current Because the potential differencebetween the terminals of a battery isconstant, the battery produces directcurrentThe battery is known as a source of emf

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Electromotive ForceThe electromotive force (emf), , of abattery is the maximum possible voltagethat the battery can provide between itsterminals

The emf supplies energy, it does not apply

a forceThe battery will normally be the sourceof energy in the circuit

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Sample CircuitWe consider thewires to have no

resistanceThe positiveterminal of thebattery is at a higher

potential than thenegative terminalThere is also aninternal resistance in

the battery

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Internal Battery ResistanceIf the internalresistance is zero,

the terminal voltageequals the emf In a real battery,there is internalresistance, r The terminalvoltage, V = - I r

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Active Figure 28.2

(SLIDESHOW MODE ONLY)

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EMF, contThe emf is equivalent to the open-circuit voltage

This is the terminal voltage when nocurrent is in the circuitThis is the voltage labeled on the battery

The actual potential difference betweenthe terminals of the battery depends onthe current in the circuit

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Load ResistanceThe terminal voltage also equals thevoltage across the external resistance

This external resistor is called the load resistanceIn the previous circuit, the load resistance

is the external resistor In general, the load resistance could beany electrical device

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Power The total power output of the battery isP = I V = I

This power is delivered to the externalresistor ( I 2 R ) and to the internalresistor ( I 2 r )P = I = I 2 R + I 2 r

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Resistors in SeriesWhen two or more resistors are connectedend-to-end, they are said to be in series

For a series combination of resistors, thecurrents are the same in all the resistorsbecause the amount of charge that passesthrough one resistor must also pass through

the other resistors in the same time intervalThe potential difference will divide among theresistors such that the sum of the potentialdifferences across the resistors is equal to

the total potential difference across thecombination

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Resistors in Series, contPotentials add

V = I R 1 + I R 2

= I (R 1+R 2)Consequence of Conservation of Energy

The equivalent

resistance has thesame effect on thecircuit as the originalcombination of

resistors

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Equivalent Resistance

SeriesR eq = R 1 + R 2 + R 3 + The equivalent resistance of a seriescombination of resistors is the algebraicsum of the individual resistances and isalways greater than any individual

resistanceIf one device in the series circuit createsan open circuit, all devices areinoperative

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Series An Example

Two resistors are replaced with their equivalent resistance

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Active Figure 28.4


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Resistors in ParallelThe potential difference across each resistor is the same because each is connected

directly across the battery terminalsThe current, I , that enters a point must beequal to the total current leaving that point

I = I 1 + I 2The currents are generally not the sameConsequence of Conservation of Charge

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Parallel, Examples

Equivalent resistance replaces the two original

resistancesHousehold circuits are wired so that electrical devicesare connected in parallel

Circuit breakers may be used in series with other circuitelements for safety purposes

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Active Figure 28.6


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ParallelEquivalent Resistance

The inverse of theequivalent resistance of two or more resistorsconnected in parallel isthe algebraic sum of theinverses of theindividual resistance

The equivalent is alwaysless than the smallestresistor in the group

1 2 3

1 1 1 1

eqR R R R K

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Resistors in Parallel, FinalIn parallel, each device operatesindependently of the others so that if one is

switched off, the others remain onIn parallel, all of the devices operate on thesame voltageThe current takes all the paths

The lower resistance will have higher currentsEven very high resistances will have somecurrents

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Combinations of

ResistorsThe 8.0- and 4.0- resistorsare in series and can bereplaced with their equivalent,12.0 The 6.0- and 3.0- resistorsare in parallel and can bereplaced with their equivalent,

2.0 These equivalent resistancesare in series and can bereplaced with their equivalent

resistance, 14.0

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Kirchhoffs RulesThere are ways in which resistors canbe connected so that the circuits formedcannot be reduced to a singleequivalent resistor Two rules, called Kirchhoffs rules ,

can be used instead

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Statement of Kirchhoffs RulesJunction Rule

The sum of the currents entering any junction

must equal the sum of the currents leaving that junctionA statement of Conservation of Charge

Loop Rule

The sum of the potential differences across all theelements around any closed circuit loop must bezero

A statement of Conservation of Energy

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Mathematical Statement of

Kirchhoffs RulesJunction Rule: I

in = I out

Loop Rule:

closedloop

0V

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More about the Junction Rule I

1 = I 2 + I 3From Conservationof ChargeDiagram (b) shows amechanical analog

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More about the Loop RuleTraveling around the loop froma to bIn (a), the resistor is traversedin the direction of the current,the potential across the resistor is I R In (b), the resistor is traversedin the direction opposite of thecurrent, the potential acrossthe resistor is is + I R

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Loop Rule, finalIn (c), the source of emf is traversed in thedirection of the emf

(from to +), and thechange in the electricpotential is + In (d), the source of emf is traversed in thedirection opposite of theemf (from + to -), andthe change in theelectric potential is -

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Junction Equations from

Kirchhoffs RulesUse the junction rule as often asneeded, so long as each time you writean equation, you include in it a currentthat has not been used in a previous

junction rule equation

In general, the number of times the junction rule can be used is one fewer thanthe number of junction points in the circuit

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Loop Equations from

Kirchhoffs RulesThe loop rule can be used as often asneeded so long as a new circuitelement (resistor or battery) or a newcurrent appears in each new equationYou need as many independent

equations as you have unknowns

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Kirchhoffs Rules Equations,

finalIn order to solve a particular circuitproblem, the number of independent

equations you need to obtain from the tworules equals the number of unknowncurrents

Any capacitor acts as an open branch in acircuitThe current in the branch containing thecapacitor is zero under steady-state conditions

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Problem-Solving Hints

Kirchhoffs RulesDraw the circuit diagram and assign labelsand symbols to all known and unknown

quantities. Assign directions to the currents.The direction is arbitrary, but you must adhere tothe assigned directions when applying Kirchhoffsrules

Apply the junction rule to any junction in thecircuit that provides new relationships amongthe various currents

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Problem-Solving Hints, contApply the loop rule to as many loops asare needed to solve for the unknowns

To apply the loop rule, you must correctlyidentify the potential difference as youcross various elements

Solve the equations simultaneously for the unknown quantities

If a current turns out to be negative, themagnitude will be correct and the directionis opposite to that which you assigned

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RC CircuitsA direct current circuit may contain capacitorsand resistors, the current will vary with time

When the circuit is completed, the capacitor starts to chargeThe capacitor continues to charge until itreaches its maximum charge ( Q = C )Once the capacitor is fully charged, thecurrent in the circuit is zero

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Charging an RC CircuitAs the plates are being charged, the potentialdifference across the capacitor increases

At the instant the switch is closed, the chargeon the capacitor is zeroOnce the maximum charge is reached, thecurrent in the circuit is zero

The potential difference across the capacitor matches that supplied by the battery

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Active Figure 28.19


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Charging a Capacitor in an

RC CircuitThe charge on thecapacitor varies with

timeq = C (1 e -t /RC ) =Q (1 e -t /RC ) is the time constant

= RC The current can befound

I( ) t RC

t eR

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Time Constant, ChargingThe time constant represents the timerequired for the charge to increase from

zero to 63.2% of its maximum has units of timeThe energy stored in the chargedcapacitor is Q = C 2

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Active Figure 28.21


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Discharging a Capacitor in an

RC CircuitWhen a chargedcapacitor is placed

in the circuit, it canbe dischargedq = Qe -t /RC

The chargedecreasesexponentially

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Discharging Capacitor At t = = RC , the charge decreases to 0.368Q max

In other words, in one time constant, the capacitor loses 63.2% of its initial charge

The current can be found

Both charge and current decay exponentiallyat a rate characterized by = RC

I t RC dq Qt edt RC

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Galvanometer A galvanometer isthe main componentin analog meters for measuring currentand voltageDigital meters are incommon use

Digital metersoperate under different principles

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Galvanometer, contA galvanometer consists of a coil of wire mounted so that it is free to rotate

on a pivot in a magnetic fieldThe field is provided by permanentmagnets

A torque acts on a current in thepresence of a magnetic field

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Active Figure 28.27


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Galvanometer, finalThe torque is proportional to the current

The larger the current, the greater the torque

The greater the torque, the larger the rotation of the coil before the spring resists enough to stopthe rotation

The deflection of a needle attached to the coil

is proportional to the currentOnce calibrated, it can be used to measurecurrents or voltages

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Ammeter An ammeter is a device that measurescurrent

The ammeter must be connected inseries with the elements beingmeasured

The current must pass directly through theammeter

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Ammeter in a CircuitThe ammeter isconnected in series withthe elements in whichthe current is to bemeasuredIdeally, the ammeter

should have zeroresistance so the currentbeing measured is notaltered

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Ammeter from Galvanometer The galvanometer typically has a

resistance of 60

To minimize theresistance, a shunt resistance , R p , isplaced in parallelwith thegalvanometer

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Active Figure 28.29


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Ammeter, finalThe value of the shunt resistor must bemuch less than the resistance of the

galvanometer Remember, the equivalent resistance of resistorsin parallel will be less than the smallest resistance

Most of the current will go through theshunt resistance, this is necessary sincethe full scale deflection of thegalvanometer is on the order of 1 mA

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Voltmeter A voltmeter is a device that measurespotential difference

The voltmeter must be connected inparallel with the elements beingmeasured

The voltage is the same in parallel

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Voltmeter in a CircuitThe voltmeter isconnected in parallelwith the element in whichthe potential difference isto be measured

Polarity must be observed

Ideally, the voltmeter

should have infiniteresistance so that nocurrent would passthrough it

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Voltmeter from Galvanometer The galvanometer typically has a

resistance of 60

To maximize theresistance, another resistor, R s , isplaced in series withthe galvanometer

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Voltmeter, finalThe value of the added resistor must bemuch greater than the resistance of the

galvanometer Remember, the equivalent resistance of resistorsin series will be greater than the largest resistance

Most of the current will go through theelement being measured, and thegalvanometer will not alter the voltagebeing measured

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Household WiringThe utility company distributes electricpower to individual homes by a pair of

wiresEach house is connected in parallel withthese wiresOne wire is the live wire and the other wire is the neutral wire connected toground

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Household Wiring, contThe potential of theneutral wire is takento be zero

Actually, the currentand voltage arealternating

The potentialdifference betweenthe live and neutralwires is about 120 V

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Household Wiring, finalA meter is connected in series with the livewire entering the house

This records the households consumption of electricity

After the meter, the wire splits so that multipleparallel circuits can be distributed throughoutthe houseEach circuit has its own circuit breaker For those applications requiring 240 V, thereis a third wire maintained at 120 V below theneutral wire

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Short CircuitA short circuit occurs when almost zeroresistance exists between two points atdifferent potentialsThis results in a very large currentIn a household circuit, a circuit breaker willopen the circuit in the case of an accidental

short circuitThis prevents any damageA person in contact with ground can beelectrocuted by touching the live wire

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Electrical SafetyElectric shock can result in fatal burnsElectric shock can cause the muscles of vital

organs (such as the heart) to malfunctionThe degree of damage depends on:

the magnitude of the currentthe length of time it actsthe part of the body touching the live wirethe part of the body in which the current exists

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Effects of Various Currents5 mA or less

can cause a sensation of shockgenerally little or no damage

10 mAmuscles contractmay be unable to let go of a live wire

100 mAif passing through the body for 1 second or less,can be fatalparalyzes the respiratory muscles

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More Effects

In some cases, currents of 1 A canproduce serious burns

Sometimes these can be fatal burnsNo contact with live wires is consideredsafe whenever the voltage is greater

than 24 V

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Ground WireElectrical equipmentmanufacturers useelectrical cords thathave a third wire,called a groundThis safety groundnormally carries nocurrent and is bothgrounded andconnected to theappliance

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Ground Wire, cont

If the live wire is accidentally shorted tothe casing, most of the current takes

the low-resistance path through theappliance to the groundIf it was not properly grounded, anyone

in contact with the appliance could beshocked because the body produces alow-resistance path to ground

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Ground-Fault Interrupters (GFI)Special power outletsUsed in hazardous areas

Designed to protect people fromelectrical shockSenses currents (of about 5 mA or

greater) leaking to groundShuts off the current when above thislevel

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