3Sources of emfThe source that maintains the current in a closed circuit is called a source of emfAny devices that increase the potential energy of charges circulating in circuits are sources of emfExamples include batteries and generatorsSI units are VoltsThe emf is the work done per unit charge
4EMF and Terminal Voltage This resistance behaves as though it were in series with the emf.
5emf and Internal Resistance A real battery has some internal resistanceTherefore, the terminal voltage is not equal to the emf
6More About Internal Resistance The schematic shows the internal resistance, rThe terminal voltage is ΔV = Vb-VaΔV = ε – IrFor the entire circuit, ε = IR + Ir
7Internal Resistance and emf, cont ε is equal to the terminal voltage when the current is zeroAlso called the open-circuit voltageR is called the load resistanceThe current depends on both the resistance external to the battery and the internal resistance
8Internal Resistance and emf, final When R >> r, r can be ignoredGenerally assumed in problemsPower relationshipI e = I2 R + I2 rWhen R >> r, most of the power delivered by the battery is transferred to the load resistor
9Resistors in SeriesWhen two or more resistors are connected end-to-end, they are said to be in seriesThe current is the same in all resistors because any charge that flows through one resistor flows through the otherThe sum of the potential differences across the resistors is equal to the total potential difference across the combination
10A series connection has a single path from the battery, through each circuit element in turn, then back to the battery.
11Resistors in Series, cont Potentials addΔV = IR1 + IR2 = I (R1+R2)Consequence of Conservation of EnergyThe equivalent resistance has the effect on the circuit as the original combination of resistors
12Equivalent Resistance – Series Req = R1 + R2 + R3 + …The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistors
13Equivalent Resistance – Series: An Example Four resistors are replaced with their equivalent resistance
14The current through each resistor is the same. The voltage depends on the resistance.The sum of the voltage drops across the resistors equals the battery voltage.
151) 12 V2) zero3) 3 V4) 4 V5) you need to know the actual value of RAssume that the voltage of the battery is 9 V and that the three resistors are identical. What is the potential difference across each resistor?9 V
16Assume that the voltage of the battery is 9 V and that the three resistors are identical. What is the potential difference across each resistor?1) 12 V2) zero3) 3 V4) 4 V5) you need to know the actual value of RSince the resistors are all equal, the voltage will drop evenly across the 3 resistors, with 1/3 of 9 V across each one. So we get a 3 V drop across each.9 V
17In the circuit below, what is the voltage across R1? 2) zero3) 6 V4) 8 V5) 4 VIn the circuit below, what is the voltage across R1?12 VR1= 4 WR2= 2 W
18In the circuit below, what is the voltage across R1? 2) zero3) 6 V4) 8 V5) 4 VThe voltage drop across R1 has to be twice as big as the drop across R2. This means that V1 = 8 V and V2 = 4 V. Or else you could find the current I = V/R = (12 V)/(6 W) = 2 A, then use Ohm’s Law to get voltages.12 VR1= 4 WR2= 2 W
19Resistors 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 be equal to the total current leaving that pointI = I1 + I2The currents are generally not the sameConsequence of Conservation of Charge
20A parallel connection splits the current; the voltage across each resistor is the same:
21Equivalent Resistance – Parallel, Example Equivalent resistance replaces the two original resistancesHousehold circuits are wired so the electrical devices are connected in parallelCircuit breakers may be used in series with other circuit elements for safety purposes
22Equivalent Resistance – Parallel The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistanceThe equivalent is always less than the smallest resistor in the group
23In the circuit to the right, what is the current through R1? 2) zero3) 5 A4) 2 A5) 7 AIn the circuit to the right, what is the current through R1?10 VR1= 5 WR2= 2 W
24In the circuit below, what is the current through R1? 2) zero3) 5 A4) 2 A5) 7 A10 VR1= 5 WR2= 2 WThe voltage is the same (10 V) across each resistor because they are in parallel. Thus, we can use Ohm’s Law, V1 = I1 R1 to find the current I1 = 2 A.
251) increases2) remains the same3) decreases4) drops to zeroPoints P and Q are connected to a battery of fixed voltage. As more resistors R are added to the parallel circuit, what happens to the total current in the circuit?
26Points P and Q are connected to a battery of fixed voltage Points P and Q are connected to a battery of fixed voltage. As more resistors R are added to the parallel circuit, what happens to the total current in the circuit?1) increases2) remains the same3) decreases4) drops to zeroAs we add parallel resistors, the overall resistance of the circuit drops. Since V = IR, and V is held constant by the battery, when resistance decreases, the current must increase.
271) all the current continues to flow through the bulb 2) half the current flows through the wire, the other half continues through the bulb3) all the current flows through the wire4) none of the aboveCurrent flows through a lightbulb. If a wire is now connected across the bulb, what happens?
281) all the current continues to flow through the bulb 2) half the current flows through the wire, the other half continues through the bulb3) all the current flows through the wire4) none of the aboveCurrent flows through a lightbulb. If a wire is now connected across the bulb, what happens?The current divides based on the ratio of the resistances. If one of the resistances is zero, then ALL of the current will flow through that path.
291) glow brighter than before 2) glow just the same as before3) glow dimmer than before4) go out completely5) explodeTwo lightbulbs A and B are connected in series to a constant voltage source. When a wire is connected across B, bulb A will:
301) glow brighter than before 2) glow just the same as before3) glow dimmer than before4) go out completely5) explodeTwo lightbulbs A and B are connected in series to a constant voltage source. When a wire is connected across B, bulb A will:Since bulb B is bypassed by the wire, the total resistance of the circuit decreases. This means that the current through bulb A increases.
311) circuit 12) circuit 23) both the same4) it depends on RThe lightbulbs in the circuit below are identical with the same resistance R. Which circuit produces more light? (brightness power)
32The lightbulbs in the circuit below are identical with the same resistance R. Which circuit produces more light? (brightness power)1) circuit 12) circuit 23) both the same4) it depends on RIn #1, the bulbs are in parallel, lowering the total resistance of the circuit. Thus, circuit #1 will draw a higher current, which leads to more light, because P = I V.
331) twice as much2) the same3) 1/2 as much4) 1/4 as much5) 4 times as muchThe three lightbulbs in the circuit all have the same resistance of 1 W . By how much is the brightness of bulb B greater or smaller than the brightness of bulb A? (brightness power)10 VABC
341) twice as much2) the same3) 1/2 as much4) 1/4 as much5) 4 times as muchThe three light bulbs in the circuit all have the same resistance of 1 W . By how much is the brightness of bulb B greater or smaller than the brightness of bulb A? (brightness power)10 VABCWe can use P = V2/R to compare the power:PA = (VA)2/RA = (10 V) 2/1 W = 100 WPB = (VB)2/RB = (5 V) 2/1 W = 25 W
351) increase2) decrease3) stay the sameWhat happens to the voltage across the resistor R1 when the switch is closed? The voltage will:VR1R3R2S
361) increase2) decrease3) stay the sameWhat happens to the voltage across the resistor R1 when the switch is closed? The voltage will:VR1R3R2SWith the switch closed, the addition of R2 to R3 decreases the equivalent resistance, so the current from the battery increases. This will cause an increase in the voltage across R1 .
371) increases2) decreases3) stays the sameWhat happens to the voltage across the resistor R4 when the switch is closed?VR1R3R4R2S[CORRECT 5 ANSWER]
38What happens to the voltage across the resistor R4 when the switch is closed? 1) increases2) decreases3) stays the sameVR1R3R4R2SABCWe just saw that closing the switch causes an increase in the voltage across R1 (which is VAB). The voltage of the battery is constant, so if VAB increases, then VBC must decrease!
391) R12) both R1 and R2 equally3) R3 and R44) R55) all the sameWhich resistor has the greatest current going through it? Assume that all the resistors are equal.VR1R2R3R5R4
40Which resistor has the greatest current going through it Which resistor has the greatest current going through it? Assume that all the resistors are equal.1) R12) both R1 and R2 equally3) R3 and R44) R55) all the sameThe same current must flow through left and right combinations of resistors. On the LEFT, the current splits equally, so I1 = I2. On the RIGHT, more current will go through R5 than R3 + R4 since the branch containing R5 has less resistance.VR1R2R3R5R4
41Problem-Solving Strategy, 1 Combine all resistors in seriesThey carry the same currentThe potential differences across them are not the sameThe resistors add directly to give the equivalent resistance of the series combination: Req = R1 + R2 + …
42Problem-Solving Strategy, 2 Combine all resistors in parallelThe potential differences across them are the sameThe currents through them are not the sameThe equivalent resistance of a parallel combination is found through reciprocal addition:
43Problem-Solving Strategy, 3 A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistorReplace any resistors in series or in parallel using steps 1 or 2.Sketch the new circuit after these changes have been madeContinue to replace any series or parallel combinationsContinue until one equivalent resistance is found
44Problem-Solving Strategy, 4 If the current in or the potential difference across a resistor in the complicated circuit is to be identified, start with the final circuit found in step 3 and gradually work back through the circuitsUse ΔV = I R and the procedures in steps 1 and 2
46An analogy using water may be helpful in visualizing parallel circuits:
47Example 1A 4.0-Ω resistor, an 8.0-Ω resistor, and a 12-Ω resistor are connected in series with a 24-V battery. What are (a) the equivalent resistance and (b) the current in each resistor? (c) Repeat for the case in which all three resistors are connected in parallel across the battery.
48Example 2A 9.0-Ω resistor and a 6.0-Ω resistor are connected in series with a power supply. (a) The voltage drop across the 6.0-Ω resistor is measured to be 12 V. Find the voltage output of the power supply. (b) The two resistors are connected in parallel across a power supply, and the current through the 9.0-Ω resistor is found to be 0.25 A. Find the voltage setting of the power supply.
49Example 3(a) Find the equivalent resistance of the circuit in Figure P18.8. (b) If the total power supplied to the circuit is 4.00 W, find the emf of the battery.
50Example 4Three 100-Ω resistors are connected as shown in Figure P The maximum power that can safely be delivered to any one resistor is 25.0 W. (a) What is the maximum voltage that can be applied to the terminals a and b? (b) For the voltage determined in part (a), what is the power delivered to each resistor? What is the total power delivered?
51Practice 1What is the equivalent resistance of the combination of resistors between points a and b in Figure P18.7? Note that one end of the vertical resistor is left free.
52Practice 2(a) You need a 45-Ω resistor, but the stockroom has only 20-Ω and 50-Ω resistors. How can the desired resistance be achieved under these circumstances? (b) What can you do if you need a 35-Ω resistor?
53Some circuits cannot be broken down into series and parallel connections.
54Gustav Kirchhoff 1824 – 1887 Invented spectroscopy with Robert Bunsen Formulated rules about radiation
55Kirchhoff’s RulesThere are ways in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistorTwo rules, called Kirchhoff’s Rules can be used instead
56Statement of Kirchhoff’s Rules Junction RuleThe sum of the currents entering any junction must equal the sum of the currents leaving that junctionA statement of Conservation of ChargeLoop RuleThe sum of the potential differences across all the elements around any closed circuit loop must be zeroA statement of Conservation of Energy
57More About the Junction Rule I1 = I2 + I3From Conservation of ChargeDiagram b shows a mechanical analog
58What is the current in branch P? [CORRECT 5 ANSWER]
59What is the current in branch P? junction6 AThe current entering the junction in red is 8 A, so the current leaving must also be 8 A. One exiting branch has 2 A, so the other branch (at P) must have 6 A.S
60Setting Up Kirchhoff’s Rules Assign symbols and directions to the currents in all branches of the circuitIf a direction is chosen incorrectly, the resulting answer will be negative, but the magnitude will be correctWhen applying the loop rule, choose a direction for transversing the loopRecord voltage drops and rises as they occur
61More About the Loop Rule Traveling around the loop from a to bIn a, the resistor is transversed in the direction of the current, the potential across the resistor is –IRIn b, the resistor is transversed in the direction opposite of the current, the potential across the resistor is +IR
62Loop Rule, finalIn c, the source of emf is transversed in the direction of the emf (from – to +), the change in the electric potential is +εIn d, the source of emf is transversed in the direction opposite of the emf (from + to -), the change in the electric potential is -ε
63Junction Equations from Kirchhoff’s Rules Use the junction rule as often as needed, so long as, each time you write an equation, you include in it a current that has not been used in a previous junction rule equationIn general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit
64Loop Equations from Kirchhoff’s Rules The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equationYou need as many independent equations as you have unknowns
65EMFs in series in the same direction: total voltage is the sum of the separate voltages
66EMFs in series, opposite direction: total voltage is the difference, but the lower-voltage battery is charged.
67EMFs in parallel only make sense if the voltages are the same; this arrangement can produce more current than a single emf.
68Problem-Solving Strategy – Kirchhoff’s Rules Draw the circuit diagram and assign labels and symbols to all known and unknown quantitiesAssign directions to the currents.Apply the junction rule to any junction in the circuitApply the loop rule to as many loops as are needed to solve for the unknownsSolve the equations simultaneously for the unknown quantitiesCheck your answers
691) both bulbs go out2) intensity of both bulbs increases3) intensity of both bulbs decreases4) A gets brighter and B gets dimmer5) nothing changesThe lightbulbs in the circuit are identical. When the switch is closed, what happens?[CORRECT 5 ANSWER]
701) both bulbs go out2) intensity of both bulbs increases3) intensity of both bulbs decreases4) A gets brighter and B gets dimmer5) nothing changesThe lightbulbs in the circuit are identical. When the switch is closed, what happens?When the switch is open, the point between the bulbs is at 12 V. But so is the point between the batteries. If there is no potential difference, then no current will flow once the switch is closed!! Thus, nothing changes.
71Which of the equations is valid for the circuit below? 1) 2 – I1 – 2I2 = 02) 2 – 2I1 – 2I2 – 4I3 = 03) 2 – I1 – 4 – 2I2 = 04) I3 – 4 – 2I = 05) 2 – I1 – 3I3 – 6 = 0Which of the equations is valid for the circuit below?2 V2 6 V4 V3 1 I1I3I2
72Which of the equations is valid for the circuit below? 1) 2 – I1 – 2I2 = 02) 2 – 2I1 – 2I2 – 4I3 = 03) 2 – I1 – 4 – 2I2 = 04) I3 – 4 – 2I = 05) 2 – I1 – 3I3 – 6 = 0Eqn. 3 is valid for the left loop: The left battery gives +2V, then there is a drop through a 1W resistor with current I1 flowing. Then we go through the middle battery (but from + to – !), which gives –4V. Finally, there is a drop through a 2W resistor with current I2.2 V2 6 V4 V3 1 I1I3I2
73Example 5The ammeter shown in Figure P18.16 reads 2.00 A. Find I1, I2, and ε.
74Example 6Determine the potential difference Є1 for the circuit in Figure P18.18.
75Example 7In the circuit of Figure P18.20, the current I1 is 3.0 A while the values of ε and R are unknown. What are the currents I2 and I3?
76Example 8Four resistors are connected to a battery with a terminal voltage of 12 V, as shown in Figure P Determine the power delivered to the 50-Ω resistor.
77Practice 3Calculate each of the unknown currents I1, I2, and I3 for the circuit of Figure P18.25.
78RC CircuitsA direct current circuit may contain capacitors and resistors, the current will vary with timeWhen the circuit is completed, the capacitor starts to chargeThe capacitor continues to charge until it reaches its maximum charge (Q = Cε)Once the capacitor is fully charged, the current in the circuit is zero
79When the switch is closed, the capacitor will begin to charge.
80The voltage across the capacitor increases with time:
81The charge follows a similar curve: This curve has a characteristic time constant:
82If an isolated charged capacitor is connected across a resistor, it discharges:
83Notes on Time ConstantIn a circuit with a large time constant, the capacitor charges very slowlyThe capacitor charges very quickly if there is a small time constantAfter t = 10 t, the capacitor is over 99.99% charged
84Example 9An uncharged capacitor and a resistor are connected in series to a source of emf. If ε = 9.00 V, C = 20.0 μF, and R = 100 Ω, find (a) the time constant of the circuit, (b) the maximum charge on the capacitor, and (c) the charge on the capacitor after one time constant.
85Example 10Consider a series RC circuit for which R = 1.0 MΩ, C = 5.0 μF, and ε = 30 V. Find the charge on the capacitor 10 s after the switch is closed.
86Practice 4A series RC circuit has a time constant of s. The battery has an emf of 48.0 V, and the maximum current in the circuit is mA. What are (a) the value of the capacitance and (b) the charge stored in the capacitor 1.92 s after the switch is closed?
87Household CircuitsThe utility company distributes electric power to individual houses with a pair of wiresElectrical devices in the house are connected in parallel with those wiresThe potential difference between the wires is about 120V
88Household Circuits, cont. A meter and a circuit breaker are connected in series with the wire entering the houseWires and circuit breakers are selected to meet the demands of the circuitIf the current exceeds the rating of the circuit breaker, the breaker acts as a switch and opens the circuitHousehold circuits actually use alternating current and voltage
89A person receiving a shock has become part of a complete circuit.
90Electrical Safety Electric shock can result in fatal burns Electric shock can cause the muscles of vital organs (such as the heart) to malfunctionThe degree of damage depends onthe magnitude of the currentthe length of time it actsthe part of the body through which it passes
91Effects of Various Currents 5 mA or lessCan cause a sensation of shockGenerally little or no damage10 mAHand muscles contractMay be unable to let go a of live wire100 mAIf passes through the body for just a few seconds, can be fatal
92Ground WireElectrical equipment manufacturers use electrical cords that have a third wire, called a case groundPrevents shocks
93Ground Fault Interrupts (GFI) Special power outletsUsed in hazardous areasDesigned to protect people from electrical shockSenses currents (of about 5 mA or greater) leaking to groundShuts off the current when above this level
94Example 11An electric heater is rated at W, a toaster at W, and an electric grill at 1 500 W. The three appliances are connected in parallel to a common 120-V circuit. (a) How much current does each appliance draw? (b) Is a 30.0-A circuit breaker sufficient in this situation? Explain.
95Example 12A lamp (R = 150 Ω), an electric heater (R = 25 Ω), and a fan (R = 50 Ω) are connected in parallel across a 120-V line. (a) What total current is supplied to the circuit? (b) What is the voltage across the fan? (c) What is the current in the lamp? (d) What power is expended in the heater?