Chapter 25 : Electric circuits
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1 Chapter 25 : Electric circuits Voltage and current Series and parallel circuits Resistors and capacitors Kirchoff s rules for analysing circuits
2 Electric circuits Closed loop of electrical components around which current can flow, driven by a potential difference Current (in Amperes A) is the rate of flow of charge Potential difference (in volts V) is the work done on charge
3 Electric circuits May be represented by a circuit diagram. Here is a simple case: R is the resistance (in Ohms Ω) to current flow
4 Electric circuits Same principles apply in more complicated cases!
5 Electric circuits How do we deal with a more complicated case? What is the current flowing from the battery?
6 Electric circuits When components are connected in series, the same electric current flows through them II Charge conservation : current cannot disappear! II
7 Electric circuits When components are connected in parallel, the same potential difference drops across them VV VV VV Points connected by a wire are at the same voltage!
8 Electric circuits When there is a junction in the circuit, the inward and outward currents to the junction are the same II 2 II 1 II 1 = II 2 + II 3 II 3 Charge conservation : current cannot disappear!
9 Consider the currents I 1, I 2 and I 3 as indicated on the circuit diagram. If I 1 = 2.5 A and I 2 = 4 A, what is the value of I 3? I 2 I 1 I A A A 4. 0 A 5. The situation is not possible 0% 0% 0% 0% 0%
10 Consider the currents I 1, I 2 and I 3 as indicated on the circuit diagram. If I 1 = 2.5 A and I 2 = 4 A, what is the value of I 3? I 2 I 1 I 3 current in = current out II 1 = II 2 + II 3 II 3 = II 1 II 2 II 3 = = 1.5 AA (Negative sign means opposite direction to arrow.)
11 A 9.0 V battery is connected to a 3 Ω resistor. Which is the incorrect statement about potential differences (voltages)? a d b c 1. V b V a = 9.0 V 2. V b V c = 0 V 3. V c V d = 9.0 V 4. V d V a = 9.0 V 0% 0% 0% 0%
12 Resistors in circuits Resistors are the basic components of a circuit that determine current flow : Ohm s law I = V/R
13 Resistors in series/parallel If two resistors are connected in series, what is the total resistance? RR 1 RR 2 same current II II Potential drop VV 1 = II RR 1 Potential drop VV 2 = II RR 2 Total potential drop VV = VV 1 + VV 2 = II RR 1 + II RR 2 = II (RR 1 + RR 2 )
14 Resistors in series/parallel If two resistors are connected in series, what is the total resistance? RR tttttttttt Potential drop VV = II RR tttttttttt = II (RR 1 + RR 2 ) Total resistance increases in series! II RR tttttttttt = RR 1 + RR 2
15 Resistors in series/parallel Total resistance increases in series!
16 Resistors in series/parallel If two resistors are connected in parallel, what is the total resistance? RR 1 RR 2
17 Resistors in series/parallel If two resistors are connected in parallel, what is the total resistance? II 1 RR 1 II RR 2 II II 2 VV Total current II = II 1 + II 2 = VV RR 1 + VV RR 2 = VV 1 RR RR 2
18 Resistors in series/parallel If two resistors are connected in parallel, what is the total resistance? RR tttttttttt II Current I = VV RR tttttttttt = VV 1 RR RR 2 1 RR tttttttttt = 1 RR RR 2 Total resistance decreases in parallel!
19 Resistors in series/parallel Total resistance decreases in parallel!
20 Resistors in series/parallel What s the current flowing? (1) Combine these 2 resistors in parallel: 1 = 1 RR pppppppp RR pppppppp = Ω (2) Combine all the resistors in series: RR tttttttttt = = Ω (3) Current II = VV RR tttttttttt = = 0.17 AA
21 If an additional resistor, R 2, is added in series to the circuit, what happens to the power dissipated by R 1? R 1 1. Increases 2. Decreases 3. Stays the same 0% 0% 0% VV = IIII PP = VVVV = II 2 RR = VV2 RR
22 If an additional resistor, R 3, is added in parallel to the circuit, what happens to the total current, I? V I 1. Increases 2. Decreases 3. Stays the same 4. Depends on R values 0% 0% 0% 0% Parallel resistors: reciprocal effective resistance is sum of reciprocal resistances 1 RR eeeeee = 1 RR RR RR 3 +
23 Series vs. Parallel CURRENT VOLTAGE RESISTANCE Same current through all series elements Voltages add to total circuit voltage Adding resistance increases total R Current splits up through parallel branches Same voltage across all parallel branches Adding resistance reduces total R String of Christmas lights connected in series Power outlets in house connected in parallel
24 Voltage divider Consider a circuit with several resistors in series with a battery. Current in circuit: II = VV RR tttttttttt = VV RR 1 + RR 2 + RR 3 The potential difference across one of the resistors (e.g. R 1 ) VV 1 = IIRR 1 RR 1 = VV RR 1 + RR 2 + RR 3 The fraction of the total voltage that appears across a resistor in series is the ratio of the given resistance to the total resistance.
25 What must be the resistance R 1 so that V 1 = 2.0 V? Ω Ω Ω 12 V R Ω VV Ω 0% 0% 0% 0%
26 Capacitors A capacitor is a device in a circuit which can be used to store charge A capacitor consists of two charged plates It s charged by connecting it to a battery Electric field E
27 Capacitors A capacitor is a device in a circuit which can be used to store charge Example : store and release energy
28 Capacitors The capacitance C measures the amount of charge Q which can be stored for given potential difference V +Q -Q CC = QQ VV QQ = CC VV (Value of C depends on geometry ) V Unit of capacitance is Farads [F]
29 Resistor-capacitor circuit Consider the following circuit with a resistor and a capacitor in series V What happens when we connect the circuit?
30 Resistor-capacitor circuit V When the switch is connected, the battery charges up the capacitor Move the switch to point a Initial current flow I=V/R Charge Q flows from battery onto the capacitor Potential across the capacitor V C =Q/C increases Potential across the resistor V R decreases Current decreases to zero
31 Resistor-capacitor circuit When the switch is connected, the battery charges up the capacitor V
32 Resistor-capacitor circuit When the battery is disconnected, the capacitor pushes charge around the circuit Move the switch to point b Initial current flow I=V C /R V Charge flows from one plate of capacitor to other Potential across the capacitor V C =Q/C decreases Current decreases to zero
33 Resistor-capacitor circuit When the battery is disconnected, the capacitor pushes charge around the circuit V
34 Capacitors in series/parallel If two capacitors are connected in series, what is the total capacitance? Same charge must be on every plate! +QQ -QQ +QQ -QQ CC 1 CC 2 QQ = CC VV Potential drop VV 1 = QQ/CC 1 Potential drop VV 2 = QQ/CC 2 Total potential drop VV = VV 1 + VV 2 = QQ CC 1 + QQ CC 2 = QQ 1 CC CC 2
35 Capacitors in series/parallel If two capacitors are connected in series, what is the total capacitance? +QQ -QQ QQ = CC VV CC tttttttttt Potential drop VV = QQ CC tttttttttt = QQ 1 CC CC 2 1 CC tttttttttt = 1 CC CC 2 Total capacitance decreases in series!
36 Capacitors in series/parallel If two capacitors are connected in parallel, what is the total capacitance? +QQ 1 QQ 1 CC 1 +QQ 2 QQ 2 QQ = CC VV VV CC 2 Total charge QQ = QQ 1 + QQ 2 = CC 1 VV + CC 2 VV = CC 1 + CC 2 VV
37 Capacitors in series/parallel If two capacitors are connected in parallel, what is the total capacitance? +QQ -QQ QQ = CC VV CC tttttttttt Total charge QQ = CC tttttttttt VV = CC 1 + CC 2 VV CC tttttttttt = CC 1 + CC 2 Total capacitance increases in parallel!
38 Two 5.0 F capacitors are in series with each other and a 1.0 V battery. Calculate the charge on each capacitor (Q) and the total charge drawn from the battery (Q total ). 1 V 5 F 5 F 1. Q = 5.0 C, Q total = 5.0 C 2. Q = 0.25 C, Q total = 0.50 C 3. Q = 2.5 C, Q total = 2.5 C 4. Q = 2.5 C, Q total = 5.0 C 0% 0% 0% 0% QQ = CCCC 1 CC tttttttttt = 1 CC CC CC 3 +
39 Two 5.0 F capacitors are in series with each other and a 1.0 V battery. Calculate the charge on each capacitor (Q) and the total charge drawn from the battery (Q total ). 1 V 5 F 5 F Potential difference across each capacitor = 0.5 V Charge on each capacitor QQ = CCCC = = 2.5 CC 1 = = 1 CC tttttttttt CC 1 CC 2 CC tttttttttt = 2 5 CC tttttttttt = 2.5 FF QQ tttttttttt = CC tttttttttt VV = = 2.5 CC
40 Two 5.0 F capacitors are in parallel with each other and a 1.0 V battery. Calculate the charge on each capacitor (Q) and the total charge drawn from the battery (Q total ). 1 V 5 F 5 F 1. Q = 5.0 C, Q total = 5.0 C 2. Q = 0.2 C, Q total = 0.4 C 3. Q = 5.0 C, Q total = 10 C 4. Q = 2.5 C, Q total = 2.5 C 0% 0% 0% 0% QQ = CCCC CC tttttttttt = CC 1 + CC 2 + CC 3 +
41 Two 5.0 F capacitors are in parallel with each other and a 1.0 V battery. Calculate the charge on each capacitor (Q) and the total charge drawn from the battery (Q total ). 1 V 5 F 5 F Potential difference across each capacitor = 1 V Charge on each capacitor QQ = CCCC = 5 1 = 5 CC QQ tttttttttt = 10 CC
42 Resistors vs. Capacitors RR 1 RR 2 RR tttttttttt = RR 1 + RR 2 CC 1 CC 2 1 CC tttttttttt = 1 CC CC 2
43 Resistors vs. Capacitors RR 1 1 RR 2 RR tttttttttt = 1 RR RR 2 CC 1 CC tttttttttt = CC 1 + CC 2 CC 2
44 Kirchoff s rules Sometimes we might need to analyse more complicated circuits, for example RR 1 = 4 Ω VV 1 = 9 VV VV 2 = 6 VV RR 2 = 1 Ω RR 3 = 2 Ω Q) What are the currents flowing in the 3 resistors? Kirchoff s rules give us a systematic method
45 Kirchoff s rules What are the currents flowing in the 3 resistors? 9 VV 4 Ω 1 Ω II 1 II 2 Kirchoff s junction rule : the sum of currents at any junction is zero 6 VV 2 Ω II 3
46 Kirchoff s rules The sum of currents at any junction is zero Watch out for directions : into a junction is positive, out of a junction is negative II 2 II 1 II 1 II 2 II 3 = 0 II 3 II 1 = II 2 + II 3
47 Kirchoff s rules What are the currents flowing in the 3 resistors? 9 VV 4 Ω 1 Ω II 1 II 2 Kirchoff s junction rule : the sum of currents at any junction is zero 6 VV 2 Ω II 3 II 2 II 1 II 3 II 1 + II 2 + II 3 = 0
48 Kirchoff s rules What are the currents flowing in the 3 resistors? 9 VV 4 Ω 1 Ω II 1 II 2 Kirchoff s loop rule : the sum of voltage changes around a closed loop is zero 4 Ω II 1 6 VV 2 Ω II 3 9 VV 1 Ω II 2
49 Kirchoff s rules Sum of voltage changes around a closed loop is zero Consider a unit charge (Q=1 Coulomb) going around this loop It gains energy from the battery (voltage change +V) It loses energy in the resistor (voltage change - I R) Conservation of energy : V - I R = 0 (or as we know, V = I R)
50 Kirchoff s rules What are the currents flowing in the 3 resistors? 9 VV 4 Ω 1 Ω II 1 II 2 Kirchoff s loop rule : the sum of voltage changes around a closed loop is zero 4 Ω II 1 6 VV 2 Ω II 3 9 VV 1 Ω II II II 2 = 0
51 Kirchoff s rules What are the currents flowing in the 3 resistors? 9 VV 4 Ω 1 Ω II 1 II 2 Kirchoff s loop rule : the sum of voltage changes around a closed loop is zero 6 VV 2 Ω II II II 3 = 0
52 Kirchoff s rules What are the currents flowing in the 3 resistors? 9 VV 4 Ω 1 Ω II 1 II 2 We now have 3 equations: II 1 + II 2 + II 3 = 0 (1) 9 4II 1 + II 2 = 0 (2) 6 II 2 + 2II 3 = 0 (3) 6 VV 2 Ω II 3 To solve for II 1 we can use algebra to eliminate II 2 and II 3 : 1 II 3 = II 1 II 2 SSSSSS. iiii 3 II 2 = II 1 SSSSSS. iiii 2 II 1 = 1.5 AA
53 Consider the loop shown in the circuit. The correct Kirchoff loop equation, starting at a is 1. II 11 RR 11 + εε 11 + II 11 RR 22 + εε 22 + II 22 RR 33 = II 11 RR 11 εε 11 II 11 RR 22 εε 33 II 33 RR 44 = II 11 RR 11 εε 11 + II 11 RR 22 εε 22 II 22 RR 33 = II 11 RR 11 εε 11 + II 11 RR 22 + εε 22 + II 22 RR 33 = 00 0% 0% 0% 0%
54 Chapter 25 summary Components in a series circuit all carry the same current Components in a parallel circuit all experience the same potential difference Capacitors are parallel plates which store equal & opposite charge Q = C V Kirchoff s junction rule and loop rule provide a systematic method for analysing circuits
= (0.400 A) (4.80 V) = 1.92 W = (0.400 A) (7.20 V) = 2.88 W
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