University Physics: Waves and Electricity
Ch27. Circuit Theory
Lecture 11
Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com
11/2January–April 2010 University Physics: Wave and Electricity
Homework IX (IT)
(a) Find the equivalent resistance between points a and b in the circuit diagram below.
(b) Calculate the current in each resistor if a potential difference of 34 V is applied between points a and b.
11/3January–April 2010 University Physics: Wave and Electricity
Solution of Homework IX (IT)
(a) eq 4 (7 ||10) 9R 7 10
4 97 10
17.12
(b)
4 4.12 9
≡eq
ViR
34
17.12 1.986 A
4 9 1.986 Ai i
4.12 7 10V V V
4.12 (1.986)(4.12)V 8.182 V
4.12 4.12 4.12V i R
11/4January–April 2010 University Physics: Wave and Electricity
Solution of Homework IX (IT)
≡
4.12 7 10V V V
77
Vi
R
8.182
7
1.169 A
1010 10
Vi
8.182
10
0.8182 A
4.12 4.12 7 7 10 10i R i R i R c d
c d
4.12 7 10 8.182 VcdV V V V
11/5January–April 2010 University Physics: Wave and Electricity
Example: Single-Loop Circuit
1 2R 2 1R 2 6 VE
1 12 VE
ba
For the circuit shown, determine Vab.
i1 2 1 2 0iR iR E E12 1 2 6 0i i
3 6i 2 Ai
• a a, cw
• a b, ccw
2 1a bV iR V +E
2 1a bV V iR E
6 (2)(2)abV
10 V
• Assume a current direction• Perform the calculation• i > 0 means that the current flows in
the assumed direction• i < 0 means that the current flows in
opposite direction
• Try again for i assumed to be ccw
11/6January–April 2010 University Physics: Wave and Electricity
2R 2V
E
1R 1V
i
Voltage Divider and Current Divider
1 2
iR R
E
1 1V iR
Voltage Divider Current Divider
11
1 2
RV
R R
E
2 2V iR 22
1 2
RV
R R
E
2R 2V
E
1R 1V
i
1i 2i
11
iR
E
22
iR
E
eqiRE
eq1
1
Ri i
R 2
11 2
Ri i
R R
12
1 2
Ri i
R R
eq
22
Ri i
R
11/7January–April 2010 University Physics: Wave and Electricity
2R
1R
Req of Two Resistance in Parallel
eq 1 2
1 1 1
R R R 2 1
1 2
R R
R R
1 2eq
1 2
R RR
R R
(3)(1)
3 10.75 eqR
2V 2
182 0.75 4
5.33 V
4V 4
182 0.75 4
10.67 V
3 1V V 18 5.33 10.67 2 V
0.75V 0.75
182 0.75 4
2 V
11/8January–April 2010 University Physics: Wave and Electricity
Example: Finding Voltage and Current
10
40
15V
6
12 1i
1V
2V
2i
For this circuit, determine i1, i2, V1, and V2.
15V
1V
4
2V
≡
1
415
4 8V
5 V
2
815
4 8V
10 V
11 12
Vi 0.417 A5
12
22 40
Vi 0.25 A10
40
6 ?i 10 ?i through the battery?i
11/9January–April 2010 University Physics: Wave and Electricity
Example: Two-Loop Circuit
1E 4 6R
2E
3E
5 4R
6 2R
1 A
2 A
1
2
If E1 = 20 V, determine E2 and E3.
1 4 2 5(1) (1) 0R R E E• Loop 1, cw
i
1 Ai 1 2i
220 (1)(6) (1)(4) 0 E
2 18 VE
5 2 6 3(1) (2) 0R R E E• Loop 2, cw
3(1)(4) 18 (2)(2) 0 E
3 10 VE
3 6 4 1(2) (1) 0R R E E• Outer Loop, ccw
3 (2)(2) (1)(6) 20 0 E
3 10 VE
11/10January–April 2010 University Physics: Wave and Electricity
Multiloop Circuits
Junction RuleThe sum of the currents entering any junction must be equal to the sum of the currents leaving that junction
• Remember again Loop Rule, Resistance Rule, and Emf Rule
Our basic tools for solving complex circuits are the loop rule and the junction rule.
1 3 2i i i • Junction Rule, at d
• Loop Rule, left-hand loop, b b, ccw
1 1 1 3 3 0i R i R E
• Loop Rule, right-hand loop, b b , ccw
3 3 2 2 2 0i R i R E• Afterward, find the unknowns
from these three equations
. . . . . . . . . .(1)
. . . . . .(2)
. . . . . .(3)
11/11January–April 2010 University Physics: Wave and Electricity
Example: Multiloop Circuits
The value of the elements in the following circuits are:
E1 = 3 V, E2 = 6 V,R1 = 2 Ω, R2 = 4 Ω
Find the magnitude and direction of the current in each of the branches.
3 1 2i i i • Junction Rule, at a
• Loop Rule, left-hand loop, a a, ccw
1 1 1 1 1 2 2 2 0i R i R i R E E
• Loop Rule, right-hand loop, a a , cw
. . . . . . . . . .(1)
. . .(2) . . .(3)1 1 22 3 2 6 4 0i i i
1 24 4 3i i
3 1 2 3 1 2 2 2 0i R i R i R E E
3 3 22 6 2 6 4 0i i i
2 34 4 0i i
11/12January–April 2010 University Physics: Wave and Electricity
Example: Multiloop Circuits
3 1 2i i i . . . . . . .(1)
. . .(2)
. . .(3)1 24 4 3i i
2 34 4 0i i
• Insert (1) into (3),
2 1 24 4( ) 0i i i
1 24 8 0i i . . .(4)
• Option A: Elimination Method
1 24 4 3i i
1 24 8 0i i
1
11 24 4 3i i
1 24 8 0i i
212 3i
• Option B: Substitution Method
21
3 4
4
ii
• From (2)
• Insert (5) into (4)
22
3 44 8 0
4
ii
2 23 4 8 0i i
212 3i
2 0.25 Ai
1
3 4( 0.25)
4i
2 0.25 Ai
. . . . . . . .(5)
0.5 A
3 0.5 ( 0.25)i 0.25 A1 2
8
4i i 0.5 A
3 0.5 ( 0.25)i 0.25 A
11/13January–April 2010 University Physics: Wave and Electricity
Homework X (IT)For the circuit depicted below, determine the currents (i1, i2, i3) and the potential difference across the resistors (V1, V2, V3).
5V
2
2V
1i 3i
2i
8
3 V
4
1V
3V