# Fundamentals of Physics Chapter 27 Circuits

```Fundamentals of Physics
Chapter 27 Circuits
1.
2.
3.
4.
5.
6.
7.
8.
Pumping Charges
Work, Energy &amp; Emf
Current in a Single Loop Circuit
Other Single Loop Circuits
Potential Differences
Multiloop Circuits
The Ammeter &amp; the Voltmeter
RC Circuits
Review &amp; Summary
Chapter Questions
Exercises &amp; Problems
2006
Physics 2112 Fundamentals of Physics
Chapter 27
1
Work, Energy and emf
emf Device:
A source of electrical energy.
Keeps one point (the + terminal) at a higher potential than the other (the
terminal).
Does work on charge carriers in a circuit to maintain a potential
difference between two terminals.
Expends an internal source of energy to do this work.
dW
dq
The emf of a device is the work done per unit charge in moving charge from its
low potential terminal to its high potential terminal.
Electromotive Force
2006
Physics 2112 Fundamentals of Physics
Chapter 27
2
Work, Energy and emf
Emf Device in a Circuit:
Transfers energy to the charge carriers moving through it.
The charge carriers then transfer energy to other elements in the circuit.
Direction of current
2006
B
&gt;
A
Physics 2112 Fundamentals of Physics
Chapter 27
3
Current in a Single Loop Circuit
i
?
Work done by the emf device:
dW
dq
i dt
Energy expended in the resistor:
i dt
i 2R dt
Energy conservation:
= iR
2006
i
R
Physics 2112 Fundamentals of Physics
Chapter 27
4
Current in a Single Loop Circuit
Kirchhoff s Loop Rule:
The algebraic sum of the changes in potential encountered in a
complete traversal of any loop of a circuit must be zero.
-iR
Resistance Rule: Passing through a resistance in the direction of i,
V = - i R; in the opposite direction +i R.
emf Rule: Passing through an ideal emf from
in the opposite direction V = clockwise from a:
- iR = 0
counterclockwise from a:
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+iR-
to +, V = +
i
= 0
Physics 2112 Fundamentals of Physics
R
Chapter 27
5
Checkpoint
a) Direction of emf arrow at B?
At point a, b, c rank:
b) magnitude of current
c) electric potential
d) electric potential of charge
carriers (greatest first)
2006
Physics 2112 Fundamentals of Physics
Chapter 27
6
Two batteries in a circuit
E1
E2
12 V
6V
direction current flow?
positive or negative work done by battery?
direction of current between A amd B?
Battery #1
Battery #2
current flows counterclockwise
Battery #1 is doing positive work (current flows from to )
Battery #2 is having work done on it (being charged)
Current flows from B to A
B is at higher potential than A
2006
Physics 2112 Fundamentals of Physics
Chapter 27
7
Problem
Current
Power
Power in Each Battery
generating heat
charging
i
discharging
Choose a direction for the current.
Define a Kirchhoff Loop:
counterclockwise starting at lower left corner.
Note the potential drops around the loop:
E1
i R2
Power:
2006
i R1
PR
E2
i2 R
0
PB
i
0.5 A
Vi
Physics 2112 Fundamentals of Physics
Chapter 27
8
Internal Resistance
Real emf Battery
i
r = internal resistance
2006
Physics 2112 Fundamentals of Physics
Chapter 27
9
Combinations of Resistors
In Series: same current through each resister.
Vac = Vab + Vbc
Vab = - i R1
Vbc = - i R2
Req =
R1 + R2
n
Req
Ri
i 1
Vac = - i Req
2006
Note: Req is larger than any
of the combining resistors.
Physics 2112 Fundamentals of Physics
Chapter 27
10
Combinations of Resistors
In Parallel: same potential drop across each resistor.
I1
I2
Vab
R1
Vab
R2
I = I1 + I2
1
Req
1
Req
i
i
2006
1
R1
Vab
Req
1
R2
n
i 1
1
Ri
Note: Req is smaller than any of
the combining resistors.
Physics 2112 Fundamentals of Physics
Chapter 27
11
Checkpoint
If R1 &gt; R2 &gt; R3, rank the three
resistances according to:
a) current through them
b) potential difference across
them, greatest first
2006
Physics 2112 Fundamentals of Physics
Chapter 27
12
Series or Parallel?
series
parallel
series
parallel
series
None in series or parallel!
2006
Physics 2112 Fundamentals of Physics
Chapter 27
13
Potential Differences in Circuits
To find the potential difference between
any two points, start at one point and
traverse the circuit to the other, following
any path, and add algebraically the
changes in potential encountered.
= 4.4 V, 2=2.1 v
r1 = 2.3 , r2=1.8 W, r3 = 5.5
a) current I
b) Potential difference between
terminals of battery 1
1
2006
Physics 2112 Fundamentals of Physics
Chapter 27
14
Problem
V1 = ?
V2 = ?
2
ground
2
2
V0 = 0
2
2
12
5
2 2 2 1 i
i
2
2
1
V1
12
1 1
V1
i
V2
5
V2
0
1A
V0
V0
11V
4 1
V0
V0
9V
2
V2
2006
V1
Physics 2112 Fundamentals of Physics
2V
Chapter 27
15
Example: Single-Loop Current
Apply Kirchhoff s Loop Rule:
fgabcdef: + E1 - I r1 - I R1 - I R2 - E2 - I r2 - I R3 = 0
Solve for I
2006
Physics 2112 Fundamentals of Physics
Chapter 27
16
Example - plugging in values
Apply Kirchhoff s Loop Rule:
fgabcdef: solve for I &amp; calculate each potential drop.
I = 0.5 A
grounded
2006
Physics 2112 Fundamentals of Physics
Chapter 27
17
Kirchhoff s Rules for Multi-Loop Circuits
Loop Rule: When any closed-circuit loop is traversed, the
algebraic sum of the changes in potential must equal zero.
Junction Rule: At any junction point in a circuit where
the current can divide, the sum of the currents into
the junction must equal the sum of the currents out
of the junction.
2006
Physics 2112 Fundamentals of Physics
Chapter 27
18
28-6
Multi-Loop Circuits
Identify the branches in the circuit and arbitrarily assign a current to each
branch:
bcd: - i2
bd: + i3
Apply Kirchhoff s Branch Rule:
2006
i2 = i1 + i3
(currents at junctions
b and d)
Physics 2112 Fundamentals of Physics
Chapter 27
19
28-6
Multi-Loop Circuits
Kirchhoff s Branch Rule:
i2 = i1 + i3
Apply Kirchhoff s Loop Rule:
badb: + E1 - i1 R1 + i3 R3 = 0
bcdb: + E2 + i2 R2 + i3 R3 = 0
badcb: + E1 - i1 R1 - i2 R2 - E2 = 0
2006
Physics 2112 Fundamentals of Physics
Chapter 27
20
DC Circuit Example
Method I: Equivalent Circuit
i1
Find i
i2
i
R1
loop 1
E
loop 2
E=6V
R1 = 6
R2 = 4
R2
R1 and R2 are in parallel.
E
1
R1
1
R2
1
6
V
loop
i
E
R2
R1
1
Req
2006
i2
i1
i
1
4
E iReq
5
12
0
1
Req
i
E
Req
loop
Req
2 .4
6V
2.4
Physics 2112 Fundamentals of Physics
2.5 A
Chapter 27
21
DC Circuit Example
iin
Method II: Kirchhoff s rules
iout
junct
junct
i
i1
i2
Find i
i1
i2
i
loop 1
R1
V
E
loop 2
R2
E=6V
R1 = 6
R2 = 4
0
loop
E
i1 R1
0
6 i1 6 0
i1
EE
i2 R2
0
6 i2 4 0
i2
i
2006
i1
i2
1
1.5
1A
1. 5 A
2 .5 A
Physics 2112 Fundamentals of Physics
Chapter 27
22
Example 2
R1 = 20
R3 = 30
, R2 =20 ,
, R4 = 8.0
=12V
What is current through the battery?
2006
Physics 2112 Fundamentals of Physics
Chapter 27
23
Example 3
R1 = 2.0
, R2 =4.0
= 3.0 V,
,
= 3.0 V
What is magnitude and direction of the
current in the three branches?
2006
Physics 2112 Fundamentals of Physics
Chapter 27
24
What are i1 and i2 ?
i8
16
8 i8
i8
2006
0
2A
8
i4
4
4
16
Physics 2112 Fundamentals of Physics
4 i4
8
0
i4
2A
Chapter 27
25
Meters
Ammeter:
Voltmeter:
RA &lt;&lt; R
RV &gt;&gt; R
Galvanometer
2006
Physics 2112 Fundamentals of Physics
Chapter 27
26
RC Circuits
Charging the capacitor
Discharging the capacitor
2006
switch S turned to point a
switch S turned to point b
Physics 2112 Fundamentals of Physics
Chapter 27
27
28-8
RC Circuits
Charging the capacitor
switch S turned to point a
VR
iR
VC
E
VR
VC
0
E
iR
q
C
0
Clockwise loop:
2006
Physics 2112 Fundamentals of Physics
q
C
Chapter 27
28
RC Circuits
VR
iR
VC
E
1st Order Linear D.E.
Solution:
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R
q
C
iR
dq
dt
qt
q
C
CE
q
C
0
E
1
e
t
RC
Physics 2112 Fundamentals of Physics
Chapter 27
29
RC Circuits
Charging the Capacitor:
q0
q
qt
C E
1
e
t
RC
0
CE
R = 2000
C = 1 mF
E = 10 V
Time Constant = RC = 2 ms
2006
Physics 2112 Fundamentals of Physics
Chapter 27
30
RC Circuits
Discharging the capacitor
b
switch S turned to point
VR
iR
VC
Kirchhoff loop:
VR
iR
R
Solution:
2006
dq
dt
qt
VC
q
C
q
C
q
C
0
0
0
q0 e
t
RC
Physics 2112 Fundamentals of Physics
Chapter 27
31
RC Circuits
Discharging the capacitor
qt
q0
CE
it
2006
q0 e
t
RC
dq
dt
q0
e
RC
t
RC
Physics 2112 Fundamentals of Physics
Chapter 27
32
Problem
= 6.0V,
2 = 5.0V, 3 = 4.0V
R1 = 100 , R2= 50
Current in each resistor?
Potential difference between a and b?
1
E1
i2
E2
i2
E2
0
i1 R1
0
0 . 05 A
Potential difference between points a and b?
Vb
E3
E2
V ab
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i2 R 2
0.06 A
i1
i1
E3
Physics 2112 Fundamentals of Physics
Va
E3
E2
Chapter 27
33
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