Short Version : 25. Electric Circuits
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Transcript Short Version : 25. Electric Circuits
Short Version :
25. Electric Circuits
Electric Circuit = collection of electrical components connected by conductors.
Examples:
Man-made circuits: flashlight, …, computers.
Circuits in nature: nervous systems, …, atmospheric circuit (lightning).
25.1. Circuits, Symbols, & Electromotive Force
Common circuit symbols
All wires ~ perfect conductors V = const on wire
Electromotive force (emf) = device that maintains fixed V across its terminals.
E.g.,
batteries (chemical),
generators (mechanical),
photovoltaic cells (light),
cell membranes (ions).
m~q
Collisions ~ resistance
g~E
Ideal emf :
no internal energy loss.
Lifting ~ emf
Energy gained by charge transversing battery = q E
( To be dissipated as heat in external R. )
Ohm’s law: I E
R
25.2. Series & Parallel Resistors
Series resistors :
I = same in every component
Same q must go
every element.
E V1 V2 I R1 I R2 I Rs
Rs R1 R2
n
For n resistors in series:
I
n=2:
V1
R1
R1 R 2
E
E
Rs
Rs R j
Vj I Rj
V2
R2
R1 R 2
E
j 1
Rj
Rs
E
Voltage divider
Real Batteries
Model of real battery = ideal emf E in series with internal resistance Rint .
I means V drop I Rint
Vterminal < E
E I Rint I RL
I
E
Rint R L
VRL
RL
R int R L
E
E
RL
Example 25.2. Starting a Car
Your car has a 12-V battery with internal resistance 0.020 .
When the starter motor is cranking, it draws 125 A.
What’s the voltage across the battery terminals while starting?
Battery terminals
E I R int I RL
RL
E
12 V
Rint
0.020
I
125 A
0.096 0.020
0.076
Voltage across battery terminals = E Vint 12 V 125A 0.020 9.5V
Typical value for a good battery is 9 – 11 V.
Parallel Resistors
Parallel resistors :
V = same in every component
I I1 I 2
E
E
E
Rp
R1 R 2
1
1
1
R p R1 R 2
For n resistors in parallel :
Rp
R1 R 2
R1 R 2
n
1
1
Rp j 1 R j
Analyzing Circuits
Tactics:
• Replace each series & parallel part by their single component equivalence.
• Repeat.
Example 25.3. Series & Parallel Components
Find the current through the 2- resistor in the circuit.
Equivalent of parallel 2.0- & 4.0- resistors:
1
1
1
3
R 2.0 4.0
4.0
R 1.33
Equivalent of series 1.0-, 1.33- & 3.0-
resistors:
RT 1.0 1.33 3.0 5.33
Total current is
I5.33
12 V
E
RT 5.33
2.25 A
Voltage across of parallel 2.0- & 4.0- resistors: V1.33 2.25 A1.33 2.99 V
Current through the 2- resistor:
I 2
2.99 V
2.0
1.5A
25.3. Kirchhoff’s Laws & Multiloop Circuits
Kirchhoff’s loop law:
V = 0 around any closed loop.
( energy is conserved )
Kirchhoff’s node law:
This circuit can’t be analyzed using
series and parallel combinations.
I=0
at any node.
( charge is conserved )
Multiloop Circuits
Problem Solving Strategy:
INTERPRET
■
Identify circuit loops and nodes.
■
Label the currents at each node, assigning a direction to each.
DEVELOP
■
Apply Kirchhoff ‘s node law to all but one nodes. ( Iin > 0, Iout < 0 )
■
Apply Kirchhoff ‘s loop law all independent loops:
Batteries: V > 0 going from to + terminal inside the battery.
Resistors: V = I R going along +I.
Some of the equations may be redundant.
Example 25.4. Multiloop Circuit
Find the current in R3 in the figure below.
Node A:
I1 I 2 I 3 0
Loop 1:
E1 I1R1 I 3R3 0
6 2 I1 I 3 0
Loop 2:
E2 I 2 R2 I3R3 0
9 4I2 I3 0
1
I1 I 3 3
2
I2
9
1 1
1
I
3
3
4
2 4
I3
4 21
3A
7 4
1
9
I3
4
4
Application: Cell Membrane
Hodgkin-Huxley (1952) circuit model of cell membrane (Nobel prize, 1963):
Resistance of cell membranes
Membrane
potential
Electrochemical effects
Time dependent effects
25.4. Electrical Measurements
A voltmeter measures potential difference between its two terminals.
Ideal voltmeter: no current drawn from circuit Rm =
Example 25.5. Two Voltmeters
You want to measure the voltage across the 40- resistor.
What readings would an ideal voltmeter give?
What readings would a voltmeter with a resistance of 1000 give?
(a)
(b)
40
V40
12 V 4V
40 80
Rparallel
40 1000
40 1000
38.5
38.5
V40
12 V 3.95V
38.5 80
Ammeters
An ammeter measures the current flowing through itself.
Ideal voltmeter: no voltage drop across it Rm = 0
Ohmmeters & Multimeters
An ohmmeter measures the resistance of a component.
( Done by an ammeter in series with a known voltage. )
Multimeter: combined volt-, am-, ohm- meter.
25.5. Capacitors in Circuits
Voltage across a capacitor cannot change instantaneously.
The RC Circuit: Charging
C initially uncharged VC = 0
Switch closes at t = 0.
VR (t = 0) = E
I (t = 0) = E / R
C charging:
VR but rate
I but rate
VC but rate
VC VR I
Charging stops when I = 0.
E I R
dI
dt
I
RC
Q
0
C
dI
I
R 0
dt
C
I
dQ
dt
t dt
dI
I0 I 0 RC
I
VC ~ 2/3 E
ln
I
t
I0
RC
I I0 e
t
RC
VC E VR
E RCt
e
R
t
E 1 e RC
Time constant = RC
I ~ 1/3 E/R
The RC Circuit: Discharging
C initially charged to VC = V0
Switch closes at t = 0.
VR = VC = V
Q
I R0
C
I
dI
dt
I
RC
I I0 e
t
RC
V V0 e
V0 RCt
e
R
t
RC
dQ
dt
I 0 = V0 / R
C discharging:
VC VR I
Disharging stops when I = V = 0.
Example 25.6. Camera Flash
A camera flash gets its energy from a 150-F capacitor & requires 170 V to fire.
If the capacitor is charged by a 200-V source through an 18-k resistor,
how long must the photographer wait between flashes?
Assume the capacitor is fully charged at each flash.
V
t RC ln 1 C
E
170 V
18 103 150 106 F ln 1
200
V
5.1 s
RC Circuits: Long- & Short- Term Behavior
For t << RC: VC const,
C replaced by short circuit if uncharged.
C replaced by battery if charged.
For t >> RC: IC 0,
C replaced by open circuit.
Example 25.7. Long & Short Times
The capacitor in figure is initially uncharged.
Find the current through R1
(a) the instant the switch is closed and
(b) a long time after the switch is closed.
(a)
(b)
I1
I1
E
R1
E
R1 R2