Transcript 25._ElectricCircuits

25. Electric Circuits
1.
2.
3.
4.
5.
Circuits, Symbols, & Electromotive Force
Series & Parallel Resistors
Kirchhoff’s Laws & Multiloop Circuits
Electrical Measurements
Capacitors in Circuits
Festive lights decorate a city.
If one of them burns out, they all go out.
Are they connected in series or in parallel?
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
GOT IT? 25.1.
The figure shows three circuits.
Which two of them are electrically equivalent?
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
Battery lifts
charges
Energy
loss
For n resistors in series:
I
E
Rs
Voltage divider
Rs   R j
Vj  I Rj 
j 1
Rj
Rs
E
Example 25.1. Voltage Divider
A lightbulb with resistance 5.0  is designed to operate at a current of 600 mA.
To operate this lamp from a 12-V battery,
what resistance should you put in series with it?
12 V
 2.4 A  0.6 A
5.0 
E  I R1  I R2
R1 
lightbulb
E
 R 2  12 V  5 
I
0.6 A
 20   5   15 
Voltage across lightbulb = V2   600  103 A   5    3 V

1
E
4
Most inefficient
GOT IT? 25.2.
The figure shows three circuits.
Which two of them are electrically equivalent?
GOT IT? 25.2.
Rank order the voltages across the identical resistors R at the top
of each circuit shown, and give the actual voltage for each.
In (a) the second resistor has the same resistance R, and
(b) the gap is an open circuit (infinite resistance).
3V
0V
6V
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 R L
I
E
R int  R L
VRL 
RL
Rint  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 R L
RL 
E
12 V
 Rint 
 0.020 
I
125 A
 0.096   0.020 
 0.076 
Voltage across battery terminals = E  Vint  12 V  125 A  0.020   9.5 V
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
GOT IT? 25.3.
The figure shows all 4 possible combinations of 3 identical resistors.
Rank them in order of increasing resistance.
3R
R/3
1
4
2R/3
3
3R/2
2
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
I 5.33 
12 V
E

RT 5.33 
 2.25 A
Voltage across of parallel 2.0- & 4.0- resistors: V1.33   2.25 A1.33   2.99 V
Current through the 2- resistor:
I 2 
2.99 V
2.0 
 1.5A
GOT IT? 25.4.
The figure shows a circuit with 3 identical lightbulbs and a battery.
(a) Which, if any, of the bulbs is brightest?
(b) What happens to each of the other two bulbs if you remove bulb C?
dimmer
brighter
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 3 R 3  0
6  2 I1  I 3  0
Loop 2:
E2  I 2 R 2  I 3 R 3  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
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 3 R 3  0
6  2 I1  I 3  0
Loop 2: E2  I 2 R 2  I 3 R 3  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.
(a) What readings would an ideal voltmeter give?
(b) What readings would a voltmeter with a resistance of 1000  give?
(a)
(b)


40 
V40  
12 V   4 V

 40   80  
R parallel 
 40  1000  
40   1000 
 38.5 


38.5 
V40  
12 V   3.95V

 38.5   80  
GOT IT? 25.5.
If an ideal voltmeter is connected between points A and B in figure, what will it read?
All the resistors have the same resistance R.
½E
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

I 
Q
I R0
C
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  106 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
GOT IT? 25.6.
A capacitor is charged to 12 V
& then connected between points A and B in the figure,
with its positive plate at A.
What is the current through the 2-k resistor
6 mA
(a) immediately after the capacitor is connected and
2 mA
(b) a long time after it’s connected?