Transcript 20. Electric Charge, Force, & Field
25. Electric Circuits
1. Circuits, Symbols, & Electromotive Force 2. Series & Parallel Resistors 3. Kirchhoff’s Laws & Multiloop Circuits 4. Electrical Measurements 5. 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
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. ) Collisions ~ resistance Ohm’s law:
I
E R
GOT IT?
25.1.
The figure shows three circuits. Which two of them are electrically equivalent?
n
= 2 :
25.2. Series & Parallel Resistors
Series resistors :
I
= same in every component Same q must go every element.
E
1
V
2
I R
1
I R
2
I R s
R s
R
1
R
2 For
n
resistors in series:
I
E R s R s
j n
1
R j V j
I R j
R j E R s V
1
R
1
R
1
R
2
E V
2
R
1
R
2
R
2
E
Voltage divider
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
5.0
2.4
A
0.6
A E
I R
1
I R
2 lightbulb
R
1 E
I
20
R
2 12
V
0.6
A
15 Voltage across lightbulb =
V
2 3
A
3
V
1 4
E
Most inefficient
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
R
int .
I
means
V
drop
I R int
V terminal
< E
E
I R
int
I R L I
R
int
E
R L V R L
R
int
R L
R L E
E R L
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 R L
E
I R
int 12
V
125
A
0.020
0.096
0.020
0.076
Voltage across battery terminals =
E
V
int 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
1
I
2 E
R
1 E
R
2 E
R p
1
R p
1
R
1 1
R
2
R p
R R R
1 2 1
R
2 For
n
resistors in parallel : 1
R p
j n
1 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 1 R/3 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
R
1 2.0
1 4.0
3 4.0
R
1.33
Equivalent of series 1.0 , 1.33 resistors: & 3.0
R T
1.0
1.33
3.0
5.33
Total current is
I
5.33
E R T
12
V
5.33
2.25
A
Voltage across of parallel 2.0 & 4.0 resistors:
V
1.33
2.25
A
1.33
Current through the 2 resistor:
I
2 2.99
V
2.0
1.5
A
2.99
V
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
This circuit can’t be analyzed using series and parallel combinations.
Kirchhoff’s loop law :
V
= 0 around any closed loop.
( energy is conserved ) Kirchhoff’s node law :
I
= 0 at any node.
( charge is conserved )
Problem Solving Strategy :
Multiloop Circuits
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. (
I
in > 0,
I
out < 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
R
3 in the figure below.
Node A:
I
1
I
2
I
3 0 Loop 1:
E
1
I R
1 1
I R
3 3 0
I
1
I
3 0 Loop 2:
E
2
I R
2 2
I R
3 3 0 9 4
I
2
I
3 0
I
1 1 2
I
3 3 1 2 4 1
I
3
I
3 7 21 4 9 4
I
2 1 4
I
3 9 4 3
A
Application: Cell Membrane
Hodgkin-Huxley (1952) circuit model of cell membrane (Nobel prize, 1963): Resistance of cell membranes Membrane potential Time dependent effects Electrochemical effects
25.4. Electrical Measurements
A voltmeter measures potential difference between its two terminals.
Ideal voltmeter: no current drawn from circuit R m =
Conceptual Example 25.1
. Measuring Voltage
What should be the electrical resistance of an ideal voltmeter?
An ideal voltmeter should not change the voltage across R 2 after it is attached to the circuit.
The voltmeter is in parallel with R 2 .
In order to leave the combined resistance, and hence the voltage across R be .
2 unchanged, R V must
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)
V
40 40 40 80 12
V
4
V R parallel
40 40 1000 1000 38.5
V
40 38.5
38.5
80 12
V
3.95
V
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
.
½ E
Ammeters
An ammeter measures the current flowing through itself.
Ideal voltmeter: no voltage drop across it R m = 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.
V R
but rate
The RC Circuit: Charging
I
but rate
C
initially uncharged
V C
= 0 Switch closes at
t
= 0.
V R
(
t
= 0) = E
I
(
t
= 0) = E /
R C
charging:
V C
Charging stops when
I
= 0.
V R
I
V C
but rate
d I I
d t RC
I I
0
d I I
0
t d t RC I
ln
I
0
t RC I V C
t I e
0
RC V R
E R t e
RC
E
1
e
t RC
Time constant =
RC E
I R
Q C
0
d I d t R
I C
0
I
d Q d t
V C ~ 2/3 E
I
~ 1/3 E /R
The RC Circuit: Discharging
Q
I R
0
C d I I
d t RC I I
t I e
0
RC
V
0
R t e
RC V t
V e
0
RC
d Q d t C
initially charged to
V C
=
V
0 Switch closes at
t
= 0.
C V R
=
V
C =
V I
0 =
V
0 /
R
discharging:
V C
V R
Disharging stops when
I
=
V
= 0.
I
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.
t
RC
ln 1
V C E
3 5.1
s
6
F
ln 1 170
V
200
V
RC Circuits: Long- & Short- Term Behavior
For
t << RC
:
V C
C
const, replaced by short circuit if uncharged.
C
replaced by battery if charged.
For
t >> RC
:
I C
C
0, replaced by open circuit.
Example 25.7
. Long & Short Times
The capacitor in figure is initially uncharged.
Find the current through
R
1 (a) the instant the switch is closed and (b) a long time after the switch is closed.
(a) (b)
I
1
E R
1
I
1
R
1
E
R
2
GOT IT?
25.6.
6 mA 2 mA 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 (a) immediately after the capacitor is connected and (b) a long time after it’s connected?