short version : 25. electric circuits. electric circuit = collection of electrical components...
TRANSCRIPT
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).
IR
EOhm’s law:
Energy gained by charge transversing battery = q ( To be dissipated as heat in external R. )
g ~ E
m ~ q
Lifting ~ emf
Collisions ~ resistance
Ideal emf : no internal energy loss.
25.2. Series & Parallel Resistors
Series resistors :
I = same in every component
1 2V V E 1 2I R I R sI R
1 2sR R R
For n resistors in series: 1
n
s jj
R R
s
IR
E
j jV I R j
s
R
R E
Voltage divider
Same q must go every element.
n = 2 :1
11 2
RV
R R
E 2
21 2
RV
R R
E
Real Batteries
Model of real battery = ideal emf in series with internal resistance Rint .
int LI R I R Eint L
IR R
E
LR
E
I means V drop I Rint
Vterminal <
intL
LR
L
RV
R R
E
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?
int LI R I R E
Voltage across battery terminals = int 12 125 0.020V V A E 9.5V
Typical value for a good battery is 9 – 11 V.
Battery terminals
intLR RI
E 12
0.020125
V
A
0.096 0.020 0.076
Parallel Resistors
Parallel resistors :
V = same in every component
1 2I I I 1 2R R
E E
pRE
1 2
1 1 1
pR R R
For n resistors in parallel :
1
1 1n
jp jR R
1 2
1 2p
R RR
R R
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
2.0 4.0R
1.33R
Total current is
Equivalent of series 1.0-, 1.33- & 3.0- resistors:
5.33
3
4.0
1.0 1.33 3.0TR
5.33T
IR E 12
5.33
V
2.25 A
Voltage across of parallel 2.0- & 4.0- resistors: 1.33 2.25 1.33V A 2.99V
Current through the 2- resistor: 2
2.99
2.0
VI
1.5A
25.3. Kirchhoff’s Laws & Multiloop Circuits
Kirchhoff’s loop law:
V = 0 around any closed loop.
( energy is conserved )
This circuit can’t be analyzed using series and parallel combinations.
Kirchhoff’s node law:
I = 0 at any node.
( charge is conserved )
Multiloop Circuits
INTERPRET
■ Identify circuit loops and nodes.
■ Label the currents at each node, assigning a direction to each.
Problem Solving Strategy:
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:
1 2 3 0I I I
Loop 1: 1 1 1 3 3 0I R I R E
3
1 1 91 3
2 4 4I
2 2 2 3 3 0I R I R E
1 36 2 0I I
2 39 4 0I I
1 3
13
2I I 2 3
1 9
4 4I I
3
4 21
7 4I 3A
Loop 2:
Application: Cell Membrane
Hodgkin-Huxley (1952) circuit model of cell membrane (Nobel prize, 1963):
Electrochemical effects
Resistance of cell membranes
Membrane potential
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?
40
4012
40 80V V
(b) 40 1000
40 1000parallelR
4V
38.5
(a)
40
38.512
38.5 80V V
3.95V
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.
The RC Circuit: Charging
C initially uncharged VC = 0
Switch closes at t = 0.
VR (t = 0) =
I (t = 0) = / R
C charging: VC VR I
Charging stops when I = 0.VR but rate I but rate
VC but rate
0Q
I RC
E
0d I I
Rd t C
dQI
d t
d I d t
I RC
0 0
I t
I
d I d t
I RC
0
lnI t
I RC
0
t
RCI I e
t
RCeR
E
C RV V E 1t
RCe
E
Time constant = RC
VC ~ 2/3
I ~ 1/3 /R
The RC Circuit: Discharging
C initially charged to VC = V0
Switch closes at t = 0.
VR = VC = V
I 0 = V0/ R
C discharging: VC VR I
Disharging stops when I = V = 0.
0Q
I RC
d I d t
I RC
dQI
d t
0
t
RCI I e
0t
RCVe
R
0
t
RCV V e
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.
ln 1 CVt RC
E
5.1 s
3 6 17018 10 150 10 ln 1
200
VF
V
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.