Download Ch 20.1, 20.5

Capacitors, Multimeters,
Circuit Analysis
Capacitor: Charging and Discharging
Charging
Discharging
Capacitor: Construction and Symbols
The capacitor in your
set is similar to a large
two-disk capacitor
s
There is no connecting path through a
capacitor
D
How is Discharging Possible?
Positive and negative charges are
attracted to each other: how can they
leave the plates?
Fringe field is not zero!
Electrons in the wire near the negative plate feel a force that
moves them away from the negative plate.
Electrons near the positive plate are attracted towards it.
Capacitor: Charging
Why does current ultimately stop flowing in the circuit?
Ultimately, the fringe field of the capacitor and the field
due to charges on the wire are such that E=0 inside the
wire. At this point, i=0.
The Effect of Different Light Bulbs
Thin filament
Thick filament
Which light bulb will glow longer?
Why?
1) Round is brighter  capacitor gets charged more?
2) Long bulb glows longer  capacitor gets charged more?
An Isolated Light Bulb
Will it glow at all?
How do electrons flow through
the bulb?
Why do we show charges near
bulb as - on the left and + on
the right?
Capacitance
-Q
+Q
Electric field in a capacitor: E  Q / A
0
 
V    E  dl
f
V  Es
i
V 
Q/ A
0
E
s
Q
0 A
s
V
In general: Q ~ V
Definition of capacitance:
Q  C V
Capacitance
s
Capacitance of a parallelplate capacitor:
0 A
C
s
Capacitance
Q  C V
Units: C/V, Farads (F)
Michael Faraday
(1791 - 1867)
Exercise
The capacitor in your
set is equivalent to a
large two-disk
capacitor
D
How large would it be?
C
A
0 A
A
s
s=1 mm
Cs
0
1 F 0.001 m 
9  10
12
C2 /N  m 2
A  1.1  10 8 m 2
D ~ 10 km (6 miles)

Exercise
A capacitor is formed by two rectangular plates 50 cm by 30
cm, and the gap between the plates is 0.25 mm. What is its
capacitance?
C
C
0 A
s
0 A
s
9  10


12

C2 /N  m 2 0.15 m 2
2.5  10
4
m
  5.4  10
9
F  5.4 nF
A Capacitor With an Insulator Between the Plates
No insulator:
E
With insulator:
Q/A
Q/A
E
K 0
0
V  Es
V 
Q
V  Es
Q/ A
0
0 A
s
s
V
C
D
0 A
s
Q/ A
V 
s
K 0
Q
K 0 A
V
s
CK
0 A
s
s
Ammeters, Voltmeters and Ohmmeters
Ammeter: measures current I
Voltmeter: measures voltage difference V
Ohmmeter: measures resistance R
Using an Ammeter
Connecting ammeter:
Conventional current must flow into the
‘+’ terminal and emerge from the ‘-’
terminal to result in positive reading.
0.150
Ammeter Design
Simple ammeter using your lab kit:
Simple commercial
ammeter
Digital ammeters: uses semiconductor elements.
ADC – analog-to-digital converter
(Combination of comparator and DAC)
Voltmeter
Voltmeters measure potential difference
VAB – add a series resistor to ammeter
V
I
R
Measure I and convert to VAB=IR
Connecting Voltmeter:
Higher potential must be connected to
the ‘+’ socket and lower one to the ‘-’
socket to result in positive reading.
Ohmmeter
How would you measure R?
A
emf
I
R
R
emf
I
Ohmmeter
Ammeter with a small voltage source
Quantitative Analysis of an RC Circuit
Vround _ trip  emf  RI  VC  0
Q
0
C
dQ emf  Q / C
I

dt
R
emf
I

Initial situation: Q=0
0
R
emf  RI 
Q and I are changing in time
dI d  emf  d  Q 
 
 

dt dt  R  dt  RC 
Q
VC 
C
d
dt
dI
1 dQ

dt
RC dt
dI
1

I
dt
RC
RC Circuit: Current
dI
1

I
dt
RC
1
1
dI  
dt
I
RC
I
t
1
1
dI


dt
I I

RC 0
0
t
ln I  ln I 0  
RC
I
t
ln  
I0
RC
I
e
I0
t

RC
Current in an RC circuit
I  I 0e  t / RC
What is I0 ?
Current in an RC circuit
I
emf t / RC
e
R
RC Circuit: Charge and Voltage
What about charge Q?
I
dQ
dt
dQ  Idt
Current in an RC circuit
emf t / RC
I
e
R
t
emf
Q   Idt 
R
0
t
t / RC
e
dt

0
Q  C emf 1  et / RC 
RC Circuit: Summary
Current in an RC circuit
I
emf t / RC
e
R
Charge in an RC circuit
Q  C emf 1  et / RC 
Voltage in an RC circuit
V  emf 1  et / RC 
The RC Time Constant
Current in an RC circuit
emf t / RC
I
e
R
When time t = RC, the current I drops by a factor of e.
RC is the ‘time constant’ of an RC circuit.
e t / RC  e 1 
1
 0.37
2.718
A rough measurement of how long it takes to reach final equilibrium
What is the value of RC?
About 9 seconds
Question
A 0.5 farad capacitor is
connected to a 1.5 volt
battery and a bulb, and
current runs until the bulb
goes out. What is the
absolute value of the
charge on one plate of the
capacitor?
A)
B)
C)
D)
E)
0.33 C
0.5 C
0.75 C
1.5 C
3.0 C
A
B
C
D
Exercise
What is the final charge on a 1 F
capacitor connected to a 1.5V battery
through resistor 100 ?
Q  CV  1.5 C
Can you apply the RC equations to the circuit below?
Current in an RC circuit
I
emf t / RC
e
R
No! Resistance depends on current.
Exercise: A Complicated Resistive Circuit
Find currents through resistors
I2
loop 1:
emf  r1I1  R1I1  R4 I 4  R7 I1  0
Loop 2
loop 2:
I1
 R2 I 2  r2 I 2  emf  R6 I 2  R3 I 3  0
I3
Loop 1
Loop 3
I4
Loop 4
loop 3:
I5
R4 I 4  R3 I 3  R5 I 5  0
nodes:
I1  I 2  I 3  I 4  0
I3  I 2  I5  0
I 4  I 5  I1  0
Five independent equations and five unknowns