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Chapter 20
A Microscopic View
of Electric Circuits
Announcements:
March 31th last day for EXAM 1 free response re-take.
Second midterm April 3rd, Thursday, 8-9:30 pm, room 203
No lecture on April 9th
Capacitance of parallel plate
capacitor:
A=area,
s-separation between plates
Select a correct formula:
1. C = A*0/s
2. C = A/0s
3. C = 0/(A*s)
4. C = s/0A
Ohmic Resistors
Ohmic resistor: resistor made of ohmic material
Ohmic materials: materials in which conductivity  is
independent of the amount of current flowing through
DV
I=
R
R=
L
sA
not a function of current
Examples of ohmic materials:
metal, carbon (at constant T!)
Is a Light Bulb an Ohmic Resistor?
Tungsten: mobility at room temperature is larger than at
‘glowing’ temperature (~3000 K)
DV
I=
R
V-A dependence:
3V
100 mA
1.5 V 80 mA
0.05 V 6 mA
R
30 
19 
8
DV
R=
I
I
V
Semiconductors
Metals, mobile electrons: slightest V produces current.
If electrons were bound – we would
need to apply some field to free some of
them in order for current to flow. Metals
do not behave like this!
Semiconductors: n depends exponentially on E
s = q nu
Conductivity depends exponentially on E
Conductivity of semiconductor rises
(resistance drops) with rising temperature
Nonohmic Circuit Elements
Semiconductors
Capacitors
DV
I=
R
|V|=Q/C, function of time
Batteries: double current, but |V|emf, hardly changes
DV
I=
R
has limited validity!
Ohmic when R is indeppendent of I!
Conventional symbols:
Series Resistance
Vbatt + V1 + V2 + V3 = 0
emf - R1I - R2I - R3I = 0
emf = R1I + R2I + R3I
emf = (R1 + R2 + R3) I
emf = Requivalent I , where Requivalent = R1 + R2 + R3
For resistors made of the same material and with the same A
it follows straight from the definition of resistance:
L
R
A
Parallel Resistance
I = I 1 + I2 + I3
emf emf emf
I


R1
R2
R3
 1
1
1
emf
I  
  emf 
Requivalent
 R1 R2 R3 
1
Requivalent
1
1
1
 

R1 R2 R3
For resistors made of the same material and with the same A
it follows straight from the definition of resistance:
L
1 A
Aequivalent  A1  A2  A3
R

A
R
L
Question
I1
I2
Identical bulbs are connected to identical set of batteries. They
produce light. Compare I1 and I2 :
A. I1 = I2
B. I1 slightly less than I2
C. I1 = 2*I2
D. I1 slightly less than 2*I2
Real Batteries: Internal Resistance
Drift speed of ions
in chemical battery:
v ~ FNC  eEC 
In usual circuit elements:
J  E
rint - internal
resistance
In a battery:
 force 
I
 FNC

   
J     
 EC 
A
 e

 unit charge 
EC 
emf
FNC
I
FNC s s


V

E
s


I
, assuming uniform field:
C
e
A
e
A
Vbattery  emf  rint I
Real Batteries: Internal Resistance
ideal battery
Vbattery  emf  rint I
Model of a real battery
Round trip
(energy conservation):
emf  rint I  RI  0
R
emf
I
R  rint
Situation 1: disconnect R (R=)
I=0
V = emf
emf
Vbattery  emf  rint I
R  rint
Situation 2: R is in place
I
Situation 3: short battery (R=0)
I = emf /rint V = 0
Real Batteries: Internal Resistance
ideal battery
Vbattery  emf  rint I
emf  rint I  RI  0
emf
I
R  rint
rint0.25 
1.5 V
R
R
100 
10 
1
0
Ideal
0.015 A
0.15 A
1.5 A
infinite
Real
0.01496 A
0.146 A
1.2 A
6A
VR=RI
1.496 V
1.46 V
1.2 V
0V
Ammeters, Voltmeters and Ohmmeters
Ammeter: measures current I
Voltmeter: measures voltage difference V
Ohmmeter: measures resistance R
Using an Ammeter
Connecting ammeter:
An ammeter must be inserted into the
circuit in series with the circuit element
whose current you want to measure.
An ammeter must have a very small
resistance, so as not to alter significantly
the circuit in which it has been inserted.
0.150
Conventional current must flow into the
‘+’ terminal and emerge from the ‘-’
terminal to result in positive reading.
Exercise: Connecting Ammeter
Is it correct connection?
A) Yes
B) No
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.
Voltmeter: Internal Resistance
VAB in absence of a voltmeter
R1
emf
VAB 
B
rint
R2
A
A
R2
emf
R1  R2
VAB in presence of a voltmeter
VAB 
R2||int 
R2||int
R1  R2||int
emf
R2 rint
R2  rint
Internal resistance of a voltmeter must be very large
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 on capacitor?
I
dQ
dt
dQ  Idt
Current in an RC circuit
I  I 0e
Current in an RC circuit
emf t / RC
I
e
R
Check: t=0, Q=0,
t
emf
Q   Idt 
R
0
 t / RC
t
t / RC
e
dt

0
Q  C emf 1  et / RC 
Q
V 
C
t--> inf, Q=C*emf
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
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