Download E - Purdue Physics

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Equilibrium vs. Steady State in a Circuit
What is "used up" in a circuit?
Kirchhoff's Current Node Law
E-field inside a wire
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Electric Field Inside the Wire
Constant current in the wire  Constant E in the wire.
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Conventional Current
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I
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I
I
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Drift Velocity controlled by |E|
Mobility (u) set by the material.
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Constant current
requires constant |E|
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Transient response when connecting a circuit
How long until steady state is reached?
Introduction to Resistors
Energy conservation in a circuit
Kirchhoff's Voltage Loop Law
Batteries
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Direction of Electric Field in a Wire
E must be parallel to the wire
E is the same along the wire
Does current fill the wire?
Is E uniform across the wire?
B
C
D
A
A
B
C
D
DVABCDA = - ò E1 ×dl - ò E3 ×dl - ò E2 ×dl - ò E3 ×dl = 0
VAB
0
E1 = E2
VCD
0
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Electric Field in a Wire
What charges make the electric field in the wires?
E
Bulb filament and wires are metals –
there cannot be excess charges in the interior
Are excess charges on the battery?
ASSUME: E due to dipole field of battery.
This cannot be the source of the E which drives current.
E
E
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Field due to the Battery
Surface charge arranges itself in such a way as to produce a
pattern of electric field that follows the direction of the wire
and has such a magnitude that current is the same along the
wire.
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Field due to Battery
E
Smooth transition from + surface charge to – to provide
constant E.
The amount of surface charge is proportional to the voltage.
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Connecting a Circuit
What happens just before and just after
a circuit is connected?
Before the circuit is connected:
++
++
---
• No current flows
• System is in equilibrium:
How is |E| = 0 maintained when there are charges here?
There must be surface charges on the wire
to prevent current from flowing before we connect the circuit.
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Connecting a Circuit
What happens just before and just after
a circuit is connected?
Before the circuit is connected:
• No current flows
• System is in equilibrium:
Think about
the gap...
E due only to
gap faces
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Connecting a Circuit
What happens just before and just after
a circuit is connected?
Before the circuit is connected:
• No current flows
• System is in equilibrium:
Think about
the gap...
E due to
everything else
cancels Egap
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Connecting a Circuit
What happens just before and just after
a circuit is connected?
Before the circuit is connected:
E due to
everything else
cancels Egap
Now close the gap ...
The gap face charge  0, and so does11Egap
Connecting a Circuit
What happens just before and just after
a circuit is connected?
Just after the circuit is connected:
There is a disturbance in the
previous (equilibrium) E-field.
Now the region next to the disturbance
updates its E-field, and the next region...
How fast does this disturbance propagate?
At the drift speed of the electrons?
At the speed of light?
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iClicker – Reality Physics!
Drift speed of electrons
Speed of light
Flip Light Switch On.
How long until electrons
from the switch reach the light bulb?
L=5m
A)
B)
C)
D)
About 1 nanosecond
About 1 microsecond
About 1 minute
About 1 day
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iClicker – Reality Physics!
Drift speed of electrons
Speed of light
Flip Light Switch On.
How long until information about the
change in E-field reaches the light bulb?
L=5m
A)
B)
C)
D)
About 16 nanoseconds
About 16 microseconds
About 16 minutes
About 16 days
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Reality Physics!
Drift speed of electrons
Speed of light
Flip Light Switch On.
How long until information about the
change in E-field reaches the light bulb?
L=5m
≈ 1 day for electrons to travel from light switch to bulb.
≈ 16 nanoseconds for the change in E-field to travel from light switch to bulb.
Because there are sooooo many electrons
in the wire, they don't have to move far to
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create a large current.
Connecting a Circuit
What happens just before and just after
a circuit is connected?
Just after the circuit is connected:
There is a disturbance in the
previous (equilibrium) E-field.
Now the region next to the disturbance
updates its E-field, and the next region...
The disturbance travels at the speed of light,
and within a few nanoseconds,
steady state is established.
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Electric field in Thin and Thick wires
After steady state is reached:
ithin = ithick
ithin = nAthin uEthin
ithick = nAthick uEthick
Ethin
Athick
=
Ethick
Athin
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Energy in a Circuit
Vwire = EL
Vbattery = ?
Energy conservation (the Kirchhoff loop rule [2nd law]):
V1 + V2 + V3 + … = 0
along any closed path in a circuit
V= U/q  energy per unit charge
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General Use of the Loop Rule
V1 + V2 + V3 + V4 = 0
(VB-VA)+ (VC-VB)+ (VF-VC)+ (VA-VF)=0
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Potential Difference Across the Battery
Coulomb force on each e
FC
non-Coulomb force on each e
1. a=FNC/m
EC
DVbatt
FC s FNC s
= EC s =
=
e
e
2. FC =eEC
FC
EC =
e
3. FC =FNC
Energy input per unit charge
emf – electromotive force
The function of a battery is to produce and maintain a charge separation.
The emf is measured in Volts, but it is not a potential difference!
The emf is the energy input per unit charge.
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chemical, nuclear, gravitational…
iClicker Questions
i  nAuE
Nichrome wire (resistive)
When wire length double, the current will be
A. Double
B. Halved
C.unchanged
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Twice the Length
Nichrome wire (resistive)
Quantitative measurement of current with a compass
DV
i = nAuE = nAu
L
Current is halved when increasing the length of the wire by a
factor of 2.
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iClicker Question
Nichrome wire (resistive)
When wire cross sectional area is doubled, the current
will be
A. Double
B. Halved
C.unchanged
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Doubling the Cross-Sectional Area
If A doubles, the current doubles.
Nichrome wire
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iClicker Questions
When plug in two batteries instead of one, the current
will be
A. Double
B. Halved
C.unchanged
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Two Batteries in Series
Why light bulb is brighter with two batteries?
Two batteries in series can drive more current:
Potential difference across two batteries in
series is 2emf  doubles electric field
everywhere in the circuit  doubles drift
speed  doubles current.
2emf - EL = 0
emf - EL = 0
2emf
Work per second:
emf
E=
E=
L
emf
i = nAuE = nAu
L
2
æ emf ö
P1batt = eLnAu ç
è L ÷ø
P  W / T  qEL / T
 (q / T ) EL  ieEL
P = nAueLE
2
L
2emf
i = nAu
L
P2batt
æ 2emf ö
= eLnAu ç
è L ÷ø
P28
2batt = 4 ´ P1batt
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How Do the Currents Know How to Divide?
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Let’s be Quantitative
Capacitors, Resistors and
Batteries
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Capacitor: Charging and Discharging
Charging
Discharging
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Capacitor: Construction and Symbols
Similar to a large
parallel plate
capacitor
D
s
There is no connecting path through a
capacitor
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Capacitor: Discharge
Electron
Current
Electric
Field
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Capacitor: Charging
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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.
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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?
After current stops,
Voltage across capacitor = Voltage across battery
no matter which bulb is used.
Capacitor charged by same amount in both
36cases.
Effect of the Capacitor Disk Size
Use two different capacitors
in the same circuit
In the first moment, which capacitor will cause the
bulb to produce more light?
Which capacitor will make the light bulb glow longer?
Fringe field: E1 
Q/ A s
2 0 R
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Effect of the Capacitor Disk Separation
In the first moment, which capacitor will cause the
bulb to produce more light?
Which capacitor will make the light bulb glow longer?
Fringe field: E1 
Q/ A s
2 0 R
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Effect of Insulator in Capacitor
Insulator
In the first moment, which capacitor will cause the
bulb to produce more light?
Which capacitor will make the light bulb glow longer?
Q/ A s
 Edipoles
Fringe field: E1 
2 0 R
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The capacitors shown are initially uncharged. When connected to
identical circuits, after 0.01 s of charging:
A) The fringe field of each capacitor is the same.
B) The fringe field of the smaller capacitor is greater.
C) The fringe field of the larger capacitor is greater.
R1
E fringe
s
Q/A s

( )
2 0 R
R2
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Parallel Capacitors
Initial moment: brighter?
Will it glow longer?
Q/ A s
Fringe field: E1 
2 0 R
Capacitors in parallel effectively increase A
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An Isolated Light Bulb
Will it glow at all?
How do electrons flow through
the bulb?
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Capacitor in a Circuit
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Charging
time
Bulb
Brightness
time
Energy conservation
Do 19.X.7!
Ecap
time
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The Current Node Rule in a Capacitor Circuit
Charge conservation:
Ii > 0 for incoming
i I i  0 Ii < 0 for outgoing
I1 = I 2+ I3
…in steady state
Capacitor transients:
not a steady state!
Cannot use Kirchhoff rule for a
part of a capacitor (area 1 or 2)
But can use for capacitor as a
whole (area 3)
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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
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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
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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
s
K 0 A
V
s
CK
0 A
s
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