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2P09 -
Class 09: Outline
Hour 1:
Conductors & Insulators
Expt. 4: Electrostatic Force
Hour 2:
Capacitors
4P09 -
Gauss’s Law
0Ssurface
closed Q
dE AE
In practice, use symmetry:
• Spherical (r)
• Cylindrical (r, )
• Planar (Pillbox, A)
6P09 -
Conductors and Insulators
A conductor contains charges that are free to move (electrons are weakly bound to atoms)
Example: metals
An insulator contains charges that are NOT free to move (electrons are strongly bound to atoms)
Examples: plastic, paper, wood
7P09 -
Conductors
Conductors have free charges
E must be zero inside the conductor
Conductors are equipotential objects
8P09 -
Conductors in Equilibrium
Conductors are equipotential objects:
1) E = 0 inside
2) Net charge inside is 0
3) E perpendicular to surface
4) Excess charge on surface
0E
10P09 -
Hollow Conductors
Charge placed INSIDE induces balancing charge INSIDE
+q-- -
-
+
+
+
-
-- -
--
-+
+
+
+ +
11P09 -
Hollow Conductors
Charge placed OUTSIDE induces charge separation on OUTSIDE
+q
----
+
+
+
E=0
15P09 -
Field Enhancement
2
2
1
1
r
qk
r
qkV ee
12
1
11 r
V
r
qkE e
22 r
VE
E field is enhanced at sharp points!
P09 -
Capacitors: Store Electric Energy
Capacitor: two isolated conductors with equal and opposite charges Q and potential difference V between them.
Units: Coulombs/Volt or Farads
QC
V
22P09 -
Calculating E (Gauss’s Law)
0S
inqd
E A
00
A
QE
0
GaussGauss
AE A
Alternatively could have superimposed two sheets
23P09 -
Parallel Plate Capacitor
C depends only on geometric factors A and d
top
bottom
V d E S
d
A
V
QC 0
0
Qd
AEd
24P09 -
Spherical CapacitorTwo concentric spherical shells of radii a and b
Gauss’s Law E ≠ 0 only for a < r < b, where it looks like a point charge:
rE ˆ4 2
0r
Q
What is E?
25P09 -
Spherical Capacitor
For an isolated spherical conductor of radius a:
20
ˆˆ
4
b
a
Qdr
r
rr
1104
baV
QC
aC 04
outside
inside
V d E S
0
1 1
4
Q
b a
Is this positive or negative? Why?
26P09 -
Capacitance of Earth
For an isolated spherical conductor of radius a:
aC 04
mF1085.8 120
m104.6 6a
mF7.0F107 4 C
A Farad is REALLY BIG! We usually use pF (10-12) or nF (10-9)
30P09 -
Energy To Charge Capacitor
1. Capacitor starts uncharged.2. Carry +dq from bottom to top.
Now top has charge q = +dq, bottom -dq3. Repeat4. Finish when top has charge q = +Q, bottom -Q
31P09 -
Work Done Charging Capacitor
At some point top plate has +q, bottom has –qPotential difference is V = q / CWork done lifting another dq is dW = dq V
32P09 -
So work done to move dq is:
Total energy to charge to q = Q:
Work Done Charging Capacitor
dW dq V
0
1Q
W dW qdqC
1qdq q dqC C
21
2
Q
C
34P09 -
Energy Stored in Capacitor
2
field energy density2o
E
Eu E
21
2U CV
Parallel-plate capacitor: andoAC V Edd
Energy stored in the E field!
2
21( )
2 2o oA E
Ed Add
( )Eu volume
37P09 -
Ideal Battery
Fixes potential difference between its terminals
Sources as much charge as necessary to do so
Think: Makes a mountain
45P09 -
Equivalent Capacitance
1 21 2
Q QV , V
C C
1 2V V V
1 2eq
Q Q QV
C C C
1 2
1 1 1
eqC C C
(voltage adds in series)
48P09 -
Dielectrics
A dielectric is a non-conductor or insulator
Examples: rubber, glass, waxed paper
When placed in a charged capacitor, the dielectric reduces the potential difference between the two plates
HOW???
49P09 -
Molecular View of Dielectrics
Polar Dielectrics :
Dielectrics with permanent electric dipole moments
Example: Water
50P09 -
Molecular View of Dielectrics
Non-Polar Dielectrics
Dielectrics with induced electric dipole moments
Example: CH4
51P09 -
Dielectric in Capacitor
Potential difference decreases because dielectric polarization decreases Electric Field!
52P09 -
Gauss’s Law for Dielectrics
Upon inserting dielectric, a charge density ’ is induced at its surface
0S
inqd EA
E A
0
'
E
What is ’?
0
' A
53P09 -
Dielectric Constant Dielectric weakens original field by a factor
Gauss’s Law with dielectrics:
0
0 0
' EE
0S inqd AE
Dielectric constants
Vacuum 1.0
Paper 3.7
Pyrex Glass 5.6
Water 80
1
' 1
54P09 -
Dielectric in a CapacitorQ0= constant after battery is disconnected
0VV
Upon inserting a dielectric:
0 00
0 0/
Q QQC C
V V V
55P09 -
Dielectric in a Capacitor
V0 = constant when battery remains connected
Upon inserting a dielectric: 0Q Q
00
0
QQC C
V V