Module 10: Capacitance
P09 - 1
Capacitors and Capacitance
Our first of 3 standard electronics devices
(Capacitors, Resistors & Inductors)
P09 - 2
Capacitors:
p
Store Electric Charge
g
Capacitor: Two isolated conductors
Equal and opposite charges ±Q
Potential difference ΔV between them.
Q
C=
ΔV
U it C
Units:
Coulombs/Volt
l b /V lt or
Farads
C is Always Positive
P09 - 3
Parallel Plate Capacitor
p
E =0
+Q = σ A
E =?
d
E =0
− Q = −σ A
P09 - 4
Parallel Plate Capacitor
p
Oppositely charged plates:
Charges move to inner surfaces to get close
Link to Capacitor Applet
P09 - 5
Calculating
g E (Gauss’s
(
Law))
r r qin
∫∫ E ⋅ dA =
S
ε0
σ AGauss
E (AGauss )=
ε0
σ
Q
E= =
ε 0 Aε 0
Note: We only “consider” a single sheet!
Doesn’t the other sheet matter?
P09 - 6
Alternate Calculation Method
Top Sheet:
σ
E=−
2ε 0
++++++++++++++
σ
E=
2ε 0
Bottom Sheet:
σ
E=
2ε 0
- - - - - - - -- - - - - -
σ
E=−
2ε 0
σ
σ
σ
Q
E=
+
= =
2ε 0 2ε 0 ε 0 Aε 0
P09 - 7
Parallel Plate Capacitor
p
r r
Q
ΔV = − ∫ E ⋅ dS = Ed =
d
Aε 0
bottom
top
ε0 A
Q
C=
=
d
ΔV
C depends only on geometric factors A and d
P09 - 8
Concept Question
Questions:
Ch
Changing
i C Dimensions
Di
i
P09 - 9
Concept Question: Changing
g g Dimensions
A parallel-plate capacitor has plates with equal and
opposite charges ±Q,
±Q separated by a distance d,
d and
is not connected to a battery. The plates are pulled
apart to a distance D > d.
d What happens?
1.
2
2.
3.
4
4.
5.
6
6.
7.
8.
9.
V increases, Q increases
V decreases,
d
Q increases
i
V is the same, Q increases
V increases,Q
i
Q is
i th
the same
V decreases, Q is thesame
V iis the
h same, Q iis the
h same
V increases, Q decreases
V decreases, Q decreases
V is the same,Q decreases
P09 - 10
Concept Question: Changing
g g Dimensions
A parallel-plate capacitor has plates with equal and
opposite charges ±Q,
±Q separated by a distance d,
d and
is connected to a battery. The plates are pulled apart
to a distance D > d.
d What happens?
1.
2
2.
3.
4.
5.
6.
7.
8.
9
9.
V increases, Q increases
V decreases
decreases, Q increases
V is the same, Q increases
V increases,, Q is the same
V decreases, Q is the same
V is the same, Q is the same
V iincreases,
Q decreases
d
V decreases, Q decreases
V is the same
same, Q decreases
P09 - 11
Demonstration:
Changing
g g C Dimensions
P09 - 12
Capacitors:
p
Review
Capacitor: Two isolated conductors
Equal and opposite charges ±Q
Potential difference ΔV between them.
Q
C=
ΔV
U it C
Units:
Coulombs/Volt
l b /V lt or
Farads
C is Always Positive
P09 - 13
Group
p Problem: Spherical
p
Shells
These two spherical
p
shells have equal
pp
charge.
g
but opposite
Find E everywhere
Find V everywhere
(assume V(∞) = 0)
P09 - 14
Spherical
p
Capacitor
p
Two concentric spherical shells of radii a and b
What is E?
Gauss’s Law Æ E ≠ 0 only for a < r < b,
where it looks like a point charge:
r
E=
Q
4πε 0 r
ˆ
r
2
P09 - 15
Spherical
p
Capacitor
p
b
r r
Qrˆ
Q ⎛ 1 1⎞
ˆ
ΔV = − ∫ E ⋅ dS = − ∫
⋅
dr
r
=
− ⎟
2
⎜
4πε 0 r
4πε 0 ⎝ b a ⎠
inside
a
outside
I this
Is
thi positive
iti or negative?
ti ? Wh
Why?
?
4πε 0
Q
C=
= −11
−11
ΔV
a −b
(
)
For an isolated spherical conductor of radius a:
C = 4πε 0 a
P09 - 16
Capacitance
p
of Earth
For an isolated spherical conductor of radius a:
C = 4πε 0 a
ε 0 = 8.85 ×10
−12
Fm
a = 6.4 × 10 m
6
−4
C = 7 ×10 F = 0.7 mF
A Farad is REALLY BIG! We usually use pF (10-12) or nF (10-9)
P09 - 17
Energy
gy Stored in Capacitor
p
Start charging capacitor
P09 - 18
Energy
gy To Charge
g Capacitor
p
+q
q
-q
q
1. Capacitor starts uncharged
1
uncharged.
2. Carry +dq from bottom to top.
Now top has charge q = +dq, bottom -dq
3. Repeat
p
4. Finish when top has charge q = +Q, bottom
-Q
P09 - 19
Work Done Charging
g g Capacitor
p
At some point top plate has +q,
+q bottom has –q
q
Potential difference is ΔV = q / C
Work done lifting another dq is dW = dq ΔV
+q
-q
P09 - 20
Work Done Charging
g g Capacitor
p
So work done to move dq is:
q 1
dW = dq
q ΔV = dq = q dq
C C
Total energy to charge to q = Q:
Q
1
W = ∫ dW = ∫ q dq
d
C0
+q
2
1Q
=
C 2
-q
q
P09 - 21
Energy
gy Stored in Capacitor
p
Q
Si
Since
C=
ΔV
2
Q
1
1
U=
= Q ΔV = C ΔV
2C 2
2
2
Where is the energy stored???
P09 - 22
Energy
gy Stored in Capacitor
p
Energy stored in the E field!
Parallel-plate capacitor:
1
1 εo A
2
U = CV =
Ed
2
2 d
( )=
2
C=
εo E
2
εo A
d
and V = Ed
2
× ( Ad) = uE × (volume)
uE = E field energy
gy densityy =
εo E
2
2
P09 - 23
Concept Question Question:
Changing C Dimensions
Energy Stored
P09 - 24
Concept Question: Changing
Dimensions
A parallel-plate capacitor, disconnected from a
battery, has plates with equal and opposite
charges,
g
separated
p
by
y a distance d.
Suppose the plates are pulled apart until separated
by a distance D > d.
d
How does the final electrostatic energy stored in
th capacitor
the
it compare to
t the
th initial
i iti l energy?
?
1. The final stored energy is smaller
1
2. The final stored energy is larger
3. Stored energy does not change.
P09 - 25
Demonstration:
Big Capacitor
E l di a Wire
Exploding
Wi
P09 - 26
MIT OpenCourseWare
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8.02SC Physics II: Electricity and Magnetism
Fall 2010
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