Chapter 25: Capacitance
Capacitors
Capacitors on a computer
motherboard
PHY2049: Chapter 25
1
Review and Overview
ÎCoulom’s
law and Gauss’ law
(equivalent)
Coulomb’s
law
Gauss’
law
E field ------ F=qE ------ Force F
(integrate)
(integrate)
(differentiate)
(differentiate)
Potential V ---- U=qV ---- Energy U
ÎApplications:
Capacitance (Ch. 25)
Electric circuits (Ch. 26)
PHY2049: Chapter 25
Current I=dq/dt
2
Subjects
ÎCapacitance:
definition and units
ÎCapacitance:
calculation
ÎCapacitors
ÎEnergy
in parallel and series
stored in electric field
ÎDielectrics
(insulators)
PHY2049: Chapter 25
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Definitions and Units
Î
Capacitor
Two conductors, electrically isolated from each other
‹ Particularly when the pair is used as device in electronic circuit to
store charge and for other purposes
‹
Î
Capacitance
q = CV
C is a constant that characterizes given pair of conductors in given
configuration
‹ For given (applied) V, larger capacitor stores more q
‹ For given q, smaller V appears in larger capacitor
‹
Î
Units
F (farad)
‹
1 F = 1 C/V (C: coulomb, not capacitance)
Note: ε0=8.854x10-12 C2/N m2=(same value) F/m
PHY2049: Chapter 25
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Why do we consider only +q and –q
forming a pair?
Let us postpone the question for the moment.
PHY2049: Chapter 25
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Capacitance calculation 1: parallel plates
q
∫SE ⋅ dA = EA + 0 = ε 0
1 q
E=
ε0 A
Gauss’ law
Solve for E
1 qd
V = − ∫ E ⋅ ds = Ed =
−
ε0 A
+
Potential difference (Do not worry about sign.
+ is always high.)
C=
q
A
= ε0
V
d
Capacitance
[capacitance]=ε0 [length]
PHY2049: Chapter 25
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Capacitance calculation 2: coaxial cylinders
Ignore ends, approximating
cylinder to be infinitely long
∫ E ⋅ dA = E (2πrL) + 0 =
S
E=
q
ε0
1 q
2πε 0 rL
1 q b dr
V = − ∫ E ⋅ ds =
−
2πε 0 L ∫a r
Gauss’ law
Solve for E
+
b
lnb − lna = ln 
a
Potential difference (Do not worry about sign.
+ is always high.)
Radii a and b.
q
L
◊ [capacitance]=ε0 [length]
C = = 2πε 0
V
ln (b a ) ◊ Depends only weakly on radii
◊ Inner conductor cannot be approximated to
Capacitance
be line with no thickness. Then C=∞.
PHY2049: Chapter 25
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Why do we consider only +q and –q
forming a pair, or why do they always
occur as a pair?
Î
Charging capacitor
Battery just moves electrons from
one side to the other until potential
difference across capacitor reaches
battery’s emf (aka “voltage”) V.
Does not add or remove charge.
Î
In general
In electronic circuits, capacitors are used in such ways that +q
and –q occur as pairs.
PHY2049: Chapter 25
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Capacitors in parallel (derivation of formula)
Î
Three capacitors in parallel are charged by battery or
power supply
q =q1 + q2 + q3
Analogy: three glasses filled
with water.
= VC1 + VC2 + VC3
q = VCeq
Definition of Ceq
Ceq = C1 + C2 + C3
Î
Generalize to more than three capacitors
Ceq = C1 + C2 + C3 + ……
PHY2049: Chapter 25
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Capacitors in series (derivation of formula)
Î
Two capacitors in series are charged by battery or power
supply
no charge
+q
no charge
still no net
charge
–q
no charge
before
Î
after
Induced charges appear immediately
+q
– q’ attracted
+q’ attracted
–q
after
If q’ ≠q, electric fields would
not be confined in capacitors.
In particular, there would be E
in connecting wire. Then
charges would move. → q’=q
in steady state.
PHY2049: Chapter 25
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(continued)
Î
Potential difference
C1
+q
– q V1
+q
C2
V2
–q
Electric fields are confined in capacitors.
→ Potential differences are present only in
capacitors.
V=V1+ V2=q/C1 + q/C2
V=q/Ceq
Definition of Ceq
q/Ceq =q/C1 + q/C2
1/Ceq =1/C1 + 1/C2
Î
Generalize to more than two capacitors in series
1/Ceq =1/C1 + 1/C2 + ……
PHY2049: Chapter 25
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