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Capacitors
A device storing electrical energy
Capacitor
A potential across connected plates causes
charge migration until equilibrium
DV
– –+ +
– –+ +
– –+ +
–q
–
+
+q
Charge stored
q = CDV
C = capacitance
Unit = C/V = farad = F
Parallel Plate Capacitance
Plate area A, separation d
A
d
Capacitance = Ae0/d
e0 = 8.8510–12
C2
N m2
Circuit Element Symbols
• Potential Source
+ –
DV
• Conductor
• Capacitor
• Resistor
or
At Equilibrium
DV
+ –
C
+ –
DV
• Capacitor charges to
potential DV
• Capacitor charge
Q = CDV
Energy in a Capacitor
• C = Q/DV so DV = Q/C
• Work to push charge DQ
W = DVDQ = (Q/C)DQ
DV
DQ
slope = 1/C
area = W
Q
Energy in a Capacitor
• Work to charge to Q is area of triangle
W = 1/2 Q(Q/C) = 1/2 Q2/C
• Work to charge to DV
W = 1/2 DV (CDV) = 1/2 C(DV)2
DV
Q/C
Q
CDV
Combining Capacitors
Parallel
and
Series
Parallel Components
• All have the same potential difference
• Capacitances add
• (conceptually add A’s)
Series Capacitors
• All have the same charge separation
• Reciprocals are additive
• (conceptually add d’s)
Gauss’s Law
• Electric flux through a closed shell is
proportional to the charge it encloses.
FE = Qin/e0
• e0 = 8.8510–12
C2
N m2
Field Around Infinite Plate
With uniform charge density s = Q/A
1 s
sA
= E(2A) , so E =
FE =
2 e0
e0
Infinite ||-Plate capacitor
Individually
Together
–q
–q
1/2 s/e0
+q
+q
0
0
s/e0
1/2 s/e0
Charge of a Capacitor
• Parallel plates of opposite charge
• Charge density s = Q/A
–
+
Fields cancel outside
Fields cancel outside
s/e0
d
Potential DV = d s/e0
= d Q/(Ae0)
Capacitance C = Q/DV
= e0 A/d
Parallel Plate Capacitance
• Plate area A, plate separation d
Q
s
• Field E =
=
e0
Ae0
Qd
• Potential DV = Ed =
Ae0
Q Ae0
Ae0
=
• Capacitance Q/DV =
Qd
d
Capacitor with a Dielectric
• If capacitance without dielectric is C,
dielectric is kC.
• k = dielectric constant
k
Dielectric Parameters
• Dielectric constant k
– Dielectric permittivity e = ke0
• Breakdown voltage
– Actually field V/m