Lecture 23
• Conductor in equilibrium
• Capacitance
• Combinations of capacitors (series and parallel)
• Energy stored in capacitor (electric field)
Conductor in Equilibrium
•
E = 0 inside conductor in equilibrium
(otherwise, charges would move)
•
Entire conductor at same V, surface is
equipotential Ē ⊥ to surface
Capacitance
(a) ∆Vc = 0 (b)∆Vwire = 0
V = Ed; E =
!0 A
C≡ d ⇒
Q
!0 A ;
Units of C:
1 farad = 1 F ≡ 1 C/V
C geometric property
(of any two electrodes)
Combinations...
Capacitors in Parallel
•
same ∆Vc ⇒
∆Q
Ceq = ∆VC =
=
Q1
∆VC
+
Q1 +Q2
∆VC
Q1
∆VC
Capacitors in Series
•
same charge Q
1
Ceq
=
∆VC
Q
=
∆V1
Q
=
+
∆V1 +∆V2
Q
∆V2
Q
•
Circuit analysis
combine elements into single
equivalent; reverse process to
calculate for each element
Energy Stored in Capacitor (Electric Field)
•
•
Potential energy of d q + capacitor increases by dU = dq∆V =
•
•
like spring (1/2k (∆x)2): discharged/released, potential to kinetic...
qdq
C
total energy transferred from battery to capacitor:
!
Q
2
Q2
1
1
UC = C 0 qdq = 2C = 2 C (∆VC )
Energy stored in E
(real!): using ∆VC = Ed and C = !0 A/d, UC =
energy stored
uE =
=
volume stored in
Uc
Ad
=
!0
2
(Ad) E 2
!0 2
2 E
Dielectrics I
•
vacuum filled charged; battery
disconnected; fill with dielectric:
∆VC < (∆VC )0 ⇒ C =
•
Q
∆VC
>
Q0
(∆VC )0
= C0
2 steps: polarization (dipoles align
separation of charge
E induced)
Dielectrics II: superposition of E
Ē = Ē0 + Ēinduced = (E0 − Einduced , from positive to negative)
E0
dielectric constant: κ ≡ E (≥ 1; κvacuum = 1)
(∆VC )0
E0
∆VC = Ed = κ d =
⇒
κ
Q
Q0
Q0
C = ∆VC = (∆Vc ) /κ = κ (∆Vc ) = κC0 (increases!)
0
0
•
if battery kept connected: ∆VC fixed, more charge flows: Q = κQ0