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Chapter 23 - Capacitors
Capacitors • A capacitor is a pair of conductors, insulated from each
other, and used to store charge and energy. • For a “charged” capacitor, one conductor is positively charged
and the other is negatively charged (net charge is always zero). • The work used in separating charge is stored as electrostatic
energy in the capacitor.
• Capacitance is the charge stored per unit potential difference: C = Q/V.
• V refers to voltage difference between conductors
• Its SI unit is the farad (F): • 1 F = 1 C/V
Parallel Plate Capacitor • What is the capacitance of a parallel plate capacitor of
area A and separation d? • Determine E:
• Determine V in terms of Q
• Capacitance is always independent of voltage and charge!
E =1�0
σ
V = Ed =1�0
σd =1�0
Qd
A
C =Q
V= �0
A
d
CT 29.C2
A parallel-plate capacitor has square plates of edge length L, separated by a distance d. If we double the dimension L and halve the dimension d, by what factor have we changed the capacitance?
A: no change B: up by 2. C: up by 4. D: up by 8 E: none of these
L
d
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Example - Spherical capacitor • What is the capacitance of a spherical capacitor,
consisting of a a inner shell of radius ra and outer radius rb?
Which group of charges took more work to bring together from a very large initial distance apart?
+1
+1
+1
d d
d
+1 +2 d
Both took the same amount of work.
Question 29.3 Work and Potential Energy
Energy stored in a capacitor • Charging a capacitor involves transferring charge
between the initially neutral plates.
• The work dW involved in moving charge dq is dW=V(q)dq • For a capacitor, q =C V(q), so
• Then the work involved in charging up to a final value of Q is
• This is therefore the electrostatic energy stored in the capacitor:
dW =q
Cdq
W =� Q
0
q
Cdq =
Q2
2C
U = W =Q2
2C=
12CV 2
CT 29.C3
How big is a one Farad Capacitor?
A) 100 million square meters B) 1 thousand square meters C) 1 square meters D) 0.001 square meters E) None of these is even close.
Assume you have two parallel plates that are separated by 1 mm. If you estimate e0
~ 10-11, what is the area of the plates to have a 1.0 Farad capacitance?
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Practical capacitors • Capacitors are manufactured
using a variety of technologies, in capacitances ranging from picofarads (pF; 10–12 F) to several farads. • Most use a dielectric material
between their plates.
• The dielectric increases capacitance by lowering the electric field and thus the potential difference required for a given charge on the capacitor.
• The dielectric constant, is a property of the dielectric material that gives the reduction in field and thus the increase in capacitance.
Connecting capacitors: parallel • Capacitors connected in parallel have their top plates
connected together and their bottom plates connected together. • Therefore the potential
difference across the two capacitors is the same.
∆V =Q1
C1=
Q2
C2
Q1 + Q2 = ∆V (C1 + C2)Cparallel = C1 + C2
Connecting capacitors: series • Capacitors connected in series are wired so that one
capacitor follows the other. • Why are both capacitors
charged the same amount?
∆V = ∆V1 + ∆V2 =Q1
C1+
Q2
C2
= Q(1C1
+1C2
)
=⇒ 1Cseries
=1C1
+1C2
Energy in the electric field • The electrostatic energy associated with a charge
distribution is stored in the electric field of the charge distribution. • Considering the uniform field of the
parallel-plate capacitor implies that the electric energy density is
• This is a universal result: • Every electric field contains
energy with this density. U =12CV 2 = uEAd
CT 29.C4
A parallel plate capacitor is charged (the plates are isolated so Q cannot change.) The plates are then pulled apart so that the plate separation d increases. The total electrostatic energy stored in the capacitor….
A:increases B:decreases C: stays same
+++++++++++++++++++++++
-----------------------------------------
+Q
-Q E d
(Hint: Did the person pulling the plates apart do positive work, negative work or no work?)
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