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 magnitude of
the 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:
1
E= σ
�0
•  Determine V in terms of Q
1
1 Qd
V = Ed = σd =
�0
�0 A
Q
A
C=
= �0
V
d
•  Capacitance is always independent of voltage and charge!
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?
L
d
A: no change
B: up by 2.
C: up by 4.
D: up by 8
E: none of these
©University of Colorado, Boulder
CT 29.C3
How big is a one Farad Capacitor?
Assume you have two parallel plates that
are separated by 1 mm. If you estimate ε0
~ 10-11, what is the area of the plates to
have a 1.0 Farad capacitance?
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.
©University of Colorado, Boulder
Example - Spherical capacitor
•  What is the capacitance of a spherical capacitor,
consisting of a a inner shell of radius ra and outer radius
r b?
Question 29.3 Work and Potential Energy
Which group of charges took more work to bring
together from a very large initial distance apart?
+2
d
+1
+1
d
+1
Both took the same amount of work.
d
d
+1
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
q
•  For a capacitor, q =C V(q), so dW =
dq
C
•  Then the work involved in charging up to a final value of Q is
W =
�
0
Q
q
Q2
dq =
C
2C
•  This is therefore the electrostatic energy stored in the capacitor:
Q2
1
U =W =
= CV 2
2C
2
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.
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.
1
U = CV 2 = uE Ad
2
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….
+++++++++++++++++++++++
d
+Q
E
-----------------------------------------
-Q
A:increases B:decreases C: stays same
©University of Colorado, Boulder
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.
Q1
Q2
∆V =
=
C1
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?
Q1
Q2
∆V = ∆V1 + ∆V2 =
+
C1
C2
1
1
+
)
= Q(
C1
C2
1
1
1
=⇒
=
+
Cseries
C1
C2