•   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

•   What is the capacitance of a parallel plate capacitor of area A and separation d ?
•   Determine E:
1
E =
�
0
σ
•   Determine V in terms of Q
V = Ed =
1
�
0
σ d =
1
�
0
Qd
A
C =
Q
V
= �
0
A 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

•   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.

Question 29.3
Work and Potential Energy
Which group of charges took more work to bring together from a very large initial distance apart?
+1 d
+2 +1 d d
+1 +1
Both took the same amount of work.
d

•   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
W =
�
0 q dq =
Q
U = W =
C
Q
2
2
C
=
1
2
C
CV
2 dW
= q dq
Q 2
C
•   Then the work involved in charging up to a final value of Q is
2
•   This is therefore the electrostatic energy stored in the capacitor:

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….
+++++++++++++++++++++++
+Q d E
----------------------------------------- -Q
A:increases B:decreases C: stays same
©University of Colorado, Boulder

V
A parallel plate capacitor is attached to a battery which maintains a constant voltage difference V between the capacitor plates.
While the battery is attached, the plates are pulled apart. The electrostatic energy stored in the capacitor
A) increases
B) decreases
C) stays constant.

•   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
u
E
=
1
2
0
E 2
•   This is a universal result:
•   Every electric field contains energy with this density.
U =
1
2
CV
2 = u
E
Ad

•   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.
Q
1
+ Q
2
∆ V =
Q
1
C
1
= ∆ V ( C
1
C parallel
= C
1
=
Q
2
C
2
+ C
2
)
+ C
2

•   Capacitors connected in series are wired so that one capacitor follows the other.
•   Why are both capacitors charged the same amount?
∆
V
= ∆
V
1
= ⇒
+ ∆
V
2
=
Q
1
C
1
1
C
1
=
Q
( series
=
C
1
1
C
1
+
Q
2
+
C
2
1
)
+
C
2
1
C
2