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PS 250: Lecture 10 Capacitance and Stored Energy
J. B. Snively September 21, 2015
Today’s Class
Combinations of Capacitors Parallel Series
Energy Storage in Capacitors Dielectrics and Breakdown Summary
d
Area = a+
-
d
Area = 2a+
-
Parallel Capacitors
CN...C2C1
CEq = C1 + C2 + · · · CN
Parallel Equivalent Capacitance CEQ:
V+
-
Since: Q = CEq�V
Potential difference across capacitors is constant:Q = CEq�V = C1�V + C2�V + · · · CN�V
Series Capacitors
...CNC2C1
1CEq
=1C1
+1C2
+ · · · 1CN
Series Equivalent Capacitance CEQ:
V+
-
Charge across each capacitor is constant:�V =
Q
CEq=
Q
C1+
Q
C2+ · · · Q
CN
Today’s Class
Combinations of Capacitors Parallel Series
Energy Storage in Capacitors Dielectrics and Breakdown Summary
Energy Storage
Capacitors store energy (via charge), and can maintain a potential difference even after the
original voltage source is removed.
The energy stored in a capacitor can be calculated by equating it with the work required to charge that capacitor to
some potential difference.
Energy StorageWork required to transfer infinitesimal charge “dq”:
dW = �V dq =q
Cdq
Work required to charge from q=0 to q=Q:
W =� Q
0
q
Cdq =
Q2
2C
Potential energy stored in the charged capacitor = = Work required to charge capacitor =
U =Q2
2C=
12Q�V
And Since Q = C�V U =
12C(�V )2
Today’s Class
Combinations of Capacitors Parallel Series
Energy Storage in Capacitors Dielectrics and Breakdown Summary
Dielectric Properties
Insulating materials (“Dielectrics”) have different properties, which can be quantified by their permittivity relative to free space.
Insulating materials have a “Dielectric strength”, or maximum electric field threshold, beyond which they begin to conduct via breakdown.
Effects on Permittivity and Capacitance
d
Area = A+Q
-Q
+
-V
Dielectric Material� = ��o
C =�A
d= �
�oA
d= �Co
Capacitance dependent on modified dielectric constant:
Capacitance SummaryCapacitance is a property of electrodes, defining their ability to store charge given an applied potential difference.
Capacitors carrying stored charge also therefore store energy, which may be released (perhaps rapidly!)
Properties determined by the dielectric material between the conductors, which vary in permittivity and breakdown threshold.
Summary / Next Class:
Mastering Physics Due Wednesday
Homework Due Friday
Begin Reading Chapter 25!
