PES 1120 Spring 2014, Spendier
Lecture 16/Page 1
Today:
- Review last time
- Capacitors in series and parallel
- Capacitors in Circuits
Capacitors in Circuits
Last lecture comment: Shouldn’t we connect the parallel-plate capacitor in a loop? Yes!
We need to start to talk about circuits. Capacitors are used in circuits extensively.
Charging a capacitor:
The positive terminal of the battery has a higher potential than the negative terminal. In
this respect the battery stores electrical energy by maintaining a potential difference via
electrochemical reactions. The battery does work in order to put charge on the plates on
the capacitor. The capacitor stores energy through the potential difference due to the
stored charge on thee plates.
The energy in the capacitor can be released very quickly by “discharging” it, whereas the
battery releases energy slowly.
What happens when the switch in the diagram is closed?
When the switch is closed, the capacitor begins to accumulate charge. Before this the
plates have no net charge. Electrons (the only charges that can move through the wires)
are attracted from the left plate l to the positive terminal of the battery
 l builds up a positive charge
Electrons are also repelled from the negative terminal of the battery and accumulate on
the right plate r
 plate r builds up a negative charge.
When V(- terminal) = V(r)
And (V+ terminal) = V(l)
Then there is no more force causing the electrons to move. (Electrons only move to
decrease their potential energy.)
Last lecture we learned that the amount of charge that can accumulate on the plate is
given by
Q = C (Vl-Vr)
PES 1120 Spring 2014, Spendier
Lecture 16/Page 2
Obviously a larger battery can supply more charge.
C….capacitance units [C/V] = [F] Farad
C
How much ch arg e stored
Q

Re lated to work done to chrage plates Vab
C 
Unit     F ....( Farad )
 V 
Last lecture we calculated capacitance for different geometries:
eA
Q
 0
Vab
d
rr
Spherical capacitor: C  4pe0 a b
(made mistake last lecture, please check!!!!)
rb  ra
Parallel-plate capacitor: C 
Vab = Va - Vb = potential of the positively charged conductor a with respect to the
negatively charged conductor b
Question last class – which geometry is more efficient in storing energy? Let’s do an
example:
PES 1120 Spring 2014, Spendier
Lecture 16/Page 3
Example 1: The plates of a spherical capacitor have radii 38.0 mm and 40.0 mm
a) Calculate the capacitance
b) What must be the plate area of a parallel-plate capacitor with the same plate separation
and capacitance?
c) Now compare area found in b) to surface area for sphere...
c)
ra2
Changing Capacitance
Since a single capacitor has such a small capacitance, we are often interested in ways of
changing its value.
To change the capacitance value:
(1) Different shape plates (spherical versus parallel-plate capacitor see above).
(2) Connecting individual capacitors together.
(3) Inserting an insulator between the plates.
The remainder of this lecture we will focus on (2)
Capacitor Combinations
Combinations of capacitors behave like a single equivalent capacitor, Ceq. Why do we
need to understand this? Capacitors are manufactured with certain standard capacitances
and working voltages. However, these standard values may not be the ones you actually
need in a particular application. You can obtain the values you need by combining
capacitors; many combinations are possible, but the simplest combinations are a series
connection and parallel connection.
PES 1120 Spring 2014, Spendier
Lecture 16/Page 4
Connect in Parallel:
–Have the same potential
V1 = V2 = V
–Charge depends on capacitance (Charge redistributes between capacitors - think about
water in a pipe - when it comes to an intersection of two pipes the water has to divide
between both)
Q = Q1 +Q2
or
Q1 = C1V
Q2 = C2V
Q Q1  Q2 Q1 Q2

 
 C1  C2
V
V
V
V
This works no matter how many capacitors we add in series. For n capacitors connected
in series:
Ceq 
n
Ceq   Ci
i 1
The equivalent capacitance of a parallel combination equals the sum of the individual
capacitances. In parallel connection the equivalent capacitance is always greater than any
individual capacitance.
PES 1120 Spring 2014, Spendier
Lecture 16/Page 5
Capacitors in Series
Connect in Series:
– Have the same charge
Q1 = Q2 = Q
– Potential difference across them add
V1 +V2 = V
Why?
V  Va  Vb  Va  Vc   Vc  Vb   V1  V2
If you wanted to replace these capacitors with just one equivalent capacitor:
Q
Q

V V1  V2
1
V  V2 V1 V2
 1
 
Ceq
Q
Q Q
Ceq 
1
1
1
 
Ceq C1 C2
This works no matter how many capacitors we add in parallel. For n capacitors connected
in parallel:
n
1
1

Ceq
i 1 Ci
The reciprocal of the equivalent capacitance of a series combination equals the sum of the
reciprocals of the individual capacitances. In a series connection the equivalent
capacitance is always less than any individual capacitance.
PES 1120 Spring 2014, Spendier
Lecture 16/Page 6
Example 2: Let C1 = 6.0 μF, C2 = 3.0 μF, and Vab = 18 V. Find the equivalent i)
capacitance, and find ii) the charge and iii) potential difference for each capacitor when
the two capacitors are connected (a) in series and (b) in parallel.
PES 1120 Spring 2014, Spendier
Lecture 16/Page 7
Capacitor Circuits
When there are combinations of series and parallel capacitors, try to reduce them to more
simple forms.
Example 3: Find the equivalent capacitor of a)