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Chapter 19
DC Circuits
Objectives: The student will be able
to:
• Determine the equivalent capacitance of
capacitors arranged in series or in parallel
or the equivalent capacitance of a series
parallel combination.
• Determine the charge on each capacitor
and the voltage drop across each
capacitor in a circuit where capacitors are
arranged in series, parallel, or a series
parallel combination.
19.4 EMFs in Series and in Parallel;
Charging a Battery
EMFs in series in the same direction: total
voltage is the sum of the separate voltages
19.4 EMFs in Series and in Parallel;
Charging a Battery
EMFs in series, opposite direction: total
voltage is the difference, but the lowervoltage battery is charged.
EMF’s in Series and in Parallel: Charging a Battery
If you put batteries in series the “right way,” their voltages
add:
+
6V
=
3V
9V
If you put batteries in series the “wrong way,” their voltages
add algebraically:
+
=
magnitudes only 
6V
3V
chosen loop direction
-6 V
+3 V
3V
-3 V  algebraically,
using chosen loop
direction
Algebraic addition of voltages for batteries in series comes
directly from Kirchoff’s loop rule.
This applies to any source of emf, not just batteries!
Why would you want to connect batteries in series?
More voltage! Brighter flashlights, etc. Chemical reactions in
batteries yield a fixed voltage. Without changing the chemical
reaction (i.e., inventing a new battery type), the only way to
change voltage is to connect batteries in series.
Go to www.howstuffworks.com to see how batteries work.
They even expose the secret of the 9 volt battery!
Click on the picture above only if you are mature enough to
handle this graphic exposé.
Go to www.howstuffworks.com to see how batteries work.
They even expose the secret of the 9 volt battery!
Shocking!
Six 1.5 V batteries in series!
Why would you want to connect batteries in series the
“wrong” way?
You probably don’t want to.
Use could use one battery to charge another—doesn’t seem
too useful, although might be in special cases.
But remember, Kirchoff’s loop rule applies to all emf’s.
You could connect a source of emf – like the alternator in your
car – so that it charges a battery.
Rechargeable batteries use an ac to dc converter as a source of
emf for recharging.
Could you connect batteries (or sources of emf) in parallel?
Sure!
a
b
3V
3V
You would still have a 3 V voltage drop across your resistor,
but the two batteries in parallel would “last” longer than a
single battery.
You could use Kirchoff’s rules to analyze this circuit and show
that Vab = 3 V.
19.4 EMFs in Series and in Parallel;
Charging a Battery
EMFs in parallel only make sense if the voltages are the same;
this arrangement can produce more current than a single emf.
It is used to provide more energy when large currents
are needed. Each of the cells in parallel has to
produce only a fraction of the total current, so the
energy loss due to internal resistance is less than for
a single cell; and the batteries will go dead less quickly.
Series
and
Parallel
EMFs;
Battery
I
Charging
Example 26-10: Jump
starting a car.
A good car battery is being used to
jump start a car with a weak battery.
The good battery has an emf of 12.5
V and internal resistance 0.020 Ω.
Suppose the weak battery has an emf
of 10.1 V and internal resistance 0.10
Ω. Each copper jumper cable is 3.0 m
long and 0.50 cm in diameter, and can
be attached as shown. Assume the
starter motor can be represented as a
resistor Rs = 0.15 Ω. Determine the
current through the starter motor (a)
if only the weak battery is connected
to it, and (b) if the good battery is
also connected.
I
Answer to Example 19-9
I
Answer to Example 19-9
I
Answer to Example 19-9
19.5 Circuits Containing Capacitors in
Series and in Parallel
Capacitors in
parallel have the
same voltage across
each one:
Circuits Containing Capacitors in Series and in
Parallel
Vab
Capacitor:
C
Capacitors connected in parallel:a
C1
C2
b
C2
+ V
The voltage drop from a to b must equal V.
Vab = V = voltage drop across each individual
capacitor.
C1
Q1
Q=CV
 Q1 = C1 V
& Q2 = C2 V
& Q3 = C3 V
a
+
Q2
C2 -
C3
Q3
+ V
Now imagine replacing the parallel
combination of capacitors by a
single equivalent capacitor.
By “equivalent,” we mean “stores
the same total charge if the voltage
is the same.”
Q1 + Q2 + Q3 = Ceq V = Q
a
Ceq
Q
+ V
Summarizing the equations on the last slide:
Q1 = C1 V
Q2 = C2 V
Q3 = C3 V
C1
C2
a
Q1 + Q2 + Q3 = Ceq V
C2
Using Q1 = C1V, etc., in the second line gives
+ -
C1V + C2V + C3V = Ceq V
C1 + C2 + C3 = Ceq
V
(after dividing both sides by V)
Generalizing:
Ceq = Ci
(capacitors in parallel)
Does this remind you of any of our resistor equations?
See Giancoli’s comment on why this makes sense, p. 533.
b
19.5 Circuits Containing Capacitors in
Series and in Parallel
In this case, the total capacitance is the sum:
(19-5)
Capacitors connected in series:
C1
C2
C3
+ +Q V -Q
An amount of charge +Q flows from the battery to the left plate
of C1. (Of course, the charge doesn’t all flow at once).
An amount of charge -Q flows from the battery to the right
plate of C3. Note that +Q and –Q must be the same in
magnitude but of opposite sign.
The charges +Q and –Q attract equal and opposite charges to
the other plates of their respective capacitors:
C1
+Q -Q
A
C2
+Q -Q
B
C3
+Q -Q
+ V
These equal and opposite charges came from the originally
neutral circuit regions A and B.
Because region A must be neutral, there must be a charge +Q
on the left plate of C2.
Because region B must be neutral, there must be a charge --Q
on the right plate of C2.
Here’s the circuit after the charges have moved and a steady
state condition has been reached:
a
C1
+Q -Q
V1
C2
A
+Q -Q
V2
C3
B
b
+Q -Q
V3
+ V
The charges on C1, C2, and C3 are the same, and are
Q = C1 V1
Q = C2 V2
Q = C3 V3
But we don’t know V1, V2, and V3 yet.
We do know that Vab = V and also Vab = V1 + V2 + V3.
Let’s replace the three capacitors by a single equivalent
capacitor.
Ceq
+Q -Q
V
+ V
By “equivalent” we mean V is the same as the total voltage
drop across the three capacitors, and the amount of charge Q
that flowed out of the battery is the same as when there were
three capacitors.
Q = Ceq V
Collecting equations:
Q = C1 V1
Q = C2 V2
Q = C3 V3
Vab = V = V1 + V2 + V3.
Q = Ceq V
Substituting for V1, V2, and V3:
Q
Q
Q
V=
+
+
C1 C 2 C 3
Substituting for V:
Q
Q
Q
Q
=
+
+
Ceq C1 C2 C3
Dividing both sides by Q:
1
1
1
1
=
+
+
Ceq C1 C2 C3
19.5 Circuits Containing Capacitors in
Series and in Parallel
In this case, the reciprocals of the
capacitances add to give the reciprocal of the
equivalent capacitance:
(19-6)
Practice Problem 1 – Similar to p. 549 #35
Six 4.5 μF capacitors are connected in parallel
and in series. Find equivalent capacitance for
each case.
Practice Problem 1 – Answer
Six 4.5 μF capacitors are connected in parallel and in
series. Find equivalent capacitance for each case.
Virtual Capacitor Lab
Homework:
Chapter 19: 34, 37, 41, 42, 43, 44
Due in 2 days
Kahoot 19-4 and 19-5