Transcript Lecture_9

Chapter 26
DC Circuits
Copyright © 2009 Pearson Education, Inc.
26-2 Resistors in Series and Parallel
Example 26-8:
Analyzing a circuit.

A 9.0-V battery whose
internal resistance r is
0.50 Ω is connected in
the circuit shown. (a)
How much current is

drawn from the
battery? (b) What is
the terminal voltage of  I
the battery? (c) What
is the current in the
6.0-Ω resistor?
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I
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Multiple configurations
Assume that in each
circuit the battery gives
12 V and each resistor
has a resistance of 4
ohms. In which circuit
does the largest current
flow through the
battery?
(e) All equal
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Multiple configurations
Assume that in each
circuit the battery gives
12 V and each resistor
has a resistance of 4
ohms. In which circuit
does the largest current
flow through the
battery? What is that
current?
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Multiple configurations
Assume that in each
circuit the battery gives
12 V and each resistor
has a resistance of 4
ohms. In which circuit
does the largest current
flow through the
battery? What is that
current?
1
1 1 2
  
R12 R R R






 R12  R 2
 R123  R12  R 
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

3R
V
2V 2 12
I


2A
2
R123 3 R 3 4
Circuit Maze
If each resistor has a
resistance of 4  and
each battery is a 4 V
battery, what is the
current flowing
through the resistor
labelled “R”?
(a) 0 A
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(b) 2 A
(c) 4 A
(d) 8 A
(e) 16 A
Circuit Maze
If each resistor has a
resistance of 4  and
each battery is a 4 V
battery, what is the
current flowing
through the resistor
labelled “R”?
I
V 8
 2A
R 4
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0  _______________________
(a) 0 A
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(b) 2 A
(c) 4 A
(d) 8 A
(e) 16 A
ConcepTest 26.6
Even More Circuits
1) R1
Which resistor has the
2) both R1 and R2 equally
greatest current going
through it? Assume that all
3) R3 and R4
the resistors are equal.
4) R5
5) all the same
V
ConcepTest 26.6
Even More Circuits
1) R1
Which resistor has the
2) both R1 and R2 equally
greatest current going
through it? Assume that all
3) R3 and R4
the resistors are equal.
4) R5
5) all the same
The same current must flow
through the left and right
combinations of resistors.
On the LEFT, the current
splits equally, so I1 = I2. On
the RIGHT, more current will
go through R5 than R3 + R4,
since the branch containing
R5 has less resistance.
V
Follow-up: Which one has the
smallest voltage drop?
ConcepTest 26.8a
Lightbulbs
Two lightbulbs operate at 120 V, but
1) the 25 W bulb
one has a power rating of 25 W while
2) the 100 W bulb
the other has a power rating of 100 W.
3) both have the same
Which one has the greater
4) this has nothing to do
with resistance
resistance?
ConcepTest 26.8a
Lightbulbs
Two lightbulbs operate at 120 V, but
1) the 25 W bulb
one has a power rating of 25 W while
2) the 100 W bulb
the other has a power rating of 100 W.
3) both have the same
Which one has the greater
4) this has nothing to do
with resistance
resistance?
Since P = V2 / R , the bulb with the lower
power rating has to have the higher
resistance.
Follow-up: Which one carries the greater current?
26-3 Kirchhoff’s Rules
Some circuits cannot be broken down into
series and parallel connections. For these
circuits we use Kirchhoff’s rules.
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26-3 Kirchhoff’s Rules
Junction rule: The sum of currents entering a
junction equals the sum of the currents
leaving it (i.e., charge does not pile up).
I in  I out
or 0  I in  I out
 I 3  I1  I 2
or 0  I 3  I1  I 2
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26-3 Kirchhoff’s Rules
Loop rule: The sum of
the changes in
potential around a
closed loop is zero.
Top:
0  45 V  I 3  1   40    I1  30  
 45  41 I 3  30 I1
 I1  1.500  1.367 I 3
Bottom: 0  45 V  I 3  1   40    80 V  I 2  1   20  
 125  41 I 3  21 I 2
 I 2  5.952  1.952 I 3
Node:
I 3  I1  I 2   1.500  1.367 I 3    5.952  1.952 I 3 

I 3  1  1.367  1.952   1.500  5.952
I 3  1.725 A

I1  1.500   1.367  1.725   0.859 A

I 2  5.952   1.952  1.725   2.584 A
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26-3 Kirchhoff’s Rules
Problem Solving: Kirchhoff’s Rules
1. Label each current, including its direction; don’t
worry if you get the direction wrong. The math will
take care of it.
2. Identify unknowns.
3. Apply junction and loop rules; you will need as
many independent equations as there are
unknowns. Each new equation MUST include a
new variable.
4. Solve the equations, being careful with signs. If the
solution for a current is negative, that current is in
the opposite direction from the one you have
chosen.
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26-3 Kirchhoff’s Rules
Example: Using Kirchhoff’s rules.
a) Calculate the currents (call them I1, I2, and
I3) through the three batteries of the circuit
in the figure.
b) What is Va-Vb?
1  2.0 V
 2   3  4.0 V
R1  1.0 
R2  2.0 
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26-3 Kirchhoff’s Rules

Left: 0  1  RI1  RI 2   2


Right: 0   2  RI 2  RI 3   3


Node: 0  I1  I 2  I 3
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R1  R1  2   R2  R
Note:
Potential change negative in specified
direction of current.
Potential change positive in specified
direction opposite to current.
26-3 Kirchhoff’s Rules

I1 
Left: 0  1  RI1  RI 2   2  I1  I 2 
1   2
R1  R1  2   R2  R
R
 
Right: 0   2  RI 2  RI 3   3  I 3  I 2  3 2
R
  
  


Node: 0  I1  I 2  I 3   I 2  1 2   I 2   I 2  3 2 
R 
R 


1     
1
1
 I2   2 1  2 3  
 4  2  4  4  A
3 R
R  3 2
3
 I1 
1  24
2
1  44 1



A
and
I


 A
3
3  2 
3
3  2  3
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Kirchhoff’s Rules
ConcepTest 26.12
1) 2 – I1 – 2I2 = 0
Which of the equations is valid
2) 2 – 2I1 – 2I2 – 4I3 = 0
for the circuit below?
3) 2 – I1 – 4 – 2I2 = 0
4) I3 – 4 – 2I2 + 6 = 0
5) 2 – I1 – 3I3 – 6 = 0
1
I2
2
6V
22 VV
4V
I1
1
I3
3
Kirchhoff’s Rules
ConcepTest 26.12
1) 2 – I1 – 2I2 = 0
Which of the equations is valid
2) 2 – 2I1 – 2I2 – 4I3 = 0
for the circuit below?
3) 2 – I1 – 4 – 2I2 = 0
4) I3 – 4 – 2I2 + 6 = 0
5) 2 – I1 – 3I3 – 6 = 0
Eq. 3 is valid for the left loop:
The left battery gives +2 V, then
there is a drop through a 1 
resistor with current I1 flowing.
Then we go through the middle
battery (but from + to – !), which
gives –4 V. Finally, there is a
drop through a 2  resistor with
current I2.
1
I2
2
6V
22 VV
4V
I1
1
I3
3
26-4 Series and Parallel EMFs;
Battery Charging
EMFs in series in the same direction: total
voltage is the sum of the separate voltages.
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26-4 Series and Parallel EMFs;
Battery Charging
EMFs in series, opposite direction: total
voltage is the difference, but the lowervoltage battery is charged.
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26-4 Series and Parallel EMFs;
Battery Charging
EMFs in parallel only make sense if the
voltages are the same; this arrangement can
produce more current than a single emf.
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26-5 Circuits Containing Resistor
and Capacitor (RC Circuits)
When the switch is
closed, the
capacitor will begin
to charge. As it
does, the voltage
across it increases,
and the current
through the resistor
decreases.
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 I t
Qt
 I t
26-5 Circuits Containing Resistor
and Capacitor (RC Circuits)
To find the voltage as a function of time, we
write the equation for the voltage changes
around the loop:
Qt
0 
 I t R
C
Since Q = dI/dt, we can integrate to find the
charge as a function of time:




Q  t   C 1  e  t RC  Qmax 1  e  t RC ; Q  t  0   0

 I  t   C  e
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 t RC

 1    t RC
 t RC

e

I
e
max

 R
R
C


26-5 Circuits Containing Resistor
and Capacitor (RC Circuits)
The voltage across the capacitor is VC = Q/C:
The quantity RC that appears in the exponent
is called the time constant of the circuit:
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26-5 Circuits Containing Resistor
and Capacitor (RC Circuits)
Example 26-11: RC circuit,
with emf.
The capacitance in the circuit shown
is C = 0.30 μF, the total resistance is
20 kΩ, and the battery emf is 12 V.
Determine (a) the time constant, (b)
the maximum charge the capacitor
could acquire, (c) the time it takes
for the charge to reach 99% of this
value, (d) the current I when the
charge Q is half its maximum value,
(e) the maximum current, and (f) the
charge Q when the current I is 0.20
its maximum value.
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26-5 Circuits Containing Resistor
and Capacitor (RC Circuits)
If an isolated charged
capacitor is
connected across a
resistor, it discharges:
0

Qt
 I t R
C
Q  t  dQ

RC
dt
 1
 R 


I t  
dQ
dt
 Q  t   Q0e  t RC  CV0e  t RC
Qt
 V0e  t RC
C

1   t RC V0  t RC
  C V0  
 e
e
R
R
C


VC  t  
I t
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Capacitor,
NOT battery!
I
26-5 Circuits Containing Resistor
and Capacitor (RC Circuits)
Example 26-12: Discharging RC circuit.
In the RC circuit shown, the battery has fully charged
the capacitor, so Q0 = CE. Then at t = 0 the switch is
thrown from position a to b. The battery emf is 20.0 V,
and the capacitance C = 1.02 μF. The current I is
observed to decrease to 0.50 of its initial value in 40
μs. (a) What is the value of Q, the charge on the
capacitor, at t = 0? (b) What is the value of R? (c) What
is Q at t = 60 μs?
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26-5 Circuits Containing Resistor
and Capacitor (RC Circuits)
Conceptual Example 26-13: Bulb in RC circuit.
In the circuit shown, the capacitor is originally
uncharged. Describe the behavior of the lightbulb
from the instant switch S is closed until a long time
later.
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26-6 Electric Hazards
Most people can “feel” a current of 1 mA; a
few mA of current begins to be painful.
Currents above 10 mA may cause
uncontrollable muscle contractions, making
rescue difficult. Currents around 100 mA
passing through the torso can cause death by
ventricular fibrillation.
Higher currents may not cause fibrillation, but
can cause severe burns.
Household voltage can be lethal if you are wet
and in good contact with the ground. Be
careful!
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26-6 Electric Hazards
A person receiving a
shock has become part
of a complete circuit.
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26-6 Electric Hazards
The safest plugs are those with three prongs;
they have a separate ground line.
Here is an example of household wiring – colors
can vary, though! Be sure you know which is the
hot wire before you do anything.
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26-7 Am-, Volt-, and Ohm-meters
An ammeter measures current; a voltmeter
measures voltage. Both are based on
galvanometers, unless they are digital.
The current in a circuit passes through the
ammeter; the ammeter should have low
resistance so as not to affect the current.
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26-7 Am-, Volt-, and Ohm-meters
Example 26-15: Ammeter design.
Design an ammeter to read 1.0 A at
full scale using a galvanometer with
a full-scale sensitivity of 50 μA and a
resistance r = 30 Ω. Check if the
scale is linear.
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26-7 Am-, Volt-, and Ohm-meters
A voltmeter should not affect the voltage across
the circuit element it is measuring; therefore its
resistance should be very large.
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26-7 Am-, Volt-, and Ohm-meters
Example 26-16: Voltmeter design.
Using a galvanometer with internal
resistance 30 Ω and full-scale
current sensitivity of 50 μA, design a
voltmeter that reads from 0 to 15 V. Is
the scale linear?
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26-7 Am-, Volt-, and Ohm-meters
Summary: An
ammeter must be in
series with the
current it is to
measure; a voltmeter
must be in parallel
with the voltage it is
to measure.
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26-7 Am-, Volt-, and Ohm-meters
An ohmmeter measures
resistance; it requires a
battery to provide a
current.
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Summary of Chapter 26
• A source of emf transforms energy from
some other form to electrical energy.
• A battery is a source of emf in parallel with an
internal resistance.
• Resistors in series:
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Summary of Chapter 26
• Resistors in parallel:
• Kirchhoff’s rules:
1. Sum of currents entering a junction
equals sum of currents leaving it.
2. Total potential difference around closed
loop is zero.
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Summary of Chapter 26
• RC circuit has a characteristic time constant:
• To avoid shocks, don’t allow your body to
become part of a complete circuit.
• Ammeter: measures current.
• Voltmeter: measures voltage.
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Questions?
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