Direct Current Circuits

1
Chapter 18

• Direct Current Circuits

2
DC Circuits
3
Sources of emf

• The source that maintains the current in a closed
  circuit is called a source of emf
• Any devices that increase the potential energy of
  charges circulating in circuits are sources of
  emf
• Examples include batteries and generators
• SI units are Volts
• The emf is the work done per unit charge

4
EMF and Terminal Voltage
This resistance behaves as though it were in
series with the emf.
5
emf and Internal Resistance

• A real battery has some internal resistance
• Therefore, the terminal voltage is not equal to
  the emf

6
More About Internal Resistance

• The schematic shows the internal resistance, r
• The terminal voltage is ?V Vb-Va
• ?V e Ir
• For the entire circuit, e IR Ir

7
Internal Resistance and emf, cont

• e is equal to the terminal voltage when the
  current is zero
• Also called the open-circuit voltage
• R is called the load resistance
• The current depends on both the resistance
  external to the battery and the internal
  resistance

8
Internal Resistance and emf, final

• When R gtgt r, r can be ignored
• Generally assumed in problems
• Power relationship
• I e I2 R I2 r
• When R gtgt r, most of the power delivered by the
  battery is transferred to the load resistor

9
Resistors in Series

• When two or more resistors are connected
  end-to-end, they are said to be in series
• The current is the same in all resistors because
  any charge that flows through one resistor flows
  through the other
• The sum of the potential differences across the
  resistors is equal to the total potential
  difference across the combination

10
A series connection has a single path from the
battery, through each circuit element in turn,
then back to the battery.
11
Resistors in Series, cont

• Potentials add
• ?V IR1 IR2 I (R1R2)
• Consequence of Conservation of Energy
• The equivalent resistance has the effect on the
  circuit as the original combination of resistors

12
Equivalent Resistance Series

• Req R1 R2 R3
• The equivalent resistance of a series combination
  of resistors is the algebraic sum of the
  individual resistances and is always greater than
  any of the individual resistors

13
Equivalent Resistance Series An Example

• Four resistors are replaced with their equivalent
  resistance

14

• The current through each resistor is the same.
• The voltage depends on the resistance.
• The sum of the voltage drops across the resistors
  equals the battery voltage.

15
1) 12 V 2) zero 3) 3 V 4) 4 V 5) you
need to know the actual value of R

• Assume that the voltage of the battery is 9 V
  and that the three resistors are identical. What
  is the potential difference across each resistor?

16

• Assume that the voltage of the battery is 9 V
  and that the three resistors are identical. What
  is the potential difference across each resistor?

1) 12 V 2) zero 3) 3 V 4) 4 V 5) you
need to know the actual value of R
Since the resistors are all equal, the voltage
will drop evenly across the 3 resistors, with 1/3
of 9 V across each one. So we get a 3 V drop
across each.
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1) 12 V 2) zero 3) 6 V 4) 8 V 5) 4 V

• In the circuit below, what is the voltage across
  R1?

18

• In the circuit below, what is the voltage across
  R1?

1) 12 V 2) zero 3) 6 V 4) 8 V 5) 4 V
The voltage drop across R1 has to be twice as
big as the drop across R2. This means that V1
8 V and V2 4 V. Or else you could find the
current I V/R (12 V)/(6 W) 2 A, then use
Ohms Law to get voltages.
19
Resistors in Parallel

• The potential difference across each resistor is
  the same because each is connected directly
  across the battery terminals
• The current, I, that enters a point must be equal
  to the total current leaving that point
• I I1 I2
• The currents are generally not the same
• Consequence of Conservation of Charge

20
A parallel connection splits the current the
voltage across each resistor is the same
21
Equivalent Resistance Parallel, Example

• Equivalent resistance replaces the two original
  resistances
• Household circuits are wired so the electrical
  devices are connected in parallel
• Circuit breakers may be used in series with other
  circuit elements for safety purposes

22
Equivalent Resistance Parallel

• Equivalent Resistance
• The inverse of the equivalent resistance of two
  or more resistors connected in parallel is the
  algebraic sum of the inverses of the individual
  resistance
• The equivalent is always less than the smallest
  resistor in the group

23
1) 10 A 2) zero 3) 5 A 4) 2 A 5) 7 A

• In the circuit to the right, what is the current
  through R1?

24

• In the circuit below, what is the current
  through R1?

1) 10 A 2) zero 3) 5 A 4) 2 A 5) 7 A
The voltage is the same (10 V) across each
resistor because they are in parallel. Thus, we
can use Ohms Law, V1 I1 R1 to find the
current I1 2 A.
25
1) increases 2) remains the same 3)
decreases 4) drops to zero

• Points P and Q are connected to a battery of
  fixed voltage. As more resistors R are added to
  the parallel circuit, what happens to the total
  current in the circuit?

26

• Points P and Q are connected to a battery of
  fixed voltage. As more resistors R are added to
  the parallel circuit, what happens to the total
  current in the circuit?

1) increases 2) remains the same 3)
decreases 4) drops to zero
As we add parallel resistors, the overall
resistance of the circuit drops. Since V IR,
and V is held constant by the battery, when
resistance decreases, the current must increase.
27
1) all the current continues to flow through the
bulb 2) half the current flows through the wire,
the other half continues through the bulb 3) all
the current flows through the wire 4) none of the
above

• Current flows through a lightbulb. If a wire is
  now connected across the bulb, what happens?

28
1) all the current continues to flow through the
bulb 2) half the current flows through the wire,
the other half continues through the bulb 3) all
the current flows through the wire 4) none of the
above

• Current flows through a lightbulb. If a wire is
  now connected across the bulb, what happens?

The current divides based on the ratio of the
resistances. If one of the resistances is zero,
then ALL of the current will flow through that
path.
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1) glow brighter than before 2) glow just the
same as before 3) glow dimmer than before 4) go
out completely 5) explode

• Two lightbulbs A and B are connected in series
  to a constant voltage source. When a wire is
  connected across B, bulb A will

30
1) glow brighter than before 2) glow just the
same as before 3) glow dimmer than before 4) go
out completely 5) explode

• Two lightbulbs A and B are connected in series
  to a constant voltage source. When a wire is
  connected across B, bulb A will

Since bulb B is bypassed by the wire, the total
resistance of the circuit decreases. This means
that the current through bulb A increases.
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1) circuit 1 2) circuit 2 3) both the same 4)
it depends on R

• The lightbulbs in the circuit below are
  identical with the same resistance R. Which
  circuit produces more light? (brightness ??
  power)

32
1) circuit 1 2) circuit 2 3) both the same 4)
it depends on R

• The lightbulbs in the circuit below are
  identical with the same resistance R. Which
  circuit produces more light? (brightness ??
  power)

In 1, the bulbs are in parallel, lowering the
total resistance of the circuit. Thus, circuit
1 will draw a higher current, which leads to
more light, because P I V.
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1) twice as much 2) the same 3) 1/2 as
much 4) 1/4 as much 5) 4 times as much

• The three lightbulbs in the circuit all have the
  same resistance of 1 W . By how much is the
  brightness of bulb B greater or smaller than the
  brightness of bulb A? (brightness ?? power)

34
1) twice as much 2) the same 3) 1/2 as
much 4) 1/4 as much 5) 4 times as much

• The three light bulbs in the circuit all have
  the same resistance of 1 W . By how much is the
  brightness of bulb B greater or smaller than the
  brightness of bulb A? (brightness ?? power)

We can use P V2/R to compare the power PA
(VA)2/RA (10 V) 2/1 W 100 W PB (VB)2/RB
(5 V) 2/1 W 25 W
35
1) increase 2) decrease 3) stay the same

• What happens to the voltage across the resistor
  R1 when the switch is closed? The voltage will

36
1) increase 2) decrease 3) stay the same

• What happens to the voltage across the resistor
  R1 when the switch is closed? The voltage will

With the switch closed, the addition of R2 to R3
decreases the equivalent resistance, so the
current from the battery increases. This will
cause an increase in the voltage across R1 .
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1) increases 2) decreases 3) stays the same

• What happens to the voltage across the resistor
  R4 when the switch is closed?

38

• What happens to the voltage across the resistor
  R4 when the switch is closed?

1) increases 2) decreases 3) stays the same
We just saw that closing the switch causes an
increase in the voltage across R1 (which is VAB).
The voltage of the battery is constant, so if
VAB increases, then VBC must decrease!
39
1) R1 2) both R1 and R2 equally 3) R3 and
R4 4) R5 5) all the same
Which resistor has the greatest current going
through it? Assume that all the resistors are
equal.
40
Which resistor has the greatest current going
through it? Assume that all the resistors are
equal.
1) R1 2) both R1 and R2 equally 3) R3 and
R4 4) R5 5) all the same
The same current must flow through 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.
41
Problem-Solving Strategy, 1

• Combine all resistors in series
• They carry the same current
• The potential differences across them are not the
  same
• The resistors add directly to give the equivalent
  resistance of the series combination Req R1
  R2

42
Problem-Solving Strategy, 2

• Combine all resistors in parallel
• The potential differences across them are the
  same
• The currents through them are not the same
• The equivalent resistance of a parallel
  combination is found through reciprocal addition

43
Problem-Solving Strategy, 3

• A complicated circuit consisting of several
  resistors and batteries can often be reduced to a
  simple circuit with only one resistor
• Replace any resistors in series or in parallel
  using steps 1 or 2.
• Sketch the new circuit after these changes have
  been made
• Continue to replace any series or parallel
  combinations
• Continue until one equivalent resistance is found

44
Problem-Solving Strategy, 4

• If the current in or the potential difference
  across a resistor in the complicated circuit is
  to be identified, start with the final circuit
  found in step 3 and gradually work back through
  the circuits
• Use ?V I R and the procedures in steps 1 and 2

45
Equivalent Resistance Complex Circuit
46
An analogy using water may be helpful in
visualizing parallel circuits
47
Example 1
A 4.0-O resistor, an 8.0-O resistor, and a 12-O
resistor are connected in series with a 24-V
battery. What are (a) the equivalent resistance
and (b) the current in each resistor? (c) Repeat
for the case in which all three resistors are
connected in parallel across the battery.
48
Example 2
A 9.0-O resistor and a 6.0-O resistor are
connected in series with a power supply. (a) The
voltage drop across the 6.0-O resistor is
measured to be 12 V. Find the voltage output of
the power supply. (b) The two resistors are
connected in parallel across a power supply, and
the current through the 9.0-O resistor is found
to be 0.25 A. Find the voltage setting of the
power supply.
49
Example 3
(a) Find the equivalent resistance of the circuit
in Figure P18.8. (b) If the total power supplied
to the circuit is 4.00 W, find the emf of the
battery.
50
Example 4
Three 100-O resistors are connected as shown in
Figure P18.12. The maximum power that can safely
be delivered to any one resistor is 25.0 W. (a)
What is the maximum voltage that can be applied
to the terminals a and b? (b) For the voltage
determined in part (a), what is the power
delivered to each resistor? What is the total
power delivered?
51
Practice 1
What is the equivalent resistance of the
combination of resistors between points a and b
in Figure P18.7? Note that one end of the
vertical resistor is left free.
52
Practice 2
(a) You need a 45-O resistor, but the stockroom
has only 20-O and 50-O resistors. How can the
desired resistance be achieved under these
circumstances? (b) What can you do if you need a
35-O resistor?
53
Some circuits cannot be broken down into series
and parallel connections.
54
Gustav Kirchhoff

• 1824 1887
• Invented spectroscopy with Robert Bunsen
• Formulated rules about radiation

55
Kirchhoffs Rules

• There are ways in which resistors can be
  connected so that the circuits formed cannot be
  reduced to a single equivalent resistor
• Two rules, called Kirchhoffs Rules can be used
  instead

56
Statement of Kirchhoffs Rules

• Junction Rule
• The sum of the currents entering any junction
  must equal the sum of the currents leaving that
  junction
• A statement of Conservation of Charge
• Loop Rule
• The sum of the potential differences across all
  the elements around any closed circuit loop must
  be zero
• A statement of Conservation of Energy

57
More About the Junction Rule

• I1 I2 I3
• From Conservation of Charge
• Diagram b shows a mechanical analog

58
1) 2 A 2) 3 A 3) 5 A 4) 6 A 5) 10 A

• What is the current in branch P?

59

• What is the current in branch P?

1) 2 A 2) 3 A 3) 5 A 4) 6 A 5) 10 A
The current entering the junction in red is 8 A,
so the current leaving must also be 8 A. One
exiting branch has 2 A, so the other branch (at
P) must have 6 A.
S
60
Setting Up Kirchhoffs Rules

• Assign symbols and directions to the currents in
  all branches of the circuit
• If a direction is chosen incorrectly, the
  resulting answer will be negative, but the
  magnitude will be correct
• When applying the loop rule, choose a direction
  for transversing the loop
• Record voltage drops and rises as they occur

61
More About the Loop Rule

• Traveling around the loop from a to b
• In a, the resistor is transversed in the
  direction of the current, the potential across
  the resistor is IR
• In b, the resistor is transversed in the
  direction opposite of the current, the potential
  across the resistor is IR

62
Loop Rule, final

• In c, the source of emf is transversed in the
  direction of the emf (from to ), the change in
  the electric potential is e
• In d, the source of emf is transversed in the
  direction opposite of the emf (from to -), the
  change in the electric potential is -e

63
Junction Equations from Kirchhoffs Rules

• Use the junction rule as often as needed, so long
  as, each time you write an equation, you include
  in it a current that has not been used in a
  previous junction rule equation
• In general, the number of times the junction rule
  can be used is one fewer than the number of
  junction points in the circuit

64
Loop Equations from Kirchhoffs Rules

• The loop rule can be used as often as needed so
  long as a new circuit element (resistor or
  battery) or a new current appears in each new
  equation
• You need as many independent equations as you
  have unknowns

65
EMFs in series in the same direction total
voltage is the sum of the separate voltages
66
EMFs in series, opposite direction total voltage
is the difference, but the lower-voltage battery
is charged.
67
EMFs in parallel only make sense if the voltages
are the same this arrangement can produce more
current than a single emf.
68
Problem-Solving Strategy Kirchhoffs Rules

• Draw the circuit diagram and assign labels and
  symbols to all known and unknown quantities
• Assign directions to the currents.
• Apply the junction rule to any junction in the
  circuit
• Apply the loop rule to as many loops as are
  needed to solve for the unknowns
• Solve the equations simultaneously for the
  unknown quantities
• Check your answers

69
1) both bulbs go out 2) intensity of both bulbs
increases 3) intensity of both bulbs
decreases 4) A gets brighter and B gets
dimmer 5) nothing changes

• The lightbulbs in the circuit are identical.
  When the switch is closed, what happens?

70
1) both bulbs go out 2) intensity of both bulbs
increases 3) intensity of both bulbs
decreases 4) A gets brighter and B gets
dimmer 5) nothing changes

• The lightbulbs in the circuit are identical.
  When the switch is closed, what happens?

When the switch is open, the point between the
bulbs is at 12 V. But so is the point between
the batteries. If there is no potential
difference, then no current will flow once the
switch is closed!! Thus, nothing changes.
71
1) 2 I1 2I2 0 2) 2 2I1 2I2 4I3
0 3) 2 I1 4 2I2 0 4) I3 4 2I2
6 0 5) 2 I1 3I3 6 0

• Which of the equations is valid for the circuit
  below?

72

• Which of the equations is valid for the circuit
  below?

1) 2 I1 2I2 0 2) 2 2I1 2I2 4I3
0 3) 2 I1 4 2I2 0 4) I3 4 2I2
6 0 5) 2 I1 3I3 6 0
Eqn. 3 is valid for the left loop The left
battery gives 2V, then there is a drop through a
1W resistor with current I1 flowing. Then we go
through the middle battery (but from to !),
which gives 4V. Finally, there is a drop
through a 2W resistor with current I2.
73
Example 5
The ammeter shown in Figure P18.16 reads 2.00 A.
Find I1, I2, and e.
74
Example 6
Determine the potential difference ?1 for the
circuit in Figure P18.18.
75
Example 7
In the circuit of Figure P18.20, the current I1
is 3.0 A while the values of e and R are unknown.
What are the currents I2 and I3?
76
Example 8
Four resistors are connected to a battery with a
terminal voltage of 12 V, as shown in Figure
P18.22. Determine the power delivered to the 50-O
resistor.
77
Practice 3
Calculate each of the unknown currents I1, I2,
and I3 for the circuit of Figure P18.25.
78
RC Circuits

• A direct current circuit may contain capacitors
  and resistors, the current will vary with time
• When the circuit is completed, the capacitor
  starts to charge
• The capacitor continues to charge until it
  reaches its maximum charge (Q Ce)
• Once the capacitor is fully charged, the current
  in the circuit is zero

79
When the switch is closed, the capacitor will
begin to charge.
80
The voltage across the capacitor increases with
time
81
The charge follows a similar curve
This curve has a characteristic time constant
82
If an isolated charged capacitor is connected
across a resistor, it discharges
83
Notes on Time Constant

• In a circuit with a large time constant, the
  capacitor charges very slowly
• The capacitor charges very quickly if there is a
  small time constant
• After t 10 t, the capacitor is over 99.99
  charged

84
Example 9
An uncharged capacitor and a resistor are
connected in series to a source of emf. If e
9.00 V, C 20.0 µF, and R 100 O, find (a) the
time constant of the circuit, (b) the maximum
charge on the capacitor, and (c) the charge on
the capacitor after one time constant.
85
Example 10
Consider a series RC circuit for which R 1.0
MO, C 5.0 µF, and e 30 V. Find the charge on
the capacitor 10 s after the switch is closed.
86
Practice 4
A series RC circuit has a time constant of 0.960
s. The battery has an emf of 48.0 V, and the
maximum current in the circuit is 0.500 mA. What
are (a) the value of the capacitance and (b) the
charge stored in the capacitor 1.92 s after the
switch is closed?
87
Household Circuits

• The utility company distributes electric power to
  individual houses with a pair of wires
• Electrical devices in the house are connected in
  parallel with those wires
• The potential difference between the wires is
  about 120V

88
Household Circuits, cont.

• A meter and a circuit breaker are connected in
  series with the wire entering the house
• Wires and circuit breakers are selected to meet
  the demands of the circuit
• If the current exceeds the rating of the circuit
  breaker, the breaker acts as a switch and opens
  the circuit
• Household circuits actually use alternating
  current and voltage

89
A person receiving a shock has become part of a
complete circuit.
90
Electrical Safety

• Electric shock can result in fatal burns
• Electric shock can cause the muscles of vital
  organs (such as the heart) to malfunction
• The degree of damage depends on
• the magnitude of the current
• the length of time it acts
• the part of the body through which it passes

91
Effects of Various Currents

• 5 mA or less
• Can cause a sensation of shock
• Generally little or no damage
• 10 mA
• Hand muscles contract
• May be unable to let go a of live wire
• 100 mA
• If passes through the body for just a few
  seconds, can be fatal

92
Ground Wire

• Electrical equipment manufacturers use electrical
  cords that have a third wire, called a case
  ground
• Prevents shocks

93
Ground Fault Interrupts (GFI)

• Special power outlets
• Used in hazardous areas
• Designed to protect people from electrical shock
• Senses currents (of about 5 mA or greater)
  leaking to ground
• Shuts off the current when above this level

94
Example 11
An electric heater is rated at 1 300 W, a toaster
at 1 000 W, and an electric grill at 1 500 W. The
three appliances are connected in parallel to a
common 120-V circuit. (a) How much current does
each appliance draw? (b) Is a 30.0-A circuit
breaker sufficient in this situation? Explain.
95
Example 12
A lamp (R 150 O), an electric heater (R 25
O), and a fan (R 50 O) are connected in
parallel across a 120-V line. (a) What total
current is supplied to the circuit? (b) What is
the voltage across the fan? (c) What is the
current in the lamp? (d) What power is expended
in the heater?
96
Practice 5