Direct Current Circuits

1
Chapter 18

• Direct Current Circuits

2
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

3
emf and Internal Resistance

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

4
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

I e/(R r)
5
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

6
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

7
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

8
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

9
Quick Quiz 18.1
When a piece of wire is used to connect points b
and c in the right Figure, the brightness of bulb
R1 (a) increases, (b) decreases but remains lit,
(c) stays the same, (d) goes out? The brightness
of bulb R2 (a) increases, (b) decreases but
remains lit, (c) stays the same, (d) goes out?
R1 (a) R2 (d)
10
Quick Quiz 18.2
In the left Fig. the current is measured with the
ammeter at the right side of the circuit. When
the switch is opened, the reading on the ammeter
(a) increases, (b) decreases, (c) does not change
Answer (b)
11
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

12
Equivalent Resistance Series An Example

• Four resistors are replaced with their equivalent
  resistance

13
Christmas light in Series
A jumper connection becomes effective, when a
bulb fails
14
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

15
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

16
Equivalent Resistance Parallel (Interact. Ex.
18.2)

• 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

17
Solution Interactive example 18.2
Power P1 I12R1 363 108 W P2 I22R2
96 54 W P3 I32R3 49 36 W Ptot P1
P2 P3 198 W

• Current
• I1 ?V/R1 18V/3O 6 A
• I2 ?V/R2 18V/6O 3A
• I3 ?V/R3 18V/9O 2 A

Power dissipated in the equivalent circuit P
(?V)2/Req 182/1.63 199 W
18
Three-Way Lightbulbs
19
Quick Quiz 18.3
In the left Fig. the current is measured with the
ammeter on the right side of the circuit
diagramm. When the switch is closed, the reading
on the ammeter (a) increases, (b) decreases, or
(c) remains the same?










Answer (a)
20
Quick Quiz 18.4
Suppose you have three identical lightbulbs, some
wire, and a battery. You connect one lightbulb to
the battery and take note of its brightness. You
add a second lightbulb, connecting it in parallel
with the previous bulbs, again taking note of the
brightness. Repeat the process with the third
bulb, connecting it in parallel with the other
two. As the lightbulb are added, what happens? to
(a) the brightness of the bulbs? (b) the
individual currents in the bulbs? (c) the power
delivered by the battery (d) the lifetime of the
battery? (Neglect the batterys internal
resistance)










Answers (a) unchanged, (b) unchanged, (c)
increase, (d) decrease
21
Quick Quiz 18.5
If the lightbulbs in Quick Quiz 18.4 are
connected, one by one, in series instead of in
parallel, what happens to (a) the brightness of
the bulbs? (b) the individual currents in the
bulbs? (c) the power delivered by the battery (d)
the lifetime of the battery? (Neglect the
batterys internal resistance)











Answers (a) decrease, (b) decrease, (c)
decrease, (d) increase
22
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

23
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

24
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

25
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

26
Equivalent Resistance Complex Circuit
27
Gustav Kirchhoff

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

28
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

29
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

30
More About the Junction Rule

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

31
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 traversing the loop
• Record voltage drops and rises as they occur

32
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

33
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

34
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

35
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

36
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

37
Example 18.4 Kirchhoffs Rules
Find the current in the circuit by using
Kirchhoffs rules.
There are 3 unknown currents, so one needs 3
independent equations.
Apply junction rule to point c I1 I2
I3 Traverse the bottom loop clockwise starting at
point a, generating an eq. with the loop rule
?Vbatt ?V4 ?V9 0 ? 6 V (4O)I1
(9O)I3 0 Traverse the top loop clockwise from
point c ?V5 ?V9 0 ? -(5O)I2 (9O)I3 0
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Example 18.4 Kirchhoffs Rules, cont.
Rewrite the 3 equations, droping units.
I1 I2 I3 (1) 4I1 9I3 6 (2) -5I2 9I3
0 (3)
From (3) I2 1.8I3, into eq. (1) ? I1 2.8I3,
this in (2) gives 42.8I3 9I3 6 ? I3
6/20.2 0.3 A From (3) ? I2 0.54 A and from
(2) ? I1 0.83 A
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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

40
Loading a capacitor
Loading a capacitor
Emf IR Q/C Emf RdQ/dt Q/C
Emf
41
Loading a capacitor, Cont.
Emf
42
Loading a capacitor, Cont. 2
e is Eulers constant 2.718 (the base of
natural logarithms) ? e-1 0.37 ? 1-e-1 0.63,
so for RC 1s the capacitor is charged to 63
after 1s. For RC 2 it is charged to 86, etc...
Leonhard Euler, Swiss Mathematician 1777 - 1783
43
Charging Capacitor in an RC Circuit

• The charge on the capacitor varies with time
• q Q(1 e-t/RC)
• The time constant, ?RC
• The time constant represents the time required
  for the charge to increase from zero to 63.2 of
  its maximum

44
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

45
Other example of exponential behavior
In many cases the growth rate of a population is
proportional to the population ?
Solution Ansatz the exponential function
Fitting the Ansatz in the differential equation
leads for the coefficient a to
And as solution to
46
Discharging a Capacitor
Discharging Capacitor in an RC Circuit
At t 0 the switch is closed. The rate at which
charge leaves the capacitor equals the negative
of the current in the resisitor ?
47
Quick Quiz 18.6
The switch is closed in the right Fig. After a
long time compared to the time constant of the
capacitor, what will the current be in the 2O
resistor? (a) 4A, (b) 3A, (c) 2A, (d) 1A, (e)
more information is needed.
Answer (c)
48
Applying Physics 18.6
Flashing yellow lights on roadways, automobile
turn signal indicators, and intermittent
windshield wipers are examples of charging and
discharging capacitors. In the right circuit,
after closing the switch the battery will charge
the capacitor. The potential difference across it
increases until, eventually, it reaches a value
at which the gas in the lamp will conduct,
causing a flash. This will discharge the
capacitor, and the process of charging begins
again.
49
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

50
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

51
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

52
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

53
Ground Wire

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

54
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

55
Electrical Signals in Neurons

• Specialized cells in the body, called neurons,
  form a complex network that receives, processes,
  and transmits information from one part of the
  body to another
• Three classes of neurons
• Sensory neurons
• Receive stimuli from sensory organs that monitor
  the external and internal environment of the body
• Motor neurons
• Carry messages that control the muscle cells
• Interneurons
• Transmit information from one neuron to another

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Diagram of a Neuron