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 (e.g. batteries, generators, etc.)
• The emf is the work done per unit charge
• SI units Volts

3
emf and Internal Resistance

• A real battery has some internal resistance r
  therefore, the terminal voltage is not equal to
  the emf
• The terminal voltage ?V Vb Va
• ?V e Ir
• For the entire circuit (R load resistance)
• e ?V Ir
• IR Ir

4
emf and Internal Resistance

• e ?V Ir IR Ir
• e is equal to the terminal voltage when the
  current is zero open-circuit voltage
• I e / (R r)
• The current depends on both the resistance
  external to the battery and the internal
  resistance
• When R gtgt r, r can be ignored
• 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

5
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

6
Resistors in Series

• The equivalent resistance has the effect on the
  circuit as the original combination of resistors
  (consequence of conservation of energy)
• For more resistors in series
• The equivalent resistance of a series combination
  of resistors is greater than any of the
  individual resistors

7
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
  (conservation of charge)
• The currents are generally not the same

8
Resistors in Parallel
9
Resistors in Parallel

• For more resistors in parallel
• 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

10
Problem-Solving Strategy

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

11
Problem-Solving Strategy

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

12
Problem-Solving Strategy

• A complicated circuit consisting of several
  resistors and batteries can often be reduced to a
  simple circuit with only one resistor
• Replace resistors in series or in parallel with a
  single resistor
• 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

13
Problem-Solving Strategy

• If the current in or the potential difference
  across a resistor in the complicated circuit is
  to be identified, start with the final circuit
  and gradually work back through the circuits (use
  formula ?V I R and the procedures describe
  above)

14
Chapter 18Problem 13

• Find the current in the 12-O resistor in the
  Figure.

15
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
• 1) Junction Rule
• 2) Loop Rule

16
Kirchhoffs Rules

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

17
Junction Rule

• I1 I2 I3
• 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

18
Loop Rule

• When applying the loop rule, choose a direction
  for transversing the loop
• Record voltage drops and rises as they occur
• If a resistor is transversed in the direction of
  the current, the potential across the resistor is
  IR
• If a resistor is transversed in the direction
  opposite of the current, the potential across the
  resistor is IR

19
Loop Rule

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

20
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
• The number of times the junction rule can be used
  is one fewer than the number of junction points
  in the circuit
• 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

21
Equations from Kirchhoffs Rules
22
Problem-Solving Strategy

• 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

23
Chapter 18Problem 17

• Determine the current in each branch of the
  circuit shown in the Figure.

24
RC Circuits

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

25
Charging Capacitor in an RC Circuit

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

• In a circuit with a large (small) time constant,
  the capacitor charges very slowly (quickly)
• After t 10 ?, the capacitor is over 99.99
  charged

26
Discharging Capacitor in an RC Circuit

• When a charged capacitor is placed in the
  circuit, it can be discharged
• q Qe -t/RC
• The charge decreases exponentially
• At t ? RC, the charge decreases to 0.368
  Qmax i.e., in one time constant, the capacitor
  loses 63.2 of its initial charge

27
Chapter 18Problem 35

• A capacitor in an RC circuit is charged to 60.0
  of its maximum value in 0.900 s. What is the time
  constant of the circuit.

28
Chapter 18Problem 54

• An emf of 10 V is connected to a series RC
  circuit consisting of a resistor of 2.0 106 O
  and a capacitor of 3.0 µF. Find the time required
  for the charge on the capacitor to reach 90 of
  its final value.

29
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

30
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

31

• Answers to Even Numbered Problems
• Chapter 18
• Problem 2
• 24 O
• 1.0 A
• 2.18 O, I4 6.0 A, I8 3.0 A, I12 2.0 A

32
Answers to Even Numbered Problems Chapter 18
Problem 6 15 O
33
Answers to Even Numbered Problems Chapter 18
Problem 18 5.4 V with a at a higher potential
than b
34
Answers to Even Numbered Problems Chapter 18
Problem 22 0.50 W
35

• Answers to Even Numbered Problems
• Chapter 18
• Problem 36
• 10.0 µF
• 415 µC

36

• Answers to Even Numbered Problems
• Chapter 18
• Problem 38
• 8.0 A
• 120 V
• 0.80 A
• 5.8 102 W