Direct Current Circuits

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Chapter 18

• Direct Current Circuits

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18.1 Sources of emf

• The source that maintains the current in a closed
  circuit is called a source of emf
• Emfelectromotive force
• Any devices that increase the potential energy of
  charges circulating in circuits are sources of
  emf
• Examples include batteries and generators

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Emf and Internal Resistance

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

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More About Internal Resistance
V

• The schematic shows the internal resistance, r
• The terminal voltage is V e Ir (source minus
  the internal loss)
• For the entire circuit,

• e IR Ir I (Rr)

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Internal Resistance and emf, cont

• e is equal to the terminal voltage when the
  current is zero (open circuit)
• 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

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18.2 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 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

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Resistors in Series, cont
V1IR1
V2IR2

• Voltages add
• V IR1 IR2
• V I (R1R2)
• Req R1R2
• The equivalent resistance Req has the same effect
  on the circuit as the original combination of
  resistors

V
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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 resistance (? see parallel
  connection of capacitors!)

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Equivalent Resistance SeriesAn Example

• Four resistors are replaced with their equivalent
  resistance

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18.3 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

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Equivalent Resistance Parallel, cont
V

• I1V/R1 and I2V/R2. The complete current
  provided by the source is given by,
  II1I2V(1/R11/R2)V/Req.
• 1/Req1/R11/R2
• Req is the equivalent resistance for a parallel
  circuit

• 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

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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 (? see series connection of
  capacitors)
• The equivalent is always less than the smallest
  resistor in the group

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Problem-Solving Strategy, 1

• When two or more resistors are connected in
  series, they carry the same current, but the
  potential differences across them are not the
  same.
• The resistors add directly to give the equivalent
  resistance of the series combination

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Problem-Solving Strategy, 2

• When two or more resistors are connected 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
• The equivalent resistance is always less than the
  smallest individual resistor in the combination

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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

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Problem-Solving Strategy, 4

• If the current in or the potential difference
  across a resistor in a 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

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ExampleCompute the equivalent resistance and the
current of the network (a) below.
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Solution
RpReq for parallel connection and RsReq for
series connection

• Step 1 1/(3 W)1/(6 W)1/Rp ? Rp2 W (b)
• Step 2 4 W2 WRs ? Rs6 W (c)
• Step 3 IV/Rs18 V/6 W3 A (d)
• Step 4 V(3 A)(4 W2 W)12 V6 V18 V (e)
• Step 5 6 V/6 W1 A and 6 V/3 W2 A (f)

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Shortcut

• For n parallel equal resistors, the equivalent Rp
  is expressed by  
• RpR1/n
• The same shortcut is valid for n equal capacitors
  in series 
• CsC1/n

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18.4 Kirchhoffs Rules

• There are ways in which resistors and batteries
  can be connected so that the circuits formed
  cannot be reduced to a single equivalent resistor
  (two examples are shown on the right)
• Two rules, called Kirchhoffs Rules can be used
  instead

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Statement of Kirchhoffs Rules

• Junction Rule (? I 0)
• 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 (? U 0)
• The sum of the potential differences across all
  the elements around any closed circuit loop must
  be zero
• A statement of Conservation of Energy

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More About the Junction Rule

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

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More About the Loop Rule

• The voltage across a battery is taken to be
  positive (a voltage rise) if traversed from to
  and and negative if traversed in the opposite
  direction.
• The voltage across a resistor is taken to be
  negative (a drop) if the loop is traversed in in
  the direction of the assigned current and
  positive if traversed in the opposite direction

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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

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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.

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Example Find the current I, r and e.

• Junction rule at a
• 2 A1 A-I0
• I3 A
• Loop (1)
• 12 V-Ir -(3 W)(2 A)0
• r12 V/(3 A)-6 V/(3 A)
• r2 W
• Loop (2)
• -e(1 W)(1 A)-( 3 W)(2 A)0
• e-5 V (polarity is opposite!)
• Check with loop (3)
• 12 V-(2 W)(3 A)-(1 W)(1 A)e 0
• e-5 V