Physics 121: Electricity

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Physics 121 Electricity Magnetism Lecture
8DC Circuits

• Dale E. Gary
• Wenda Cao
• NJIT Physics Department

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emf and emf devices

• The term emf comes from the outdated phrase
  electromovitive force.
• emf devices include battery, electric generator,
  solar cell, fuel cell,
• emf devices are sources of charge, but also
  sources of voltage (potential difference).
• emf devices must do work to pump charges from
  lower to higher terminals.
• Source of emf devices chemical, solar,
  mechanical, thermal-electric energy.

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Emf

• We need a symbol for emf, and we will
  use a script to represent emf. is the
  potential difference between terminals of an emf
  device.
• The SI unit for emf is Volt (V).
• We earlier saw that there is a relationship
  between energy, charge, and voltage
• So,
• Power

Real emf device V (open loop) V lt
(close loop)
Ideal emf device V (open or close loop)
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Work, Energy, and EMF

• The following circuit contains two ideal
  rechargeable batteries A and B, a resistance R,
  and an electric motor M that can lift an object
  by using energy it obtains from charge carriers
  in the circuit. Which statement is correct?
• Battery B lost chemical energy
• Battery B charge Battery A
• Battery B provides energy to Motor M
• Battery B provides energy to heat R
• All of the above are true

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Kirchhoffs Rules

• Junction Rule At any junction, the sum of the
  currents must equal zero

• For a move through a resistance in the direction
  of current, the change in potential is iR in
  the opposite direction it is iR.
• For a move through an ideal emf device in the
  direction of the emf arrow, the change in
  potential is ? in the opposite direction it is
  - ?.

• Loop Rule The sum of the potential differences
  across all elements around any closed circuit
  loop must be zero

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A Single-Loop Circuit

• Travel clockwise from a
• Travel counterclockwise from a

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Resistances in Series

• Junction Rule When a potential difference V is
  applied across resistances connected in series,
  the resistances have identical currents i
• Loop Rule The sum of the potential differences
  across resistances is equal to the applied
  potential difference V
• (a)
• (b)
• The equivalent resistance of a series combination
  of resistors is the numerical sum of the
  individual resistances and is always greater than
  any individual resistance.

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Resistances in Parallel

• When a potential difference V is applied across
  resistances connected in parallel, the
  resistances all have that same potential
  difference V.
• (a) Junction Rule
• (b) Loop Rule
• The inverse of the equivalent resistance of two
  or more resistors in a parallel combination is
  the sum of the inverse of the individual
  resistances. Furthermore, the equivalent
  resistance is always less than the smallest
  resistance in the group.

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Resistors in series and parallel

• 2. Four resistors are connected as shown in
  figure. Find the equivalent resistance between
  points a and c.
• 4 R.
• 3 R.
• 2.5 R.
• 0.4 R.
• Cannot determine
• from information given.

c
R
R
R
R
a
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Capacitors in series and parallel

• 3. Four capacitors are connected as shown in
  figure. Find the equivalent capacitance between
  points a and c.
• 4 C.
• 3 C.
• 2.5 C.
• 0.4 C.
• Cannot determine
• from information given.

c
C
C
C
C
a
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Example Real Battery

• Real battery has internal resistance to the
  internal movement of charge.
• Current
• Potential difference
• clockwise
• Power

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Example Multiple Batteries

• What is the potential difference and power
  between the terminals of battery 1 and 2?
• Current i in this single-loop (counterclockwise)
• Potential difference (clockwise)
• Power
• A battery (EMF) absorbs power (charges up)
  when i is opposite to ?.

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Potential difference and battery

• 4. According to the circuit shown, which
  statement is correct?
• Vb 8.0 V
• Va - 8.0 V
• Vb-Va - 8.0 V
• Va - 12.0 V
• Vb 12.0 V

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Multiloop Circuits

• Determine junctions, branches and loops.
• Label arbitrarily the currents for each branch.
  Assign same current to all element in series in a
  branch.
• The directions of the currents are assumed
  arbitrarily negative current result means
  opposite direction.
• Junction rule
• You can use the junction rule as often as you
  need. In general, the number of times you can use
  the junction rule is one fewer than the number of
  junction points in the circuit.

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Multiloop Circuits

• Determine loop and choose moving direction
  arbitrarily.
• When following the assumed current direction, ir
  is negative and voltage drops Reverse when going
  against the assumed current Emf is positive when
  traversed from to , negative otherwise.
• Loop rule
• badb left-hand loop in counterclockwise
• bdcb right-hand loop in counterclockwise
• You can apply the loop rule as often as needed as
  long as a new circuit element or a new current
  appears in each new equation.

• Equivalent loop and wise
• badcb big loop in counterclockwise
• bcdb right-hand loop in clockwise

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Multiloop Circuits

• In general, to solve a particular circuit
  problem, the number of independent equations you
  need to obtain from the two rules equals the
  number of unknown currents.
• Solution

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RC Circuits Charging a Capacitor

• RC circuits time-varying currents, switch to a
• Start with,
• Loop rule
• Then,
• Substituting and rearranging,
• Boundary condition,
• Therefore,
• Integrating,

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RC Circuits Charging a Capacitor

• Charge,
• For charging current,
• A capacitor that is being charged initially acts
  like
• ordinary connecting wire relative to the
  charging current. A long time later, it acts like
  a broken wire.
• Potential difference,
• t 0 q 0, Vc 0, i ?/R
• t gt ? q c?, Vc ?, i 0
• t RC q c?(1-e-1) 0.632c? i ?/Re-1
  0.368 ?/R

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Unit of RC

• 5. If ? RC, what it the unit of ? ?
• ??F (ohm?farad)
• C/A (coulomb per ampere)
• ??C/V (ohm?coulomb per volt)
• V?F/A (volt?farad per ampere)
• s (second)

? RC (V/i)(Q/V)Q/Q/tt
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RC Circuits Discharging a Capacitor

• RC circuits time-varying currents, switch to b
• Start with,
• Loop rule
• Then,
• Boundary condition,
• Therefore,
• Hence,
• t 0 q q0 CV0, i q0/RC
• t gt ? q 0, i 0

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Resistors in series and parallel

• 6. Consider the circuit shown and assume the
  battery has no internal resistance. Just after
  the switch is closed, what is the current in the
  battery?
• 0.
• ?/2R.
• 2?/R.
• ?/R.
• impossible to determine

C
R
R
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Resistors in series and parallel

• 7. Consider the circuit shown and assume the
  battery has no internal resistance. After a very
  long time, what is the current in the battery?
• 0.
• ?/2R.
• 2?/R.
• ?/R.
• impossible to determine

C
R
R
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Summary

• An emf device does work on charges to maintain a
  potential difference between its output
  terminals.
• Kirchhoffs rules
• Loop rule. The algebraic sum of the changes
  in potential encountered in a complete traversal
  of any loop of a circuit must be zero.
• Junction rule. The sum of the current
  entering any junction must be equal to the sum of
  the currents leaving that junction.
• Series resistances when resistances are in
  series, they have the same current. The
  equivalent resistance that can replace a series
  combination of resistance is
• Parallel resistance when resistances are in
  parallel, they have the same potential
  difference. The equivalent resistance that can
  replace a parallel combination of resistance is,
• Single loop circuits the current in a single
  loop circuit is given by
• Power when a real battery of emf and internal
  resistance r does work on the charges in a
  current I through the battery,
• RC Circuits when an emf is applied to a
  resistance R and capacitor C in series,
• RC Circuits when a capacitor discharges through
  a resistance R, the charge decays according to
• And the current is