Physics 121: Electricity

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Physics 121 Electricity Magnetism Lecture
13E-M Oscillations and AC Current

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

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Electromagnetic Oscillations
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Oscillating Quantities

• We will write oscillating quantities with a
  lower-case symbol, and the corresponding
  amplitude of the oscillation with upper case.
• Examples

Oscillating Quantity Oscillating Quantity Amplitude
Voltage v V
Current i I
Charge q Q
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Derivation of Oscillation Frequency

• We have shown qualitatively that LC circuits act
  like an oscillator.
• We can discover the frequency of oscillation by
  looking at the equations governing the total
  energy.
• Since the total energy is constant, the time
  derivative should be zero
• But and , so making these
  substitutions
• This is a second-order, homogeneous differential
  equation, whose solution is
• i.e. the charge varies according to a cosine wave
  with amplitude Q and frequency w. Check by taking
  
  two time derivatives of charge
• Plug into original equation

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Which Current is Greatest?

• The expressions below give the charge on a
  capacitor in an LC circuit. Choose the one that
  will have the greatest maximum current?
• q 2 cos 4t
• q 2 cos(4tp/2)
• q 2 sin t
• q 4 cos 4t
• q 2 sin 5t

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Time to Discharge Capacitor

• The three circuits below have identical inductors
  and capacitors. Rank the circuits according to
  the time taken to fully discharge the capacitor
  during an oscillation, greatest first.
• I, II, III.
• II, I, III.
• III, I, II.
• III, II, I.
• II, III, I.

I. II. III.
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Charge, Current Energy Oscillations

• The solution to the equation
  is , which gives the
  charge oscillation.
• From this, we can determine the corresponding
  oscillation of current
• And energy
• But recall that , so
  .
• That is why our graph for the energy oscillation
• had the same amplitude for both UE and UB.
• Note that

Constant
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Damped Oscillations

• Recall that all circuits have at least a little
  bit of resistance.
• In this general case, we really have an RLC
  circuit, where the oscillations get smaller with
  time. They are said to be damped oscillations.

• Then the power equation becomes

Damped Oscillations
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Resonant Frequency

• How does the resonant frequency w for an ideal LC
  circuit (no resistance) compare with w for a
  non-ideal one where resistance cannot be ignored?
• The resonant frequency for the non-ideal circuit
  is higher than for the ideal one (w gt w).
• The resonant frequency for the non-ideal circuit
  is lower than for the ideal one (w lt w).
• The resistance in the circuit does not affect the
  resonant frequencythey are the same (w w).

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Alternating Current

• The electric power out of a home or office power
  socket is in the form of alternating current
  (AC), as opposed to the direct current (DC) of a
  battery.
• Alternating current is used because it is easier
  to transport, and easier to transform from one
  voltage to another using a transformer.
• In the U.S., the frequency of oscillation of AC
  is 60 Hz. In most other countries it is 50 Hz.

• The figure at right shows one way to make an
  alternating current by rotating a coil of wire in
  a magnetic field. The slip rings and brushes
  allow the coil to rotate without twisting the
  connecting wires. Such a device is called a
  generator.
• It takes power to rotate the coil, but that power
  can come from moving water (a water turbine), or
  air (windmill), or a gasoline motor (as in your
  car), or steam (as in a nuclear power plant).

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RLC Circuits with AC Power

• When an RLC circuit is driven with an AC power
  source, the driving frequency is the
  frequency of the power source, while the circuit
  can have a different resonant frequency
  .
• Lets look at three different circuits driven by
  an AC EMF. The device connected to the EMF is
  called the load.
• What we are interested in is how the voltage
  oscillations across the load relate to the
  current oscillations.
• We will find that the phase relationships
  change, depending on the type of load (resistive,
  capacitive, or inductive).

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A Resistive Load

• Phasor Diagram shows the instantaneous phase of
  either voltage or current.
• For a resistor, the current follows the voltage,
  so the voltage and current are in phase (f 0).
• If
• Then

f
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Power in a Resistive Circuit

• The plot below shows the current and voltage
  oscillations in a purely resistive circuit.
  Below that are four curves. Which color curve
  best represents the power dissipated in the
  resistor?
• The green curve (straight line).
• The blue curve.
• The black curve.
• The red curve.
• None are correct.

PR
t
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A Capacitive Load

• For a capacitive load, the voltage across the
  capacitor is proportional to the charge
• But the current is the time derivative of the
  charge
• In analogy to the resistance, which is the
  proportionality constant between current and
  voltage, we define the capacitive reactance as
• So that .
• The phase relationship is that f -90º, and
  current leads voltage.

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An Inductive Load

• For an inductive load, the voltage across the
  inductor is proportional to the time derivative
  of the current
• But the current is the time derivative of the
  charge
• Again in analogy to the resistance, which is the
  proportionality constant between current and
  voltage, we define the inductive reactance as
• So that .
• The phase relationship is that f 90º, and
  current lags voltage.

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Units of Reactance

• We just learned that capacitive reactance is
  and inductive reactance is .
  What are the units of reactance?
• Seconds per coulomb.
• Henry-seconds.
• Ohms.
• Volts per Amp.
• The two reactances have different units.

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Summary Table
Circuit Element Symbol Resistance or Reactance Phase of Current Phase Constant Amplitude Relation
Resistor R R In phase with vR 0º (0 rad) VRIRR
Capacitor C XC1/wdC Leads vR by 90º -90º (-p/2) VCICXC
Inductor L XLwdL Lags vR by 90º 90º (p/2) VLILXL
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Summary

• Energy in inductor
• LC circuits total electric magnetic energy is
  conserved
• LC circuit
• LRC circuit
• Resistive, capacitive,
  inductive

Energy in magnetic field
Charge equation
Current equation
Oscillation frequency
Charge equation
Oscillation frequency
Reactances
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Thoughts on Clickers

• How did you like using the clickers in this
  class?
• Great!
• It had its moments.
• I could take it or leave it.
• I would rather leave it.
• It was the worst!

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Thoughts on Clickers

• Which answer describes the most important way
  that the clicker aided you in learning the
  material?
• It helped me to think about the material
  presented just before each question.
• It broke up the lecture and kept me awake.
• It tested my understanding.
• It provided immediate feedback.
• It showed me what others were thinking.

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Thoughts on Clickers

• Which answer describes the second most important
  way that the clicker aided you in learning the
  material?
• It helped me to think about the material
  presented just before each question.
• It broke up the lecture and kept me awake.
• It tested my understanding.
• It provided immediate feedback.
• It showed me what others were thinking.

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Thoughts on Clickers

• How would you react to clickers being used in
  other classes at NJIT?
• I think it would be excellent.
• I think it is a good idea.
• I wouldnt mind.
• I would rather not.
• I definitely hope not.

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Thoughts on Clickers

• What problems did you have with your clicker?
• I had no problems with my clicker.
• It was too big or bulky, a pain to carry around.
• I had trouble remembering to bring it to class.
• My clicker had mechanical problems.
• I lost or misplaced it (for all or part of the
  semester).

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Thoughts on Clickers

• If you had the choice between using a clicker
  versus having a lecture quiz where you had to
  fill in a scantron, which would you prefer?
• I would prefer the clicker.
• I would prefer the scantron quiz.

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Have a Nice Day

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