Physics 1502: Lecture 22 Today

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Physics 1502 Lecture 22Todays Agenda

• Announcements
• RL - RV - RLC circuits
• Homework 06 due next Wednesday
• Induction / AC current

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Induction

• Self-Inductance, RL Circuits

long solenoid
Energy and energy density
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e on e off
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Charging Discharging
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Mutual Inductance

• Suppose you have two coils with multiple turns
  close to each other, as shown in this
  cross-section

• We can define mutual inductance M12 of coil 2
  with respect to coil 1 as

It can be shown that
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Inductors in Series

• What is the combined (equivalent) inductance of
  two inductors in series, as shown ?

Note the induced EMF of two inductors now adds
Since
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Inductors in parallel

• What is the combined (equivalent) inductance of
  two inductors in parallel, as shown ?

Note the induced EMF between points a and be is
the same !
Also, it must be
We can define
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LC Circuits

• Consider the LC and RC series circuits shown

• Consider from point of view of energy!
• In the RC circuit, any current developed will
  cause energy to be dissipated in the resistor.
• In the LC circuit, there is NO mechanism for
  energy dissipation energy can be stored both in
  the capacitor and the inductor!

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RC/LC Circuits
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LC Oscillations(qualitative)
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Energy transfer in a resistanceless, nonradiating
LC circuit. The capacitor has a charge Qmax at t
0, the instant at which the switch is closed.
The mechanical analog of this circuit is a
blockspring system.
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LC Oscillations(quantitative)

• What do we need to do to turn our qualitative
  knowledge into quantitative knowledge?
• What is the frequency w of the oscillations
  (when R0)?
• (it gets more complicated when R finiteand R is
  always finite)

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LC Oscillations(quantitative)

• Begin with the loop rule

• Guess solution (just harmonic oscillator!)

• where w0 determined from equation
• f, Q0 determined from
  initial conditions

• Procedure differentiate above form for Q and
  substitute into loop equation to find w0.

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Review LC Oscillations

• Guess solution (just harmonic oscillator!)

• where w0 determined from equation
• f, Q0 determined from
  initial conditions

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The energy in LC circuit conserved !
When the capacitor is fully charged
When the current is at maximum (Io)
The maximum energy stored in the capacitor and in
the inductor are the same
At any time
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Lecture 22, ACT 1

• At t0 the capacitor has charge Q0 the resulting
  oscillations have frequency w0. The maximum
  current in the circuit during these oscillations
  has value I0 .
• What is the relation between w0 and w2 , the
  frequency of oscillations when the initial charge
  2Q0 ?

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Lecture 22, ACT 1

• At t0 the capacitor has charge Q0 the resulting
  oscillations have frequency w0. The maximum
  current in the circuit during these oscillations
  has value I0 .

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Summary of EM

• J. C. Maxwell (1860) summarized all of the work
  on electric and magnetic fields into four
  equations, all of which you now know.
• However, he realized that the equations of
  electricity magnetism as then known (and now
  known by you) have an inconsistency related to
  the conservation of charge!

I dont expect you to see that these equations
are inconsistent with conservation of charge, but
you should see a lack of symmetry here!
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Amperes Law is the Culprit!

• Gauss Law

• Symmetry both E and B obey the same kind of
  equation (the difference is that magnetic charge
  does not exist!)

• Amperes Law and Faradays Law

• If Amperes Law were correct, the right hand side
  of Faradays Law should be equal to zero -- since
  no magnetic current.
• Therefore(?), maybe there is a problem with
  Amperes Law.
• In fact, Maxwell proposes a modification of
  Amperes Law by adding another term (the
  displacement current) to the right hand side of
  the equation! ie

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Displacement current
FE
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Maxwells Displacement Current

• Can we understand why this displacement current
  has the form it does?

• Consider applying Amperes Law to the current
  shown in the diagram.
• If the surface is chosen as 1, 2 or 4, the
  enclosed current I
• If the surface is chosen as 3, the enclosed
  current 0! (ie there is no current between the
  plates of the capacitor)

Big Idea The Electric field between the plates
changes in time. displacement current ID e0
(dfE/dt) the real current I in the wire.
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Maxwells Equations

• These equations describe all of Electricity and
  Magnetism.
• They are consistent with modern ideas such as
  relativity.
• They even describe light