Physics%20122B%20%20Electricity%20and%20Magnetism

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Physics 122B Electricity and Magnetism
Lecture 25 (Knight 34.1 to 34.5) Maxwells
Equations

• Martin Savage

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Lecture 25 Announcements

• Midterm 3 is graded and can be picked up at
  the end of lecture
• The Final Exam is Tuesday June 5 at 2.30 4.20 pm

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About the Final Examination

• Final is at 230 pM on Tuesday (June 5)
• Total point value of Final 150 points
• Tutorial multiple-choice question (20 pts)
• Lecture multiple-choice questions (60 pts)
• Lecture long answer questions (25 pts)
• Lab multiple-choice question (20 pts)
• Tutorial long-answer question (25 pts)
• You may bring three 8½ x 11 pages on which
  you may write anything on both sides. Also be
  sure to bring a Scantron sheet (pre-filled-out as
  much as possible) and a calculator with a good
  battery.
• There will be assigned seating. Look up your
  seat assignment on Tycho before coming to the
  Final.

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The Series RLC Circuit
The figure shows a resistor, inductor, and
capacitor connected in series. The same current
i passes through all of the elements in the loop.
From Kirchhoffs loop law, E vR vL vC.
Because of the capacitive and inductive
elements in the circuit, the current i will not
in general be in phase with E, so we will have i
I cos(wt-f) where f is the phase angle between
current i and drive voltage E. If vLgtvC then the
current i lags E and fgt0. If vCgtvL then i leads
E and flt0.
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Analyzing an LRC Circuit
Draw the current vector I at some arbitrary
angle. All elements of the circuit will have
this current.
Draw the resistor voltage VR in phase with the
current. Draw the inductor and capacitor
voltages VL and VC 900 before and behind the
current, respectively.
The phasors VR and VL-VC form the sides of a
right triangle, with E0 as the hypotenuse.
Therefore, E02 VR2(VL-VC)2.
Draw the emf E0 as the vector sum of VR and
VL-VC. The angle of this phasor is wt, where the
time-dependent emf is E0 cos wt.
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Impedance and Phase Angle
We can define the impedance Z of the circuit
as
From the phasor diagram ,we see that the
phase angle f of the current is given by
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Resonance
The current I will be a maximum when
wL1/wC. This defines the resonant frequency of
the system w0
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ExampleDesigning a Radio Receiver

• An AM radio antenna picks up a 1000 kHz
  signal with a peak voltage of 5.0 mV. The tuning
  circuit consists of a 60 mH inductor in series
  with a variable capacitor. The inductor coil
  has a resistance of 0.25 W, and the resistance of
  the rest of the circuit is negligible.
• To what capacitance should the capacitor be
  tuned to listen to this radio station.
• What is the peak current through the circuit at
  resonance?
• A stronger station at 1050 kHz produces a 10 mV
  antenna signal. What is the current in the radio
  at this frequency when the station is tuned to
  1000 kHz.

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Electromagnetic Fields and Forces
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Field Lines
Field lines start and stop on charges (if any).
Q
-Q
Field lines never cross.
weak
Field line spacing indicates field strength.
strong
Field lines form closed loops only when there is
a current or a flux change in the other field
(i.e., energy flow).
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Gausss Law Revisited
(magnetic monopoles go here)
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The Lorentz Force
Coulombs electric force law
Magnetic force on a moving charge
Lorentz Force Law The most general statement of
electromagnetic forces on a charge.
E and B may be frame-dependent (see the
later part of this lecture), but the Lorentz
Force does not change with frame.
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ExampleThe Motion of a Proton
A proton is launched with velocity v0j into
a region of space where an electric field E0i and
a magnetic field B0i are parallel. How many
cyclotron orbits will the proton make while
traveling a distance L along the x axis? Find an
algebraic expression and evaluate your answer for
E0 10 kV/m, B0 0.1 T, v0 1.0x105 m/s, and
L 10 cm.



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Question
In what direction is the net force on the moving
charge?
(a) Left (b) Right (c) Into
page (d) Up and left at 450 (e) Down and
left at 450
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The Amperian Surface
Amperes Law
Question What restricts the shape and extent of
the surface bounded by the integration path?
Answer The shape of the surface does not
matter. Any surface should be valid. If the
surface intersects no current, the line integral
is zero. Otherwise, it has a non-zero value.
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Something is Missing !!!!
Maxwells Paradox Consider a capacitor that is
being charged by a battery, with a current flow
to the positive plate and from the negative
plate. If the Amperes Law surface goes
through the wire, a current passes through it.
If the Amperes Law surface goes through the
capacitor gap, no current passes through it.
Thus there is a paradox. The line integral of
Amperes Law appears to depend on which surface
is used, bringing its validity into question.
Maxwells Solution Add a displacement current
term that depends on the changing electric field
in the gap.
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Displacement Current
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Induced Magnetic Field
Thus, the situation is symmetric a
changing magnetic field induces an electric
field, and a changing electric field induces a
magnetic field. In both cases, the induced
field lines are in closed loops, and represent
potential sources of energy. Note, however,
that there is a sign difference. The loops are
in opposite directions.
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ExampleFields in a Charging Capacitor
A 2.0 cm diameter parallel plate capacitor
with a 1.0 mm gap is being charged at the rate of
0.50 C/s. What is the magnetic field
strength in the gap at a radius of 0.5 cm?
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A Prelude toMaxwells Equations
Suppose you come across a vector field that
looks something like this. What are the
identifiable structures in this field?
1. An outflow structure
2. An inflow structure
3. An clockwise circulation structure
4. An counterclockwise circulation
structure
Maxwells Equations will tell us that the
flow structures are charges ( and -) and the
circulation structures are energy flows in the
field.
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Maxwells Equations
(magnetic monopole charge goes here)
Gausss Law
Gausss Law for magnetism
Ampère-Maxwell Law
Faradays Law
(magnetic monopole current goes here)
Lorentz Force Law
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A Prelude to Waves
Maxwells formulation of electricity and
magnetism has an interesting consequence. The
equations can be manipulated to give a wave
equations for E and B of the form
This can be recognized as describing an
electromagnetic wave traveling through space with
a velocity of
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E or B? Its frame dependent.
Now turn on a magnetic field into the
diagram. From Bills perspective the charge
experiences a upward vxB force. But from
Sharons perspective, the charge is not moving
and should experience no magnetic force. Do we
have a paradox?
Sharon runs past Bill carrying a positive
charge. From Bills perspective the charge is
moving, but from Sharons perspective the charge
is at rest.
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Galilean Relativity
Consider a reference frame S that is at
rest, and another reference frame S that is
moving at a constant velocity V with respect to S.
Therefore, a force F as observed in S must
have the same magnitude and direction when
observed in S.
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Transformation of E and B
Consequently, in the reference frames of
Bill and Sharon, it wasnt the force that changes
with the motion. Therefore, it must have
been the fields. In Sharons frame, if there was
no magnetic force, there must have been an
electric force. In other words, in her moving
frame there must have been an induced electric
field that produced a force in the upward
direction.
More generally, if an electric field E is
present in S, then in S
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ExampleTransforming the Electric Field
In a laboratory at rest there are fields of
E 10 kV/m and B 0.10 T , both in the x
direction in the laboratory frame. What is
the electric field in a reference frame moving
with velocity V 1.0x105 m/s in the y direction.
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Producing B from Moving E
Now consider Sharon and Bill again. Now the
charge is at rest in Bills reference frame.
From Bills perspective B0, and there is only an
electric field E
From Sharons perspective there is the same
electric field E, since q and r are the same as
in Bills frame
However, Sharon also sees a magnetic field
B produced by the charge moving at -V
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ExampleTwo Views of a Magnetic Field
A 1.0 T magnetic field points upward. A
rocket flies by the laboratory, parallel to the
ground, with a velocity of 1000 m/s. What
are the fields between the magnets pole tips, as
viewed from a scientist aboard the rocket?
.
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Question
Reference frame S observes E and B fields as
shown. Which diagram shows the fields in
reference frame S?
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Lecture 25 Announcements

• Midterm 3 is graded and can be picked up at
  the end of lecture
• The Final Exam is Tuesday June 5 at 2.30 4.20 pm