Physics 272
March 20
Spring 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Spring 14 - von Doetinchem - 129
Summary
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No magnetic forces act on the charges in the
stationary U part, but sliding rod creates potential
difference (source of emf)
Phys272 - Spring 14 - von Doetinchem - 130
Bar on inclined plane
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Bar on inclined plane
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Bar on inclined plane
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Bar on inclined plane
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Summary
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Only the magnetic flux through the loop is changing
Phys272 - Spring 14 - von Doetinchem - 135
Summary
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What does the electric field look like?
Line integral has to be negative when magnetic flux
is increasing (Lenz's law)
Phys272 - Spring 14 - von Doetinchem - 136
Nonelectrostatic electric fields
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Faraday's law works for two different situations:
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Induced current from magnetic forces when conductor
moves through magnetic field
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Time-varying magnetic field induces electric field in a
stationary conductor and induces a current
The electric field of the 2nd case is also induced when
no conductor is present
–
It is not conservative
–
Field does non-zero amount of work on charges particle on
closed path
–
This is a non-electrostatic electric field in contrast to a
electrostatic electric field
A change of magnetic field acts as a source of electric
field that cannot be produced with a static distribution
Phys272 - Spring 14 - von Doetinchem - 137
Eddy currents
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Induced currents are not necessarily confined to
well-defined paths in conductors
Induced eddy-like currents can form in any type of
metal in changing magnetic fields or by moving
through a magnetic field
Applications:
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Currents causes heating → induction furnace
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Eddy currents causes braking effect → trains
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Metal detector at the airport:
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Magnetic field creates eddy current in objects
Eddy current creates induced magnetic field
Induced magnetic field creates eddy currents in receiver coil
Phys272 - Spring 14 - von Doetinchem - 141
Direction of eddy currents
upper current: falls through region of increased magnetic field
→ builds up induced magnetic field against external field
(counter-clockwise current)
metallic disk falling
through magnetic field
- currents to the right
are induced
- induced currents feel
upward magnetic force
→ slow down velocity
stationary magnetic field
only disk is moving magnetic
field is stationary
lower current: falls through region of decreased magnetic field
→ builds up induced magnetic field trying to maintain the external field
(clockwise current)
Phys272 - Spring 14 - von Doetinchem - 142
Eddy currents
https://www.youtube.com/watch?v=Pl7KyVIJ1iE
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solid metal ring placed on iron core whose base is wrapped in wire
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when DC current is passed through the wire, a magnetic field is formed in the iron core
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this sudden magnetic field induces a current in the metal ring, which in turn creates another magnetic field that opposes the
original field
–
ring briefly jumps upwards
cut in the ring
–
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cannot form current inside → will not jump
ring is cooled in liquid nitrogen → resistance of the metal is lowered → more current to flow.
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ring jump jumps higher
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magnetic field curves away at the top of the iron coil → with DC power ring will never fly off the top
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When AC current is passed through wire → ring flies off the top of the iron core.
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current lags the emf by 90 degrees in inductors
–
forces on the ring are always pointing upwards
Phys272 - Spring 14 - von Doetinchem - 143
Eddy currents
https://www.youtube.com/watch?v=7_-RqkYatWI
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solid copper pendulum mounted between poles of an electromagnet
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pendulum is set into motion
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then the magnets are turned on
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magnets induce eddy currents in the copper opposing the motion of the pendulum
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pendulum quickly slows to a stop
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eddy current braking
copper pendulum with strips cut into it is not slowed nearly as much as the solid
pendulum
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cuts in the copper prevent large eddy currents from forming
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only eddy currents smaller than strips of copper can be formed
Phys272 - Spring 14 - von Doetinchem - 144
Displacement current and Maxwell's equations
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A varying magnetic field creates an induced electric
field
Varying electric fields also create magnetic fields
Essential feature to understand electromagnetic
waves
To understand relationship: look at charging of
capacitor
Phys272 - Spring 14 - von Doetinchem - 145
Displacement current and Maxwell's equations
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Look at charging of capacitor:
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Conducting current ic charges capacitor and builds up electric field
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No conducting current between plates
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Applying Ampere's law to both situations reveals contradiction:
Phys272 - Spring 14 - von Doetinchem - 146
Displacement current and Maxwell's equations
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Electric flux increases while conducting current is
decreasing
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Charge on capacitor:
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Charging capacitor → current changes:
Phys272 - Spring 14 - von Doetinchem - 147
Displacement current and Maxwell's equations
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Discrepancy from last slides can be resolved by having the
change in conducting current translate into a change of electric
flux
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Ampere's law becomes:
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Displacement current density:
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In this sense the displacement current is going through the
capacitor
Phys272 - Spring 14 - von Doetinchem - 148
The reality of displacement current
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Physical significance of displacement current?
This magnetic field can me measured and has a
real physical meaning
Phys272 - Spring 14 - von Doetinchem - 149
Conducting and displacement current
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Rod of pure silicon is carrying a current. Electric
field varies sinusoidal with time.
Phys272 - Spring 14 - von Doetinchem - 150
Conducting and displacement current
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Rod of pure silicon is carrying a current. Electric
field varies sinusoidal with time.
Phys272 - Spring 14 - von Doetinchem - 151
Maxwell's equations of electromagnetism
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Gauss's law for electric fields (surface integral)
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Electric field is related to total charge in an enclosed
surface
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Electric charges are sources of magnetic fields
Gauss's law for magnetism (surface integral)
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No magnetic monopoles exist
→ magnetic flux through closed surface is always zero
Phys272 - Spring 14 - von Doetinchem - 152
Maxwell's equations of electromagnetism
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Ampere's law (line integral)
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Conducting and displacement current act as sources of
magnetic fields
Faraday's law (line integral)
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A changing magnetic field or magnetic flux induces an
electric field
Phys272 - Spring 14 - von Doetinchem - 153
Maxwell's equations of electromagnetism
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Electric field in Maxwell's equation is a
superposition of
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the conservative part from the electrostatic field caused
by a charge distribution
(does not contribute to line integral in Faraday's law)
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The non-conservative part caused by induced currents
(does not contribute to surface integral in Gauss's law as
it is not caused by static charges)
Time-varying field of either kind induce field of the
other kind
Starting point for electromagnetic wave discussion
→ physical basis for light, X-ray, etc.
Phys272 - Spring 14 - von Doetinchem - 154
Review
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Faraday's law: induced emf is the negative of the time rate
of change of magnetic flux through a loop
Lenz's law: induced current/emf tends to cancel the change
that caused it.
Motional emf: moving conductor in magnetic field causes
motional emf
Induced electric fields:
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emf induced by a changing magnetic flux through a stationary
conductor
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induced electric field of non-conservative nature
Phys272 - Spring 14 - von Doetinchem - 155
Review
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Maxwell's equations
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Surface integrals:
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Line integrals:
Phys272 - Spring 14 - von Doetinchem - 156
Discussion
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The angular speed of the loop is doubled, then the
frequency with which the induced current changes
direction doubles, and the maximum emf also doubles.
Why?
Does the torque required to turn the loop change?
Phys272 - Spring 14 - von Doetinchem - 157
Discussion
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The angular speed of the loop is doubled, then the
frequency with which the induced current changes direction
doubles, and the maximum emf also doubles. Why?
Does the torque required to turn the loop change?
when angular speed is doubled the rate of change of the flux
doubles and this causes the induced emf and induced
current to double
torque required is proportional to the current in the loop, so
the torque also doubles
Phys272 - Spring 14 - von Doetinchem - 158
Discussion
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An airplane flies over Antarctica, where the
magnetic field of the earth mostly directed upward
away from the ground.
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As viewed by a passenger facing toward the front of the
plane, is the left or the right wingtip at higher potential?
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Does the answer depend on the direction the plane is
flying?
Phys272 - Spring 14 - von Doetinchem - 159
Discussion
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An airplane flies over Antarctica, where the magnetic field
of the earth mostly directed upward away from the ground.
–
As viewed by a passenger facing toward the front of the plane, is
the left or the right wingtip at higher potential?
–
Does the answer depend on the direction the plane is flying?
positive charge collects toward the right; the righthand
wing tip is at higher potential
does not depend on the direction the plane is flying
Phys272 - Spring 14 - von Doetinchem - 160
Discussion
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Would it be appropriate to ask how much energy an
electron gains during a complete trip around the
wire loop with a certain induced current?
Would it be appropriate to ask what potential
difference the electron moves through during such a
trip?
Phys272 - Spring 14 - von Doetinchem - 161
Discussion
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The work done on an electron by the induced electric
field during a complete trip around the loop is e ε
energy can be removed from the electron due to the
resistance of the loop
The induced electric field is a non-conservative field
→ path does matter in this case, not just the potential
difference
Phys272 - Spring 14 - von Doetinchem - 162