Physics 272
October 23
Fall 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Fall 14 - von Doetinchem - 170
Motional electromotive force
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Origin of electromotive force is of non-electrostatic
nature (similar to battery → chemical)
Charges are brought to a higher potential
Concept can be generalized to conductors of any
shape and in any field (can be non-uniform, but not
varying with time)
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take the perpendicular projection of the velocity with
respect to the magnetic field (cross product)
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Use the parallel projection of the former along a line
element of the conductor (scalar product)
Phys272 - Fall 14 - von Doetinchem - 171
The Faraday disk dynamo
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As before for the slidewire generator:
assume positive free charge carriers
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Positive charges accumulate at the edges
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Radially outward current flow
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Electric field builds up → emf is created
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Difference: velocity depends on the
distance to the center
Phys272 - Fall 14 - von Doetinchem - 172
Induced electric fields
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We understand the concept of induction for moving
charges
Where is induced current coming from if the flux is
changing in a stationary conductor?
Phys272 - Fall 14 - von Doetinchem - 175
Induced electric fields
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Green wire loop is not in a magnetic field
(magnetic field outside solenoid is negligible)
Only the magnetic flux through the loop is changing
Phys272 - Fall 14 - von Doetinchem - 176
Induced electric fields
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Before: charges were pushed through conductor
because of magnetic forces
Conclusion for stationary case: changing magnetic
flux generates an induced electric field in the wire loop
Furthermore:
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induced electric field in the loop is not conservative
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charges gain electric potential
Stationary setup
Phys272 - Fall 14 - von Doetinchem - 177
Induced electric fields
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What does the electric field look like?
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Cylindrical symmetry
→ electric field has
the same magnitude
on the circle
→ has to be tangential
to cancel out according
to Gauss's law (no net
charge present inside)
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Line integral has to be
negative when magnetic
flux is increasing (Lenz's law)
Phys272 - Fall 14 - von Doetinchem - 178
Induced electric fields
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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 - Fall 14 - von Doetinchem - 179
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
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It is not conservative
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Field does non-zero amount of work on charges particle on
closed path
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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 - Fall 14 - von Doetinchem - 180
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 - Fall 14 - von Doetinchem - 184
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 - Fall 14 - von Doetinchem - 185
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 - Fall 14 - von Doetinchem - 186
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 - Fall 14 - von Doetinchem - 187
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 - Fall 14 - von Doetinchem - 188
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 - Fall 14 - von Doetinchem - 189
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 - Fall 14 - von Doetinchem - 190
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 - Fall 14 - von Doetinchem - 191
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 - Fall 14 - von Doetinchem - 192
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 - Fall 14 - von Doetinchem - 193
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 - Fall 14 - von Doetinchem - 194
Additional material
Phys272 - Fall 14 - von Doetinchem - 195
Induced electric fields
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aolenoid with 500 turns, A=4.0cm2, current in
windings is increasing with 100A/s
Phys272 - Fall 14 - von Doetinchem - 196
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
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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
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forces on the ring are always pointing upwards
Phys272 - Fall 14 - von Doetinchem - 197
Conducting and displacement current
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Rod of pure silicon is carrying a current. Electric
field varies sinusoidal with time.
Phys272 - Fall 14 - von Doetinchem - 198
Conducting and displacement current
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Rod of pure silicon is carrying a current. Electric
field varies sinusoidal with time.
Phys272 - Fall 14 - von Doetinchem - 199