Download Induced electric fields

Physics 272
March 17
Spring 2015
www.phys.hawaii.edu/~philipvd/pvd_15_spring_272_uhm
go.hawaii.edu/KO
Prof. Philip von Doetinchem
[email protected]
PHYS272 - Spring 15 - von Doetinchem - 133
Direction of induced electromagnetic fields
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For increasing external magnetic field the induced magnetic field is in the opposite direction and
works against the external field
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For decreasing external magnetic field the induced magnetic field is in the same direction
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Sign rules for the direction of induced emf:
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Define positive direction of area
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Determine the sign of the magnetic flux from the area and the magnetic field
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If flux is increasing → induced emf is negative
If flux is decreasing → induced emf is positive
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Right hand rule:
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align area vector with thumb
Positive emf → current is in the same direction as curled fingers
Negative emf → current is in the opposite direction of curled fingers
Not magnetic flux, but
changing magnetic flux
causes induction effects
PHYS272 - Spring 15 - von Doetinchem - 134
Lenz's law
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Alternative method for determining
the direction of induced current or
emf
Lenz's law can be derived from
Faraday's law
The direction of any magnetic
induction effect is such as to
oppose the cause of the effect
Cause can be
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Changing flux due to varying magnetic field
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Changing flux due to motion of conductors
Source: http://de.wikipedia.org/wiki/Emil_Lenz
Heinrich F. E. Lenz
1804-1865
Think about it like: induced current tries keeping the
system in the state it was before the flux change happened.
PHYS272 - Spring 15 - von Doetinchem - 135
Slidewire generator
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Look at individual charge in slidewire:
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Feels magnetic force
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Separates charges
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Builds up electric field
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Equilibrium between electric force and
magnetic force
(→ also see Hall effect)
No magnetic forces act on the charges in
the stationary U part, but sliding rod
creates potential difference (source of emf)
→ establishes current
PHYS272 - Spring 15 - von Doetinchem - 139
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 - Spring 15 - von Doetinchem - 140
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 - Spring 15 - von Doetinchem - 143
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 - Spring 15 - von Doetinchem - 144
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 - Spring 15 - von Doetinchem - 145
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 - Spring 15 - von Doetinchem - 146
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 - Spring 15 - von Doetinchem - 147
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 - Spring 15 - von Doetinchem - 148
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 15 - von Doetinchem - 152
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 15 - von Doetinchem - 153
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 15 - von Doetinchem - 154
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 15 - von Doetinchem - 155
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 15 - von Doetinchem - 156
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 15 - von Doetinchem - 157
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 15 - von Doetinchem - 158
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 15 - von Doetinchem - 159
Additional material
PHYS272 - Spring 15 - von Doetinchem - 160
Induced electric fields
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solenoid with 500 turns, A=4.0cm2, current in
windings is increasing with 100A/s
PHYS272 - Spring 15 - von Doetinchem - 161
Motional emf in the slidewire generator
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Suppose the moving rod in the slidewire configuration
is 0.1m long and 2.5m/s fast with a resistance of 0.03
and a uniform magnetic field of 0.6T:
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Induced current in the loop:
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Force on the rod
PHYS272 - Spring 15 - von Doetinchem - 162
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 - Spring 15 - von Doetinchem - 163
Bar on inclined plane
PHYS272 - Spring 15 - von Doetinchem - 164
Bar on inclined plane
PHYS272 - Spring 15 - von Doetinchem - 165
Bar on inclined plane
PHYS272 - Spring 15 - von Doetinchem - 166
Bar on inclined plane
PHYS272 - Spring 15 - von Doetinchem - 167
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 15 - von Doetinchem - 168
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 15 - von Doetinchem - 169
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 15 - von Doetinchem - 170
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 - Spring 15 - von Doetinchem - 171
Conducting and displacement current
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Rod of pure silicon is carrying a current. Electric
field varies sinusoidal with time.
PHYS272 - Spring 15 - von Doetinchem - 172
Conducting and displacement current
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Rod of pure silicon is carrying a current. Electric
field varies sinusoidal with time.
PHYS272 - Spring 15 - von Doetinchem - 173