Physics 272
March 18
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 - 96
Midterm
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I will reduce homework
load, but you should take
it even more serious
You will get more points
for less homework
→ I really want you do the
homework
→ please do not just copy
from somewhere
→ deep understanding of the
homework will help
you for the final
→ copying homework not
remaining HW is 10% of the grade, final is 50%
Final will be cumulative in the sense that the lecture just builds on the
concepts from the first half.
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Conceptual questions on first half are going to be part of the final
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Problems will focus more on the second part
Phys272 - Spring 14 - von Doetinchem - 97
Suggestions
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Really try do the homework yourself!
Work with the solutions after
submission and make sure you
understand what is going on!
Get the concepts right
→ use “test your understanding
questions from the book”
Phys272 - Spring 14 - von Doetinchem - 98
Last question from the midterm
Phys272 - Spring 14 - von Doetinchem - 99
Last question from the midterm
Outside of the charge distribution → potential of a point charge
Phys272 - Spring 14 - von Doetinchem - 100
Last question from the midterm
Phys272 - Spring 14 - von Doetinchem - 101
Summary
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Basic concept of induction: changing magnetic
flux through a circuit is inducing a current
Faraday's law of induction:
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The induced electromotive force in a closed loop
equals the negative of the time rate of change of
magnetic flux through the loop.
Phys272 - Spring 14 - von Doetinchem - 102
Summary
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steadily increasing
uniform magnetic
field
Induces current
Current direction
depends on flux
change
Increasing flux
→ negative emf
Induced magnetic field
works against external
magnetic field
induced
magnetic field
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A simple alternator
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An alternator is a device that generates emf
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A simple alternator
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Emf is sinusoidal with time → alternating current
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Plane perpendicular to magnetic field:
maximum(minimum) flux
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Plane (anti)parallel: zero flux
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Fastest change when plane (anti)parallel
Careful:
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Electromotive force is not created out of nowhere
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Energy must be conserved and energy has to be
supplied to make the loop spin
→ energy conversion
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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 14 - von Doetinchem - 106
Lenz's law and the slidewire generator
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Metal rod slides on U-shaped conductor
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Uniform magnetic field
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Metal rod slides right → increases the area
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Induced magnetic field is in the opposite direction of
external magnetic field
Phys272 - Spring 14 - von Doetinchem - 107
Lenz's law and the response to flux changes
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Lenz's law gives only the direction of induced
current, not the magnitude
Magnitude of current depends on resistance of
circuit
Lower circuit resistance
→ greater induced current
→ stronger induced magnetic field opposes flux
change
→ more difficult to change flux through the circuit
Phys272 - Spring 14 - von Doetinchem - 108
Lenz's law
http://www.youtube.com/watch?v=sPLawCXvKmg
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Motional electromotive force
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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 14 - von Doetinchem - 114
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 14 - von Doetinchem - 115
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 0.03Ω and a uniform magnetic field of
0.6T:
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Induced current in the loop:
Phys272 - Spring 14 - von Doetinchem - 116
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 0.03Ω and a uniform magnetic field of
0.6T:
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Force on the rod:
opposing the motion of the rod (Lenz's law)
Phys272 - Spring 14 - von Doetinchem - 117
The Faraday disk dynamo
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Like 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
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The Faraday disk dynamo
Velocity increases with distance from center
→ integrate over small segments:
Phys272 - Spring 14 - von Doetinchem - 119
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?
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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
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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/lose electric potential
Phys272 - Spring 14 - von Doetinchem - 124
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)
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Induced electric fields
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aolenoid with 500 turns, A=4.0cm2, current in
windings is increasing with 100A/s
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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
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