Physics 272
March 13
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 - 63
Summary
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General form of Ampere's law:
This also true if multiple conductors are present
within the closed integration path
(magnetic field is the sum of the individual magnetic
field of the different conductors)
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Summary
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Field of a toroidal solenoid
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Tightly wound turns
in donut shape
magnetic field concentric
with toroid axis
Each turn is
perpendicular to the
circular axis of the
toroid
Ideal toroid confines
magnetic field to the
space between windings
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Field of a toroidal solenoid
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Field of a toroidal solenoid
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Magnetic materials
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So far currents were surrounded by vacuum
Similar to electric fields, the magnetic field changes due
to material
Particularly iron is making magnetic fields stronger
Moving electrons in atoms cause current loops
→ currents are typically completely random in material
→ in some materials the current loops can be oriented
in an external magnetic field (material is magnetized
→ atomic magnetic field adds to the external magnetic
field
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Paramagnetism
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In an atom most magnetic
moments from the moving
electrons cancel out
BUT: sometimes a net
magnetic dipole moment
remains
Source: http://de.wikipedia.org/wiki/Paramagnetismus
When you place the material in a magnetic field
→ field exerts torque
→ tries to align magnetic moments
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Magnetic field of a current loop is proportional to the magnetic
dipole moment
→ Magnetization:
(total magnetic
moment per unit
volume)
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Paramagnetism
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If a magnetized material completely surrounds a current-carrying wire:
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Materials that can be magnetized are called paramagnetic
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Magnetic field at any point in such a material is enhanced by a
dimensionless factor with respect to vacuum
→ relative permeability: Km
Change of magnetic dipole moment in material:
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Paramagnetism
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Be careful of the different µ:
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The magnetic dipole moment is generally a vector
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The permeability is a dimensionless constant
Magnetic susceptibility:
→ see table of materials in book
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Two competing effects:
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Alignment of magnetic dipole moments in external field
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Random thermal motion → randomizes orientation of dipole moments
→ increasing temperature decreases magnetic susceptibility
→ paramagnetic bodies feel stronger attraction to magnets at
cold temperatures
Phys272 - Spring 14 - von Doetinchem - 73
Diamagnetism
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Some atomic materials
have a zero total magnetic
moment when no magnetic
field is present
http://www.youtube.com/watch?v=IFv4VOrWecI
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BUT: magnetic effects can
be caused by external magnetic fields altering the
electron motions inside the atom (diamagnetic)
→ additional current loops are created
→ additional field is in the opposite direction of
external field (more later)
→ weaken the external magnetic field
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Ferromagnetism
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Examples: iron, nickel, cobalt, ...
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Strong interactions of atomic magnetic dipole moments
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magnetic domains: Complete regions with lined up/parallel magnetic moments
(also present without any external magnetic field)
Domains can be aligned with external field
Permeability is much higher than for paramagnetic materials (1,000-100,000x)
→ ferromagnetic materials are much stronger attracted by a magnet
→ for instance: magnets pick up iron nails, but no aluminum cans
→ use in electromagnets, transformers, generators,...
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A ferromagnetic material
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Consider a cube (side length 2cm) shaped
permanent magnet with magnetization of 8x10 5 A/m
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More complicated wire calculation
3
3
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More complicated wire calculation
2
2
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More complicated wire calculation
B2,x
B2,y=
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More complicated wire calculation
B3,z=
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More complicated wire calculation
2,x
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Review
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Magnetic field of a moving charge:
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Magnetic field of a conductor (law of Biot-Savart)
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Magnetic field of a long, straight, current-carrying
conductor
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Review
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Magnetic field of a current loop
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Ampere's law
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How to find monopoles?
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A topic of current interest in physics research is the
search for an isolated magnetic pole (monopole). If
such an entity were found, how could it be
recognized? What would its properties be?
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analogous to a point charge
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magnetic field lines would terminate on it
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magnetic flux through a closed surface would be
proportional to the net number of magnetic monopoles in
the volume enclosed by the surface
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Paramagnetism vs. Diamagnetism
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The magnetic susceptibility of paramagnetic
materials is quite strongly temperature dependent,
but that of diamagnetic materials is nearly
independent of temperature. Why the difference?
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Paramagnetism is due to partial alignment of magnetic
moments of individual atoms
→ alignment is disrupted by random motion of atoms
→ motion increases when the temperature increases
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Diamagnetic susceptibility depends on how easy it is to
induce a net magnetic moment in an atom with no
magnetic moment in the absence of external fields
→ effect is independent of the initial orientation of the
atom
→ not affected much by temperature
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Electromagnetic induction
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Demo 1:
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magnet moved in
→ magnetic flux through
solenoid changes
→ induced current
appears
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The faster the magnet
the higher the induced
current
http://www.youtube.com/watch?v=hajIIGHPeuU
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If solenoid is approached
first with the other magnetic pole, the direction of the induced
current changes
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When magnet is moved away from the solenoid the direction of
the current changes again.
Demo 2:
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Same as demo 1, but using a different coil and a digital multimeter.
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Electromagnetic induction
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Demo 3:
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two solenoids: one large
one connected in a
simple circuit and a
second, smaller one,
connected to an ammeter
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When switch is closed
→ a DC current is
established in the circuit
→ steady magnetic field is
http://www.youtube.com/watch?v=hajIIGHPeuU
produced in the large
solenoid
→ no induced current in the small solenoid as the magnetic flux through
it does not change
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when switch is switched on or off
→ an induced current is produced
→ for a short period of time the current changes
→ magnetic field is produced by the large solenoid changes as well
→ induced current in the small solenoid.
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Changing magnetic flux
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The key component is the changing magnetic flux
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Flux changes caused by
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magnetic field changes with time
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coil moves through a non-uniform magnetic field
The changing flux causes an induced electromotive
force
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Proportional to the rate of change of magnetic flux
through the coil
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Direction of the induced emf depends on if the flux is
increasing or decreasing
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No flux change = no induced emf
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Induction applications
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Induction is a very
important effect
that is widely used
Electric generators
produces emf by
varying magnetic
flux through coils of
wire
Source: http://de.wikipedia.org/wiki/Elektrischer_Generator
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In this sense any appliance you plug plugged into a wall
jack uses induced emfs
Be careful:
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Magnetically induced emfs result from non-electrostatic forces
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Distinguish from forces cause by electrostatic electric fields
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Faraday's law
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Basic concept: changing magnetic flux through a
circuit
Recall magnetic flux:
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Magnetic flux ΦB describes
the number of field lines
poking through an area A
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Every surface can be
separated in little surface
elements dA
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Magnetic flux is a scalar quantity
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Faraday's law
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Basic concept: changing magnetic flux through a
circuit
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.
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Emf and current induced in a loop
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Uniform magnetic field
between poles of
electromagnet, but
magnitude is
increasing by 0.020T
per second
Coil with area of
2
120cm is in this field,
total resistance 5Ω
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Emf and current induced in a loop
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Uniform magnetic field between poles of
electromagnet, but magnitude is increasing by
0.020T/s
Coil with 120cm2 is in this field, total resistance 5Ω
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Direction of induced electromagnetic fields
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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
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Induced magnetic field in conductor
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For increasing external magnetic field the induced
magnetic field is in the opposite direction and works
against the external field
For decreasing external magnetic field the induced
magnetic field is in the same direction
One more time: not magnetic flux, but changing
magnetic flux causes induction effects
Phys272 - Spring 14 - von Doetinchem - 95