Download Magnetic dipole in a nonuniform magnetic field

Physics 272
February 25
Spring 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html
Prof. Philip von Doetinchem
[email protected]
Phys272 - Spring 14 - von Doetinchem - 372
Magnetic forces on moving charges
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Magnitude of the force is proportional to amount of
charge
Magnitude of the force is proportional to the
magnetic field strength.
Magnitude of the force is proportional to the velocity
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electric force is always the same: no matter if charge
moves or not!
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Particle at rest does not feel magnetic force
Force is perpendicular to the velocity and magnetic
field
Phys272 - Spring 14 - von Doetinchem - 373
Thomson's e/m experiment
http://www.youtube.com/watch?v=o1z2S3ME0cI
Phys272 - Spring 14 - von Doetinchem - 374
Forces on a current-carrying Wire
http://www.youtube.com/watch?v=43AeuDvWc0k
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When wires are connected in series and power is applied they
will repel each other
→ currents are going in opposite directions and repel
when they are connected in parallel they will attract one another
→ the currents in each are going in the same direction and
attract
Phys272 - Spring 14 - von Doetinchem - 375
Magnetic force on a current-carrying conductor
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How does an electric motor work?
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Magnets exert force on moving charges (currents) in wires
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These forces make the motor turn
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Average charge on each charge:
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Total force on all moving charges
Phys272 - Spring 14 - von Doetinchem - 376
Magnetic force on a current-carrying conductor
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General case: B field is not perpendicular to wire:
Non straight wire → divide into infinitesimal small
sections:
Negative charges move the opposite direction and
the force goes in the same direction as for positive
charges
Phys272 - Spring 14 - von Doetinchem - 377
Loud speaker
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Radial magnetic field of
permanent magnets
exerts force on voice coil
Source: http://en.wikipedia.org/wiki/Loudspeaker
magnet
Current in voice coil depends
on the signal from the
amplifier
Direction of current decides
the direction of the force
Speaker cone starts vibrating
Volume knob turns up the
current amplitude
voicecoil
suspension
diaphragm
Phys272 - Spring 14 - von Doetinchem - 378
Magnetic force on a curved conductor
Phys272 - Spring 14 - von Doetinchem - 379
Magnetic force on a curved conductor
1
Phys272 - Spring 14 - von Doetinchem - 380
Magnetic force on a curved conductor
Phys272 - Spring 14 - von Doetinchem - 381
Force and torque on a current loop
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Current-carrying conductors often form closed loops
Calculate torque on a loop in a magnetic field
application: loud speaker
Example:
rectangular loop
in a uniform
magnetic field
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The total force
on the loop is
zero
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But the total torque is
generally not zero
0
µ
Phys272 - Spring 14 - von Doetinchem - 382
Force and torque on a current loop
Phys272 - Spring 14 - von Doetinchem - 383
Magnetic torque: vector form
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Greatest torque when magnetic dipole moment and
magnetic field are perpendicular
Torque is zero when magnetic dipole moment and
magnetic field are (anti)parallel
Analogue to electric dipole moment and electric field
Phys272 - Spring 14 - von Doetinchem - 384
Potential energy for a magnetic dipole
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If magnetic dipole changes orientation in magnetic field
→ the field does work on it
In analogy to the potential energy of an electric dipole
→ potential energy for a magnetic dipole:
Potential energy is zero when magnetic moment is
perpendicular to the field
torque tries to align magnetic moment and magnetic field
Derived equations are also true for any type of plane
loop and not only for rectangular loops
Phys272 - Spring 14 - von Doetinchem - 385
Magnetic torque: loops and coils
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Solenoid:
helical winding of wire
Close spacing of
windings
→ approximate as
circular loops
Total magnetic torque
of a solenoid in a uniform
magnetic field is just the
sum of the torque of the individual windings:
Solenoids are important as source of magnetic fields
Phys272 - Spring 14 - von Doetinchem - 386
MRI: Magnetic resonance imaging
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Body consists of a lot of water → hydrogen atoms
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Hydrogen atoms have a magnetic dipole moment
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If you place a human body in a strong magnetic field
→ magnetic dipole moments of hydrogen align with
the field
illuminating the aligned moments with radio waves
can locally flip the magnetic moments
(quantum mechanics give the explanation)
Measure how many radio waves are absorbed
→ tells you how much hydrogen is present
Ideal for analyzing soft tissue that is transparent for
X-ray imaging
Phys272 - Spring 14 - von Doetinchem - 387
Magnetic dipole in a nonuniform magnetic field
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Forces in radial direction
cancel out
Non-uniform components
create a net force in
direction of the magnetic
field
Phys272 - Spring 14 - von Doetinchem - 388
How to pick up an unmagnetized object
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Picture an electron as spinning around the atom
nucleus (again QM needed for deeper understanding)
→ charged electron creates a current
→ can be approximated as current-carrying loop
→ creates magnetic dipole moment
Large fraction of magnetic
dipole moments of electrons
in iron atom align
→ iron has non-zero magnetic
dipole moment
Piece of iron: dipole moments
of individual iron atoms are not
aligned
Phys272 - Spring 14 - von Doetinchem - 389
How to pick up an unmagnetized object
→ placing it in a strong
magnetic field causes
alignment and generates
a magnetic dipole moment
of the iron
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Dropping the iron piece or
heating can randomize the
magnetic dipole moments of
the atoms again
→ total magnetic dipole moment of iron piece goes
back to zero
Phys272 - Spring 14 - von Doetinchem - 390
How to pick up an unmagnetized object
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Picking up an unmagnetized object:
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Magnet aligns magnetic dipole moments in
unmagnetized object with its magnetic field
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Non-uniform magnetic field attracts magnetic dipole
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Effect does not depend on holding the magnetized object
close to the south or north pole
→ magnetic moment always tends to align with the
magnetic field → attraction
Other materials have smaller tendencies to align
their magnetic dipole moments
→ more in future lectures
Phys272 - Spring 14 - von Doetinchem - 391
The direct-current motor
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Magnetic torque is converted into mechanical energy
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Direct current motor:
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Loop in a magnetic field
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Magnetic dipole moment is
generated by external
current source every time
the current loop aligns with
the magnetic field
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Torque is created and loops
start spinning
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After the spinning 180deg
→ magnetic dipole moment is
reversed with respect to the
loop, but stays the same with
respect the magnetic field
→ loop continues to spin in the
same direction
Phys272 - Spring 14 - von Doetinchem - 392
Magnetic motor
http://www.youtube.com/watch?v=STnsB5DE9pk
toroid with three different wire windings is connected to 220 VAC 3-phase
voltage
● voltage phase of each of the three windings lags 120 degrees behind the next
→ changing induced magnetic field
→ changing field causes metal objects to rotate when placed inside.
● Motors using this principle are very common
● power lines are often seen in sets of three to provide three phases
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Phys272 - Spring 14 - von Doetinchem - 393
The Hall effect
http://www.youtube.com/watch?v=AcRCgyComEw
Phys272 - Spring 14 - von Doetinchem - 394
The Hall effect
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Conductor strip perpendicular to a magnetic field
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Magnetic force causes polarization effect in material
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Electric field between lower and upper side of strip builds
up
Phys272 - Spring 14 - von Doetinchem - 395
The Hall effect
Phys272 - Spring 14 - von Doetinchem - 396
The Hall effect
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copper strip: 2mm thick, 1.5cm long
current through strip: 75A
magnetic field perpendicular to strip 2T
AH=−5.3·10−11m3/C
→ Hall voltage: VH=4.0µV
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Hall voltages are small and are another example to
study the sign of the charge carriers
Phys272 - Spring 14 - von Doetinchem - 397
Review
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Magnetic force is a fundamental interaction between moving charged
particles
Magnetic field can be represented by magnetic field lines
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Magnetic field direction is a tangent to the line
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The resulting magnetic force is always perpendicular to the magnetic field
direction
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Magnetic flux through a closed surface is always zero
In a uniform magnetic field charged particles move in a circle at
certain radius
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Current-carrying conductors feel magnetic force
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Torque on a current loop is a common application of magnetic force
Phys272 - Spring 14 - von Doetinchem - 398
Current loop as a compass
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How might a loop of wire carrying a current be used
as a compass? Could such a compass distinguish
between north and south?
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current loop pivoted about a vertical diameter
→ earth’s magnetic field will provide a torque that aligns
the normal to the loop with the earth’s field
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earth’s magnetic field points toward the north geographic
pole
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current direction in the loop is known
→ direction of the magnetic dipole moment known
→ alignment of the loop determines the direction earth’s
field
Phys272 - Spring 14 - von Doetinchem - 399
Lightning strike and earth's magnetic field
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A lightning strike hits a metal flagpole. A typical
current can be as high as 100,000A. Can such a
strike bend the flagpole?
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Flagpole length 10m
magnetic field maximum 0.1mT
current along the pole 100,000A
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Force = 10m x 0.1mT x 100,000A = 100N
→ not enough force to bend a flagpole
→ magnetic field is also not perpendicular to Earth
surface (most flagpoles are) → effect further reduced
Phys272 - Spring 14 - von Doetinchem - 400