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VII. Magnetostatics
VII. Magnetostatics
A. Magnetic Field
B. Current is a source of
Magnetic Field
Dr. Bill Pezzaglia
C. Electrodynamics
Updated 2012Feb28
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A. Magnetic Field
1. Discovery of Magnets
•
900 BC: Attributed to shepherd Magnus,
who found nails of his sandals pulled out
by rocks atop Mount Ida
2) Pole Strength
•
(writings of Pliny the elder,
23-79 AD )
3) Magnetic Field
•
Ore “Magnetite” (Iron Oxide) is a
common in Magnesia, Thessaly (Greece).
1) Magnets
Thales of Miletos (624-454 BC)
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• Famous theorems of similar triangles
(a) Loadstone: Magnetite
•
Loadstones (“Leading
Magnets”) used in early
navigation by Chinese
perhaps as early as
1200 BC !
•
Appear in Europe
around 1190 AD
•
Current thought is that
they are magnetized by
lightening strikes
• Amber rubbed with fur attracts straw
• The magnet has a “soul” because it
moves iron.
Here is a narrow
tomb Great Thales
lies; yet his renown
for wisdom reached
the skies
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(b). Magnet Laws
•
Peter de Maricourt (aka Peter
Peregrinus) wrote
famous letter on magnets
August 8, 1269 (31 copies still exist)
•
When you break a magnet you get 2
magnets
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(c). William Gilbert (1544-1603)
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•“Father of Science” (i.e. use
experiments instead of citing
ancient authority)
•1600 Book “De Magnete”
“Magnus magnes ipse est
globus terrestris” (the whole
earth is a magnet)
•Compass points to North Pole of
earth, not to North Star (and
hence N pole is really a South
magnetic pole!)
Earth’s Magnetic Field
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Pole is Moving
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• 1831 Sir James Ross discovers pole near Hudson Bay (70.5°N, 95°W).
• It is now closer to (83°N, 114°W).
•Chinese (720 Ad?)
tabulate that compass does
not point to true north.
•The magnetic axis is
slightly tilted (11°) with
respect to the rotational
axis of the Earth.
•Near San Francisco, the
“magnetic deviation” of a
compass from true north is
about 15° east
Earth’s magnetic field traps charged particles
ejected from the Sun (the solar wind)
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Charged Particles spiraling around magnetic field lines
near north pole makes the Aurora Borealis
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2. Magnetic “Pole Strength”
2b. Wanted: Magnetic Monopole
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(a) Definition
• 1831 Pierre Curie: Why are there no
magnetic monopoles?
• Recall an “electric dipole” is a “stick” of
length “L” with opposite charges ±Q on the
end, Dipole Moment: p=LQ
(other references say 1894?)
• If you break a magnet, you can’t get a “N”
pole by itself, you get another “dipole”
with N & S ends.
• Define “Magnetic Dipole” by same type of
formula: m=Lqm
• 1931 Paul Dirac (using quantum mechanics)
derives what the fundamental magnetic
charge would be in relation to fundamental
charge e (and permeability of free space 0
and Planck’s constant “h”). Experimental
limits say mass is at least 600x of proton.
• Pole Strength qm is “magnetic charge”.
•
Old cgs units: 1 “pole strength” repels another with 1 dyne of
force at 1 cm.
•
New units: Amp-meter (10 of old cgs pole strengths)
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2c. Magnetic Force
• 1750 John Michell comes up with
an inverse-square force law for
magnetic poles (note 38 years
before Coulomb’s similar law for
charge)
• Unit system has been adjusted so
that the Permeability of Free
Space 0 is exactly:
• We could also state that the energy stored
in a dipole magnet would be (in analogy to
electrostatic energy formula):
F
 0 qmQm
4 r 2
 0  4 10 7
1  0 qm
2 4 L
3. The Magnetic Field
•
(a) Discovery:
• 1821 Michael Faraday First
proposes ideas of “Lines of
Force”
•
Example: iron filings over a
magnetic show field lines
e 0
2
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The force between dipoles (along a line) can
be shown to be:
F 6
N
A2
 0 m1m2
4 z 4
Dipoles will twist until they are parallel. The torque of
the first on the second would be given by a cross
product
 

 2
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h
2.c.ii Force between Dipoles
•
U
qm 
 0 m1  m2
4 z 3
3.a.ii Magnetic Flux is Conserved
•
Because there are no
magnetic monopoles, there
are no “sources” of
magnetic field lines.
•
Magnetic Field Lines must
be continuous (i.e. continue
through magnet)
•
Gauss’s law for magnetism: total
magnetic flux through a closed
surface is ZERO.
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http://www.youtube.com/watch?v=uj0DFDfQajw&feature=related
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3b. Definition of Field
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• Definition: Analogous to electric field, except using the
magnetic charge (pole strength)
• Ouch, our definition is in terms of “pole
strength”, which is an abstraction (magnetic
monopoles don’t exist). Instead we usually
measure the magnetic field in terms of:


F  qm B
• Units of Tesla:
T
• Torque on a known dipole:
Weber
N

m2
A m
• Old cgs units: Gauss:
(field of earth is ~1 G)
G
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3.b.ii Definition of Field



  m B
• Change in energy of dipole in a field
Dyne
 10  4 T
pole strength
 
U  m  B  mB cos 
Note: field of our big permanent magnet is only around 0.1 Tesla !
3c. Field of Dipole
•
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The field of a dipole “m” as a function of position vector
“r” is rather messy.
  
 
  3m  r m 
B (r )  0  5  3 
4  r
r 
 0 2m
4 z 3
 m
Bz ( x,0,0)   0 3
4 x
3.c.ii Magnetic Dip
•
Dipole field of earth will not be
parallel to surface, except at
equator!
•
1581 Robert Norman of London,
makes first device in Europe to
measure the “dip” (he had to clip
off one end of a good compass to
make it level). Note was probably
measured 500 years earlier in
Persia.
•
Early Navigators used magnetic
dip to estimate latitude (proposed
by Gilbert)
•
Dip near San Francisco is nearly 60°
downward!
Bz (0,0, z ) 
•
Along z axis simplifies to:
•
Along x axis simplifies to:
•
From this last formula, knowing magnetic field of earth
is about 0.8 G at equator, we get m=1023 Amp-m2.
B. Current is Source of Magnetic Field
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Introduction
•
Recall the source of electric fields is electric
charge (Gauss’s Law)
•
But there are no magnetic charges (aka
monopoles) to create Magnetic Field
•
Source of all magnetism is the movement of
electric charge-either macroscopic current or
microscopic “atomic” currents.
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1) Force of Current on Magnets
2) Electromagnets
3) Ampere’s Law
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1. Force of Current on Magnets
An early clue…
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(a) Oersted’s Experiment (1819)
1751
Benjamin Franklin:
Current in a wire will
deflect a magnet!
electricity can
magnetize needles.
Hans Oersted
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(b) Michael Faraday
•
•
(c) Biot-Savart Law
•
1821: lines of force
circle the wire. There
is no “north” or
“south” pole.
(a) Field of a Coil
tan  
B  0 NI

Be 2r Be
B
Their more general formula
shows the magnetic field
from a small piece of wire L
carrying current is:
  I  
B  0 3 L  r
4 r
2. Electromagnets
• 1820 Johann Schweigger (with Ampere)
invent the (tangent) Galvanometer, a
coil around a compass needle. The
tangent of the angle of deflection is
proportional to the current in the coil
(i.e. primitive current meter).
1820 Biot & Savart show
magnetic field around a is
inversely proportional to
distance. In modern notation:
•
Direction is
determined by
right hand rule
• Applying Biot-Savart Law, the field at
the center of a loop of current of radius r
is easily calculated. The field is
increased by “N” the number of turns.
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BN
0 I
2r
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0 I
2 r
Jean-Baptiste Biot
Felix Savart
(b). Magnetic Moment of Coil
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• Ampere shows that the field of a coil
of “N” turns, loop area “A”, is
equivalent to that of a magnet, with
dipole moment:
m  NIA
• For a Solenoid, the field inside is
nearly constant with value (where N
is the number of turns, and “L” is the
length of the coil)
B  0
NI
L
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(c). Magnetic Cores
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3. Ampere’s Circulation Law
• 1823 Sturgeon invents the electromagnetic
(coil around magnetic core).
Although it is based upon Ampere’s work, the law was
actually formulated by Maxwell 1856.
• 1827 Joseph Henry improves design using
insulating wire.
(a) The General Laws of Maxwell (3 of the 4)
•
• Presence of magnetic core increases field
by a factor of Km (over 100 for Iron, over
10,000 for “mu metal”).
•
• Equivalently, replace 0 by  in all formulas
where permeability of the medium is:
•
  K m 0
b. Statement of Ampere’s Law
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I
L
Discussion/details on the board in class!
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Introduction
•
Term “Electrodynamics was given by
Ampere
•
Reciprocity: By Newton’s 3rd law, if current
makes a force on a magnet, then a magnet
should make a force on a current
•
And since an electromagnet is equivalent
to a magnet, we can deduce there should
be a force between currents.
1) Ampere: Force between wires
2) Lorentz Force Law
3) Torques on Current Loops
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• Field around coaxial cable (i.e. shielding effects)
• Derive Oersted’s result
• Derive field inside of a solenoid
This form is really only useful for very symmetric
situations, for example where B is constant over the
loop. Then you can simply write (where “L” is the
perimeter of the path)
Magnetic Force on Electric Charge
c. Application of Ampere’s Law
• Field around thin wire carrying current I is
independent of thickness of wire.
[This is analogous to electric field around a ball of charge
is independent of radius of ball]
 
B
  dr   0 I
C. Electrodynamics
Gauss’s Law shows that charge is the source of
electric fields (electric flux through a closed surface
is proportional to net enclosed charge)
Gauss’s Law for magnetism states that there are no
magnetic charges (magnetic flux through a closed
surface is zero).
“Ampere’s Circulation Law” is the general statement
that current is the source of magnetic fields.
But we can use Ampere’s Law to show:
In general: The sum of magnetic field about a
closed loop is proportional to the sum of
current passing through the loop
B  0
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1. Forces on Currents
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(b) The “Motor Rule”
(a) Ampere’s Force Law (1820-22)
• Currents in same direction attract
• Currents in opposite direction repel
• Force (per unit length) between current
carrying wires depends on distance “r”:
F  0 I1 I 2

L2 4 r
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• Faraday’s explanation:
• First wire creates B field
B1 
André-Marie Ampère
(1775 -1836)
 0 I1
4 r
Force on second wire carrying
current “I” due to magnetic field
must “B” from first current must be
(aka “Motor Rule”):
• Two wires carrying “1 amp” of current
separated by 1 cm attract with force of
2107 Newtons/meter (definition of
Amp)

 
F  I LB
http://www.youtube.com/watch?v=kapi6ZDvoRs
http://www.stmary.ws/highschool/physics/home/notes/electricity/magnetism/MagForcesBetweenWires.htmhttp://www.stmary.ws/highschool/physics/home/notes/electricity/magnetism/MagForcesBetweenWires.htm
(c) The Electric Motor
•
1820 Faraday invents the
first “homopolar” motor
•
It’s simply a wire that
rotates in a vat of
mercury
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(c) The Electric Motor: History
•
•
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1822 “Barlow’s Wheel”
Copper disk rotates in a
magnetic field when current
flows radially outward
Peter Barlow (1776-1862)
Ignore the sound track: http://www.youtube.com/watch?v=KsuF01pwFfM
Demonstration
•
Needs new rare-earth magnet
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2. Force on Moving Charge
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(a) Lorentz Force Law (1892)
• First ideas probably done (wrong)
by Weber 1846
• First done by Maxwell 1861
• Current is just moving charges
• Force on charge “q” moving with
velocity “v” in magnetic field “B”:

 
F  qvB
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(b) (Edwin) Hall Effect (1879)
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• If current is + charges moving to right
then they will be deflected to the front
and a positive voltage measured.
• Since magnetic force is
perpendicular to velocity, we get
centripetal acceleration.
• If current is - charges moving to left
then they will be deflected to the front
and a negative voltage measured.
• Experiment proves that current in
metals is really negative charges
moving. The voltage induced gives
you the drift velocity (“d” is width).
V  dvD B
• But, the frequency (period) of orbit
turns out to be independent of
speed! Cyclotron frequency
depends only on charge to mass
ratio.
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c.ii. The Cyclotron
•
m
• Electric charges will spiral around
magnetic field lines with radius:
(with higher speed, bigger!)
“Hall Probes” are used to
measure magnetic fields.
Knowing properties of
conductor, the measured
voltage will be proportional to
the presence of magnetic field.
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c. Cyclotron Equation
v2
 qvB
R
R
f 
mv
qB
v
B q

 
2R 2  m 
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c.iii. The Cyclotron
• The largest one in
the United States is
Femi Lab. It is 3
miles in
circumference, and
can produce over
400 GeV.
Invented in 1932 by E.O. Lawrence
and M.S. Livingston. Protons are
injected into the center of two "D"
shaped hollow conductors called
"dees". The perpendicular magnetic
field makes them go in circular orbits.
• They are accelerated across the
space between the Dees by a varying
electric field. That way it accelerates
one way and then as it goes one-half
circle it is accelerated across to the
other side.
• The path gets larger and large and
eventually after the particle gains
enough energy it is ejected to the
target. These generate from 1 to 10
MeV (Million electron Volts) of
energy.
Good Video: http://www.youtube.com/watch?v=M_jIcDOkTAY
Mr Ion: http://www.youtube.com/watch?v=6BxyqFK2KRI
http://www.stkate.edu/physics/phys100/Chapt6.html
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c.iii. The Mass Spectrometer
• Heavier mass isotope will follow BIGGER radius
path in magnetic field
r
mv 1 2Vm

eB B
e
This was how we measured masses of
nuclei (and found out there are “isotopes”)
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(a) Equivalent Magnetic Moment of Loop
• 1919 Thomson’s student Francis Aston constructs
first function mass spectrometer.
• Ionized element is accelerated through a voltage
giving it speed:
3. Torques on Current Loops
v  2V
e
m
• Let “L” be side of square loop
• Force on side wires: F=ILB
• Torque:



L
2
  r  F  2  ILB  IL2 B
• Hence its our old formula for
torque on a magnetic moment,
where moment of loop is:



  m B
m  NIA
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(b) d’Arsonval Galvanometer
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(c) Modern Motor Design
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1832 Sturgeon added the “commutator” which switches the polarity as
the loop turns so that the motion will be continuous.
• 1882 design, the multiple loop
coil in a very strong magnet
made the first very sensitive
ammeter.
http://www.youtube.com/watch?v=Xi7o8cMPI0E&feature=player_embedded
• The coil has a spring on it to pull
it back to center. When current
is added, the torque twists the
coil, moving the pointer.
D. Appendix: Right Hand Rule
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E. References
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• There are many conventions for the right hand rule (and even “left
hand rules”). See http://en.wikipedia.org/wiki/Right-hand_rule
• The convention I am using is the following picture:
• See AJP 67, 448 (1999) which discusses that Ampere is NOT the author of the circulation law that is
named after him!
• http://www.phy6.org/earthmag/lodeston.htm
• http://www.phy6.org/earthmag/demagrev.htm
• http://www.seds.org/messier/xtra/Bios/michell.html
• http://en.wikipedia.org/wiki/Galvanometer
• http://iesfgcza.educa.aragon.es/depart/fisicaquimica/fisicasegundo/videosmagnetismo.html
• http://www.animations.physics.unsw.edu.au/jw/homopolar.htm (includes field rotation
paradox and animations)
• http://chss.montclair.edu/~pererat/impersci.htm (museum of old instruments)
• Interactive Barlow Wheel http://demonstrations.wolfram.com/BarlowsWheel/
• Old Films: http://www.archive.org/details/academic_films
• http://hyperphysics.phy-astr.gsu.edu/Hbase/magnetic/cyclot.html
• Another DC motor video: http://www.youtube.com/watch?v=FjNnRyLexNM&feature=related
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