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VII. Magnetostatics

Dr. Bill Pezzaglia

Updated 2012Feb28

VII. Magnetostatics

A. Magnetic Field

B. Current is a source of Magnetic Field

C. Electrodynamics

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A. Magnetic Field

1) Magnets

2) Pole Strength

3) Magnetic Field

31. Discovery of Magnets

• 900 BC: Attributed to shepherd Magnus, who found nails of his sandals pulled out by rocks atop Mount Ida

• (writings of Pliny the elder, 23-79 AD )

• Ore “Magnetite” (Iron Oxide) is acommon in Magnesia, Thessaly (Greece).

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Thales of Miletos (624-454 BC)

• Famous theorems of similar triangles

• Amber rubbed with fur attracts straw

• The magnet has a “soul” because it moves iron.

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Here is a narrow tomb Great Thales lies; yet his renown for wisdom reached the skies

(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

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(b). Magnet Laws• Peter de Maricourt (aka Peter

Peregrinus) wrotefamous letter on magnetsAugust 8, 1269 (31 copies still exist)

• When you break a magnet you get 2 magnets

7(c). William Gilbert (1544-1603)

•“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!)

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Earth’s Magnetic Field

•Chinese (720 Ad?) tabulate that compass does not point to true north.

•The magnetic axis is slightly tilted (11°) withrespect to the rotational axis of the Earth.

•Near San Francisco, the “magnetic deviation” of a compass from true north is about 15° east

9 Pole is Moving 10

• 1831 Sir James Ross discovers pole near Hudson Bay (70.5°N, 95°W).• It is now closer to (83°N, 114°W).

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”

(a) Definition

• Recall an “electric dipole” is a “stick” of length “L” with opposite charges ±Q on the end, Dipole Moment: p=LQ

• Define “Magnetic Dipole” by same type of formula: m=Lqm

• Pole Strength qm is “magnetic charge”.

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• 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)

2b. Wanted: Magnetic Monopole

• 1831 Pierre Curie: Why are there no magnetic monopoles?(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.

• 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.

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0eh

qm

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):

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20

4 r

QqF mm

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0 104A

N

L

qU m

20

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2.c.ii Force between Dipoles

• The force between dipoles (along a line) can be shown to be:

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4210

46

z

mmF

• Dipoles will twist until they are parallel. The torque of the first on the second would be given by a cross product

3210

42

z

mm

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

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http://www.youtube.com/watch?v=uj0DFDfQajw&feature=related

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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3b. Definition of Field

• Definition: Analogous to electric field, except using the magnetic charge (pole strength)

• Units of Tesla:

• Old cgs units: Gauss:(field of earth is ~1 G)

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BqF m

mA

N

m

WeberT

2

Tstrengthpole

DyneG 410

Note: field of our big permanent magnet is only around 0.1 Tesla !

3.b.ii Definition of Field

• 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:

• Torque on a known dipole:

• Change in energy of dipole in a field

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Bm

cosmBBmU

3c. Field of Dipole

• The field of a dipole “m” as a function of position vector “r” is rather messy.

• 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.

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350 3

4)(

r

m

r

rmrB

30 2

4),0,0(

z

mzBz

30

4)0,0,(

x

mxBz

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!

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B. Current is Source of Magnetic Field

1) Force of Current on Magnets

2) Electromagnets

3) Ampere’s Law

23Introduction

• 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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An early clue…

1751Benjamin Franklin:

electricity can magnetize needles.

251. Force of Current on Magnets

(a) Oersted’s Experiment (1819)

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Current in a wire will deflect a magnet!

Hans Oersted

(b) Michael Faraday

• 1821: lines of force circle the wire. There is no “north” or “south” pole.

• Direction is determined by right hand rule

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(c) Biot-Savart Law

• 1820 Biot & Savart show magnetic field around a is inversely proportional to distance. In modern notation:

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Jean-Baptiste Biot

Felix Savart

• Their more general formula shows the magnetic field from a small piece of wire Lcarrying current is:

rLr

IB

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4

r

IB

2

0

2. Electromagnets

(a) Field of a Coil

• 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.

• 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).

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r

INB

20

ee B

NI

rB

B

2tan 0

(b). Magnetic Moment of Coil

• Ampere shows that the field of a coil of “N” turns, loop area “A”, is equivalent to that of a magnet, with dipole moment:

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NIAm

L

NIB 0

• For a Solenoid, the field inside is nearly constant with value (where Nis the number of turns, and “L” is the length of the coil)

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(c). Magnetic Cores

• 1823 Sturgeon invents the electromagnetic (coil around magnetic core).

• 1827 Joseph Henry improves design using insulating wire.

• 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:

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0 mK

3. Ampere’s Circulation Law

Although it is based upon Ampere’s work, the law was actually formulated by Maxwell 1856.

(a) The General Laws of Maxwell (3 of the 4)• 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.

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b. Statement of Ampere’s Law

In general: The sum of magnetic field about a closed loop is proportional to the sum of current passing through the loop

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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)

IrdB 0

L

IB 0

c. Application of Ampere’s Law

But we can use Ampere’s Law to show:

• 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]

• Field around coaxial cable (i.e. shielding effects)

• Derive Oersted’s result

• Derive field inside of a solenoid

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Discussion/details on the board in class!

C. Electrodynamics

Magnetic Force on Electric Charge

1) Ampere: Force between wires

2) Lorentz Force Law

3) Torques on Current Loops

35Introduction

• 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.

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1. Forces on Currents

(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”:

• Two wires carrying “1 amp” of current separated by 1 cm attract with force of 2107 Newtons/meter (definition of Amp)

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André-Marie Ampère(1775 -1836)

r

II

L

F 210

2 4

http://www.youtube.com/watch?v=kapi6ZDvoRshttp://www.stmary.ws/highschool/physics/home/notes/electricity/magnetism/MagForcesBetweenWires.htmhttp://www.stmary.ws/highschool/physics/home/notes/electricity/magnetism/MagForcesBetweenWires.htm

(b) The “Motor Rule”

• Faraday’s explanation:

• First wire creates B field

Force on second wire carrying current “I” due to magnetic field must “B” from first current must be (aka “Motor Rule”):

r

IB 10

1 4

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BLIF

(c) The Electric Motor

• 1820 Faraday invents the first “homopolar” motor

• It’s simply a wire that rotates in a vat of mercury

39(c) The Electric Motor: History

• 1822 “Barlow’s Wheel”• Copper disk rotates in a

magnetic field when current flows radially outward

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Peter Barlow (1776-1862)

Ignore the sound track: http://www.youtube.com/watch?v=KsuF01pwFfM

Demonstration

• Needs new rare-earth magnet

41 2. Force on Moving Charge

(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”:

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BvqF

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(b) (Edwin) Hall Effect (1879)

• If current is + charges moving to right then they will be deflected to the front and a positive voltage measured.

• 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).

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BdvV D

“Hall Probes” are used to measure magnetic fields. Knowing properties of conductor, the measured voltage will be proportional to the presence of magnetic field.

c. Cyclotron Equation

• Since magnetic force is perpendicular to velocity, we get centripetal acceleration.

• Electric charges will spiral around magnetic field lines with radius:(with higher speed, bigger!)

• 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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qvBR

vm

2

qB

mvR

m

qB

R

vf

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c.ii. The Cyclotron

• 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.

45 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.

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http://www.stkate.edu/physics/phys100/Chapt6.html

Good Video: http://www.youtube.com/watch?v=M_jIcDOkTAYMr Ion: http://www.youtube.com/watch?v=6BxyqFK2KRI

c.iii. The Mass Spectrometer

• 1919 Thomson’s student Francis Aston constructs first function mass spectrometer.

• Ionized element is accelerated through a voltage giving it speed:

• Heavier mass isotope will follow BIGGER radius path in magnetic field

m

eVv 2

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e

Vm

BeB

mvr

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This was how we measured masses of nuclei (and found out there are “isotopes”)

3. Torques on Current Loops

(a) Equivalent Magnetic Moment of Loop

• Let “L” be side of square loop

• Force on side wires: F=ILB

• Torque:

• Hence its our old formula for torque on a magnetic moment, where moment of loop is:

BILILBL

Fr 2

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NIAm

Bm

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(b) d’Arsonval Galvanometer

• 1882 design, the multiple loop coil in a very strong magnet made the first very sensitive ammeter.

• 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.

49 (c) Modern Motor Design1832 Sturgeon added the “commutator” which switches the polarity as

the loop turns so that the motion will be continuous.

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http://www.youtube.com/watch?v=Xi7o8cMPI0E&feature=player_embedded

D. Appendix: Right Hand Rule• 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:

51 E. References

• 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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