VII. Magnetostatics. VII. Magnetostatics. 1. Discovery of Magnets. Thales of Miletos ( BC) (a) Loadstone: Magnetite. A.

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VII. Magnetostatics VII. Magnetostatics A. Magnetic Field Dr. Bill Pezzaglia B. Current is a source of Magnetic Field Updated 01Feb8 C.

Discovery of Magnets 4 1) Magnets 900 BC: Attributed to shepherd Magnus, who found nails of his sandals pulled out by rocks atop Mount Ida ) Pole

Thales of Miletos (64-454 BC) 5 (a) Loadstone: Magnetite 6 Faous theores of siilar triangles Aber rubbed with fur attracts straw The agnet has a soul

1 VII. Magnetostatics VII. Magnetostatics A. Magnetic Field Dr. Bill Pezzaglia B. Current is a source of Magnetic Field Updated 01Feb8 C. Electrodynaics A. Magnetic Field 1. Discovery of Magnets 4 1) Magnets 900 BC: Attributed to shepherd Magnus, who found nails of his sandals pulled out by rocks atop Mount Ida ) Pole Strength (writings of Pliny the elder, -79 AD ) ) Magnetic Field Ore Magnetite (Iron Oxide) is a coon in Magnesia, Thessaly (Greece). Thales of Miletos ( BC) 5 (a) Loadstone: Magnetite 6 Faous theores of siilar triangles Aber rubbed with fur attracts straw The agnet has a soul because it oves iron. Loadstones ( Leading Magnets ) used in early navigation by Chinese perhaps as early as 100 BC! Appear in Europe around 1190 AD Here is a narrow tob Great Thales lies; yet his renown for wisdo reached the skies Current thought is that they are agnetized by lightening strikes 1

wrote faous letter on agnets August 8, 169 (1 copies still exist) Father

ce (i.e. use experients instead of citing ancient authority) When you

globus terrestris (the whole earth is a agnet) Copass points to North Pole

pole!) Chinese (70 Ad?) tabulate that copass does not point to true north.

pole near Hudson Bay (70.5 N, 95 W). It is now closer to (8 N, 114 W).

2 (b). Magnet Laws 7 (c). Willia Gilbert ( ) 8 Peter de Maricourt (aka Peter Peregrinus) wrote faous letter on agnets August 8, 169 (1 copies still exist) Father of Science (i.e. use experients instead of citing ancient authority) When you break a agnet you get agnets 1600 Book De Magnete Magnus agnes ipse est globus terrestris (the whole earth is a agnet) Copass points to North Pole of earth, not to North Star (and hence N pole is really a South agnetic pole!) Chinese (70 Ad?) tabulate that copass does not point to true north. Earth s Magnetic Field 9 Pole is Moving Sir Jaes Ross discovers pole near Hudson Bay (70.5 N, 95 W). It is now closer to (8 N, 114 W). The agnetic axis is slightly tilted (11 ) with respect to the rotational axis of the Earth. Near San Francisco, the agnetic deviation of a copass fro true north is about 15 east Earth s agnetic field traps charged particles ejected fro the Sun (the solar wind) 11 1 Charged Particles spiraling around agnetic field lines near north pole akes the Aurora Borealis

onopoles? (other references say 1894?) Define Magnetic Dipole by sae type of forula: =Lq If you break a agnet, you can t get a N pole by itself, you get another dipole with N & S ends.

New units: Ap-eter (10 of old cgs pole strengths) 191 Paul Dirac (using quantu echanics) derives what the fundaental agnetic charge would be in relation to fundaental charge e (and pereability of

Magnetic Force 1750 John Michell coes up with an inverse-square force law for agnetic poles (note 8 years before Coulob s siilar law for charge) Unit syste has been adjusted so that the Pereability

3 . Magnetic Pole Strength 1 b. Wanted: Magnetic Monopole 14 (a) Definition Recall an electric dipole is a stick of length L with opposite charges ±Q on the end, Dipole Moent: p=lq 181 Pierre Curie: Why are there no agnetic onopoles? (other references say 1894?) Define Magnetic Dipole by sae type of forula: =Lq If you break a agnet, you can t get a N pole by itself, you get another dipole with N & S ends. Pole Strength q is agnetic charge. Old cgs units: 1 pole strength repels another with 1 dyne of force at 1 c. New units: Ap-eter (10 of old cgs pole strengths) 191 Paul Dirac (using quantu echanics) derives what the fundaental agnetic charge would be in relation to fundaental charge e (and pereability of free space 0 and Planck s constant h ). Experiental liits say ass is at least 600x of proton. q h e 0 c. Magnetic Force 1750 John Michell coes up with an inverse-square force law for agnetic poles (note 8 years before Coulob s siilar law for charge) Unit syste has been adjusted so that the Pereability of Free Space 0 is exactly: We could also state that the energy stored in a dipole agnet would be (in analogy to electrostatic energy forula): 0 q Q F 4 r U 1 0 q 4 L 7 N A 15.c.ii Force between Dipoles The force between dipoles (along a line) can be shown to be: 0 F 6 4 z 1 4 Dipoles will twist until they are parallel. The torque of the first on the second would be given by a cross product z 16. The Magnetic Field 17.a.ii Magnetic Flux is Conserved 18 (a) Discovery: 181 Michael Faraday First proposes ideas of Lines of Force Exaple: iron filings over a agnetic show field lines Because there are no agnetic onopoles, there are no sources of agnetic field lines. Magnetic Field Lines ust be continuous (i.e. continue through agnet) Gauss s law for agnetis: total agnetic flux through a closed surface is ZERO.

b. Definition of Field 19.b.ii Definition of Field 0 Definition: Analogous to electric field, except using the agnetic charge (pole strength) Units of Tesla: Old cgs units: Gauss: (field of earth is

Instead we usually easure the agnetic field in ters of: Torque on a known dipole: Change in energy of dipole in a field B U B Bcos Note: field of our big peranent agnet is only around 0.1 Tesla! c.

4 b. Definition of Field 19.b.ii Definition of Field 0 Definition: Analogous to electric field, except using the agnetic charge (pole strength) Units of Tesla: Old cgs units: Gauss: (field of earth is ~1 G) F q B T Weber N A G Dyne 10 4 T pole strength Ouch, our definition is in ters of pole strength, which is an abstraction (agnetic onopoles don t exist). Instead we usually easure the agnetic field in ters of: Torque on a known dipole: Change in energy of dipole in a field B U B Bcos Note: field of our big peranent agnet is only around 0.1 Tesla! c. Field of Dipole 1.c.ii Magnetic Dip The field of a dipole as a function of position vector r is rather essy. 0 r B( r ) 5 4 r r Along z axis siplifies to: Along x axis siplifies to: B z B z 0 (0,0, z) 4 z 0 ( x,0,0) 4 x Dipole field of earth will not be parallel to surface, except at equator! 1581 Robert Noran of London, akes first device in Europe to easure the dip (he had to clip off one end of a good copass to ake it level). Note was probably easured 500 years earlier in Persia. Early Navigators used agnetic dip to estiate latitude (proposed by Gilbert) Fro this last forula, knowing agnetic field of earth is about 0.8 G at equator, we get =10 Ap-. Dip near San Francisco is nearly 60 downward! B. Current is Source of Magnetic Field Introduction 4 1) Force of Current on Magnets ) Electroagnets ) Apere s Law Recall the source of electric fields is electric charge (Gauss s Law) But there are no agnetic charges (aka onopoles) to create Magnetic Field Source of all agnetis is the oveent of electric charge-either acroscopic current or icroscopic atoic currents. 4

1751 Benjain Franklin: electricity can agnetize needles. An early clue 5 1.

6 Hans Oersted (b) Michael Faraday 7 (c) Biot-Savart Law 8 181: lines of force circle the wire. There is no north or south pole.

In odern notation: Their ore general forula shows the agnetic field fro a sall piece of wire L carrying current is: 0 I B 4 r L r B 0 I r Jean-Baptiste Biot Felix

Magnetic Moent of Coil 0 (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.

5 1751 Benjain Franklin: electricity can agnetize needles. An early clue 5 1. Force of Current on Magnets (a) Oersted s Experient (1819) Current in a wire will deflect a agnet! 6 Hans Oersted (b) Michael Faraday 7 (c) Biot-Savart Law 8 181: lines of force circle the wire. There is no north or south pole. Direction is deterined by right hand rule 180 Biot & Savart show agnetic field around a is inversely proportional to distance. In odern notation: Their ore general forula shows the agnetic field fro a sall piece of wire L carrying current is: 0 I B 4 r L r B 0 I r Jean-Baptiste Biot Felix Savart. Electroagnets 9 (b). Magnetic Moent of Coil 0 (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 nuber of turns. 180 Johann Schweigger (with Apere) invent the (tangent) Galvanoeter, a coil around a copass needle. The tangent of the angle of deflection is proportional to the current in the coil (i.e. priitive current eter). B 0 NI tan B r e B e I B N 0 r Apere shows that the field of a coil of N turns, loop area A, is equivalent to that of a agnet, with dipole oent: NIA For a Solenoid, the field inside is nearly constant with value (where N is the nuber of turns, and L is the length of the coil) NI B 0 L 5

Presence of agnetic core increases field by a factor of K (over 100 for Iron, over 10,000 for u etal ).

6 (c). Magnetic Cores 1. Apere s Circulation Law 18 Sturgeon invents the electroagnetic (coil around agnetic core). Although it is based upon Apere s work, the law was actually forulated by Maxwell Joseph Henry iproves design using insulating wire. Presence of agnetic core increases field by a factor of K (over 100 for Iron, over 10,000 for u etal ). Equivalently, replace 0 by in all forulas where pereability of the ediu is: K 0 (a) The General Laws of Maxwell ( 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 agnetis states that there are no agnetic charges (agnetic flux through a closed surface is zero). Apere s Circulation Law is the general stateent that current is the source of agnetic fields. b. Stateent of Apere s Law c. Application of Apere s Law 4 In general: The su of agnetic field about a closed loop is proportional to the su of current passing through the loop dr I B 0 This for is really only useful for very syetric situations, for exaple where B is constant over the loop. Then you can siply write (where L is the perieter of the path) B 0 I L But we can use Apere 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 Discussion/details on the board in class! C. Electrodynaics 5 Introduction 6 Magnetic Force on Electric Charge 1) Apere: Force between wires Ter Electrodynaics was given by Apere ) Lorentz Force Law Reciprocity: By Newton s rd law, if current akes a force on a agnet, then a agnet should ake a force on a current ) Torques on Current Loops And since an electroagnet is equivalent to a agnet, we can deduce there should be a force between currents. 6

1. Forces on Currents 7 (b) The Motor Rule 8 (a) Apere s Force Law (180-) Currents in sae direction attract Currents

4 Two wires carrying 1 ap of current separated by 1 c attract with force of 10 7 Newtons/eter (definition of Ap)

ws/highschool/physics/hoe/notes/electricity/agnetis/magforcesbetweenwires.hthttp://www.stary.

ht Faraday s explanation: First wire creates B field 0 I 4 r 1 B1 Force on second wire carrying current I due to

Motor: History 40 180 Faraday invents the first hoopolar otor It s siply a wire that rotates in a vat of ercury 18

Ignore the sound track: http://www.youtube.co/watch?v=ksuf01pwffm Deonstration 41.

7 1. Forces on Currents 7 (b) The Motor Rule 8 (a) Apere s Force Law (180-) Currents in sae direction attract Currents in opposite direction repel Force (per unit length) between current carrying wires depends on distance r : F 0 I1I L 4 Two wires carrying 1 ap of current separated by 1 c attract with force of 10 7 Newtons/eter (definition of Ap) r André-Marie Apère ( ) Faraday s explanation: First wire creates B field 0 I 4 r 1 B1 Force on second wire carrying current I due to agnetic field ust B fro first current ust be (aka Motor Rule ): F I L B (c) The Electric Motor 9 (c) The Electric Motor: History Faraday invents the first hoopolar otor It s siply a wire that rotates in a vat of ercury 18 Barlow s Wheel Copper disk rotates in a agnetic field when current flows radially outward Peter Barlow ( ) Ignore the sound track: Deonstration 41. Force on Moving Charge 4 Needs new rare-earth agnet (a) Lorentz Force Law (189) First ideas probably done (wrong) by Weber 1846 First done by Maxwell 1861 Current is just oving charges Force on charge q oving with velocity v in agnetic field B : F q v B 7

(b) (Edwin) Hall Effect (1879) 4 c. Cyclotron Equation 44 If current is + charges oving to right then they will be deflected to the front and a positive voltage easured.

The voltage induced gives you the drift velocity ( d is width). V dv D B Hall Probes are used to easure agnetic fields.

Electric charges will spiral around agnetic field lines with radius: (with higher speed, bigger!) But, the frequency (period) of orbit turns out to be independent of speed!

Protons are injected into the center of two "D" shaped hollow conductors called "dees". The perpendicular agnetic field akes the go in circular orbits.

8 (b) (Edwin) Hall Effect (1879) 4 c. Cyclotron Equation 44 If current is + charges oving to right then they will be deflected to the front and a positive voltage easured. If current is - charges oving to left then they will be deflected to the front and a negative voltage easured. Experient proves that current in etals is really negative charges oving. The voltage induced gives you the drift velocity ( d is width). V dv D B Hall Probes are used to easure agnetic fields. Knowing properties of conductor, the easured voltage will be proportional to the presence of agnetic field. Since agnetic force is perpendicular to velocity, we get centripetal acceleration. Electric charges will spiral around agnetic 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 ass ratio. f v qvb R v R qb v B R q c.ii. The Cyclotron 45 c.iii. The Cyclotron 46 Invented in 19 by E.O. Lawrence and M.S. Livingston. Protons are injected into the center of two "D" shaped hollow conductors called "dees". The perpendicular agnetic field akes the 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 largest one in the United States is Fei Lab. It is iles in circuference, and can produce over 400 GeV. The path gets larger and large and eventually after the particle gains enough energy it is ejected to the target. These generate fro 1 to 10 MeV (Million electron Volts) of energy. Good Video: Mr Ion: c.iii. The Mass Spectroeter 47. Torques on Current Loops Thoson s student Francis Aston constructs first function ass spectroeter. Ionized eleent is accelerated through a voltage giving it speed: e v V (a) Equivalent Magnetic Moent of Loop Let L be side of square loop Force on side wires: F=ILB Heavier ass isotope will follow BIGGER radius path in agnetic field v 1 r eb B V e This was how we easured asses of nuclei (and found out there are isotopes ) Torque: L r F ILB IL B Hence its our old forula for torque on a agnetic oent, where oent of loop is: B NIA 8

(b) d Arsonval Galvanoeter 188 design, the ultiple loop coil in a very strong agnet ade the first very sensitive aeter. The coil has a spring on it to pull it back to center.

49 (c) Modern Motor Design 18 Sturgeon added the coutator which switches the polarity as the loop turns so that the otion will be continuous. http://www.youtube.co/watch?

hi! http://www.phy6.org/earthag/lodeston.ht http://www.phy6.org/earthag/deagrev.ht http://www.seds.org/essier/xtra/bios/ichell.htl http://en.org/wiki/galvanoeter http://iesfgcza.educa.

9 (b) d Arsonval Galvanoeter 188 design, the ultiple loop coil in a very strong agnet ade the first very sensitive aeter. The coil has a spring on it to pull it back to center. When current is added, the torque twists the coil, oving the pointer. 49 (c) Modern Motor Design 18 Sturgeon added the coutator which switches the polarity as the loop turns so that the otion will be continuous D. Appendix: Right Hand Rule 51 E. References 5 There are any conventions for the right hand rule (and even left hand rules ). See The convention I a using is the following picture: See AJP 67, 448 (1999) which discusses that Apere is NOT the author of the circulation law that is naed after hi! (includes field rotation paradox and aniations) (useu of old instruents) Interactive Barlow Wheel Old Fils: Another DC otor video: 9

Pearson Physics Level 30 Unit VI Forces and Fields: Chapter 12 Solutions

Concept Check (top) Pearson Physics Level 30 Unit VI Forces and Fields: Chapter 12 Solutions Student Book page 583 Concept Check (botto) The north-seeking needle of a copass is attracted to what is called