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Electromagnetism

Electromagnetism. Zhu Jiongming Department of Physics Shanghai Teachers University. Electromagnetism. Chapter 1 Electric Field Chapter 2 Conductors Chapter 3 Dielectrics Chapter 4 Direct-Current Circuits Chapter 5 Magnetic Field Chapter 6 Electromagnetic Induction

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Electromagnetism

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  1. Electromagnetism Zhu Jiongming Department of Physics Shanghai Teachers University

  2. Electromagnetism • Chapter 1 Electric Field • Chapter 2 Conductors • Chapter 3 Dielectrics • Chapter 4 Direct-Current Circuits • Chapter 5Magnetic Field • Chapter 6 Electromagnetic Induction • Chapter 7 Magnetic Materials • Chapter 8 Alternating Current • Chapter 9 Electromagnetic Waves

  3. Chapter 5 Magnetic Field • §1. Introduction to Basic Magnetic Phenomena • §2. The Law of Biot and Savart • §3. Magnetic Flux • §4. Ampere’s Law • §5. Charged Particles Moving in a Magnetic Field • §6. Magnetic Force on a Current-Carrying Conductor • §7. Magnetic Field of a of a Current Loop

  4. Moving charge  §1. Basic Magnetic Phenomena • Comparing with Electric Fields： • E： charge  electric field charge • （produce） （force） • M： Moving charge  magnetic field • Permanent Magnets • Magnetic Effect of Electric Currents • Molecular Current

  5. Permanent Magnets • Two kinds of Magnets：natural、manmade • Two Magnetic Poles： south S、north N • Force on each other： repel（N-N, S-S） • attract（N-S） • Magnetic Monopole ？

  6. S N N S I I Magnetic Effect of Electric Currents • Experiments Show • Straight Line Current • Two Parallel Lines • Circular Current • Solenoid and Magnetic Bar • Molecular Current—— Ampere’s Assumption

  7. I I M M’ Magnetic Field B • Experiment：Helmholts coils in a • hydrogen bulb，an electron gun • Conclusion ：moving charge • F= q vB （ Definition of B ） • ( Electric Field ： F= qE ) • Unit ：Tesla • Magnetic Field Lines ：（curve with a direction） • Tangent at any point on a line is in the direction of the magnetic field at that point • Number of field linesthrough unit areaperpendicular to B equals the magnitude of B

  8. §2. The Law of Biot and Savart • 1. The Law of Biot and Savart • 2. Magnetic Field of a Long Straight Line Current • 3. Magnetic Field of a Circular Current Loop • 4. Magnetic Field on the Axis of a Solenoid • 5. Examples

  9. dB P r  Idl 1. The Law of Biot and Savart • The field of a current element Idl • dB  Idl，1/r2，sin • r： Idl P • ：angle between Idl and r • Proportionality constant：0/4 = 10-7 • Direction：dBIdl，dB  r • Integral ： • Compare with：

  10. I 1 O P a dl 2 2. Field of a Long Straight Line Current • currentI，distance a • all dB in same direction l sin = a/r ctg = - l/a  r = a/sin l = - a ctg dl= ad/sin2 r  Infinite long：  1= 0，  2= ， Direction ：right hand rule

  11. I P o z R r dB dl 3. Magnetic Field of a Circular Current • currentI，radius R， • P on axis , distance a a  • component dB||= dBcos • symmetry，dB cancel， B = 0  = 90o cos = R/r r2 = R2 + a2

  12. I  dl P 4. Field on the Axis of a Solenoid • currentI, radius R, Length L, • n turns per unit length • dB at Ponaxis caused by nIdl • （ as circular current ） l R L ctg = l/R  l = R ctg dl= - Rd/sin2 R2+ l2 = R2/sin2

  13. I 1 B 2 B P R L O L Field on the Axis of a Solenoid • Direction ： • right hand rule • (1) center（or R << L） • 1= 0， 2= ，  B = 0nI • (2) ends（Ex.：left） • 1= 0， 2= /2， B = 0nI/2 • (3) outside, cos1、cos2 same sign, minus, B small • inside, opposite sign, plus，B large

  14. I1 B I I2 O I C Example （p.345/5 - 3 -11） • Uniform ring with current，find B at the center. • Sol.： • Straight lines：  1 2 • （ circular current： ） • arc 1：  B1= B2 opposite direction  B = 0 • arc 2： • parallel：I1R1 = I2R2

  15. Exercises • p.212 / 5-2- 3, 8, 12, 13, 16

  16. §3. Magnetic Flux • 1. Magnetic Flux • 2. Magnetic Flux on Closed Surface • 3. Magnetic Flux through Closed Path

  17. dS B 1. Magnetic Flux • Flux on area element dS • dB = B · dS = B dS cos • Flux on surface S ( integral ) • if B anddS in same direction (  = 0 ), write dS  B：Flux per unit area perpendicular to B • define number of B lines through dS = B · dS = dB • then line density = • = Magnitude of B • Unit ： 1 Web = 1 T · m2

  18. dB Idl 2. Magnetic Flux on Closed Surface • Show：(1) dB of current element Idl • B lines areconcentric circles • these circles and the surface S • either not intercross（no contribution to flux） • or intercross 2 times（in/out，flux +/-）

  19. Magnetic Flux on Closed Surface • Show： (2) magnetic field of any currents • superposition： B = B1 + B2 + … • B lines are continual，closed，or    • —— called The field without sources • Compare with： • E lines from +q or ，into -q or  • —— called The field with sources

  20. n S2 S1 n L 3. Magnetic Flux through Closed Path • Any surfaces bounded by the closed path L have the same flux • Show： • Turn the normal vector of S1 • opposite, same as that of S2 • then —— called Magnetic Flux through Closed Path L

  21. Exercises • p.214 / 5-3- 1, 3

  22. §4. Ampere’s Law • 1. Ampere’s Law • 2. Magnetic Field of a Uniform Long Cylinder • 3. Magnetic Field of a Long Solenoid • 4. Magnetic Field of a Toroidal Solenoid

  23. I L 1. Ampere’s Law • Ampere’s Law : • L：any closed loop • I：net current enclosed by L • Three steps to show the law： • L encloses a Long Straight Current I • L encloses no Currents • L encloses Several Currents

  24. I L ds d dl B I L L Encloses a Long Straight Current I • Field of a long line current I： （direction: tangent）

  25. I L Encloses no Currents • Current I is outside L  L1 L2

  26. L Encloses Several Currents • L encloses several currents • Principle of superposition： B = B1 + B2 + … • I is algebraic sum of the currents enclosed by L • direction of Iiwith direction of L（integral）： right hand rule，take positive sign

  27. P B L r R 0 2. Field of a Uniform Long Cylinder • radius R，current I（outgoing）， • find Bat P a distance r from the axis • concentric circle L with radius r， • symmetry：same magnitude of B on L， • direction：tangent outside： inside：

  28. B P Direction is along the Tangent • radius R，current I（outgoing）， • field Bat P a distance r from the axis • symmetry： B in direction of tangent

  29. z B B’ a b B’’ z’ c d 3. Magnetic Field of a Long Solenoid • Field inside is along axis • Show：turn 180 o round zz’：BB’ • I opposite： B’B’’ • B’’ should coincide with B direction： right hand rule

  30. 4. Field of a Toroidal Solenoid • Symmetry：B on the circle L • magnitude： same • direction： tangent • （ L >> r，N turns） • in： • out： direction：right hand rule if L ，becomes a long solenoid

  31. dB   z 5. Field of a Uniform Large Plane • Surface current  （width l，thickness d ） • Direction：parallel • opposite on two sides • ( right hand rule ) l

  32. Exercises • p.215 / 5-4- 2, 3, 4, 5

  33. §5. Charged Particles Moving in B • 1. Motion of Charged Particles in a Magnetic Field • 2. Magnetic Converging • 3. Cyclotrons • 4. Thomson’s e/m Experiment（skip） • 5. The Hall Effect

  34. m, q=- e v F O R 1. Motion of Charged Particles • Lorents Force ：F = q ( E + v B ) • if E = 0 ， F = q v B • if v  B ， q moves in a circle • with constant speed • Centripetal force： • Radius ： R = mv/qB • Period ： T = 2R/v = 2 m/qB • Frequency ： f = 1/ T = qB/2 m • Ratio of charge to mass： q/ m= v/BR = /B

  35. R h P’ P 2. Magnetic Converging • v making an angle  with B： • v|| = v cos • v = v sin • Helical path • radius： • pitch： • Magnetic Converging： • different R，same h • from P to P’ • distance h

  36. 3. Cyclotrons • Principle：uniform field，outward • 2 Dees，alternating emf • q accelerated as crossing the gap （not depend on v, r） • Take period of emf same as that of q • accelerated 2 times per revolution • v  r  ( • ), T not changed • Application：accelerating proton、 etc. to slam into a solid target to learn it’s structure

  37. Cyclotrons • Compare with straight line accelerator • Str.： • Cyc.： • To gain the same v，need • Ex.：deuteron q/m ~ 10 7，B ~ 2，R ~ 0.5 • need U ~ 10 7 （volts） • frequency of emf f = qB/2m ~ B magnetic field • relativity：v  m  f  • varying frequency —— Synchrotrons

  38. z y fL l fe x I d 5. The Hall Effect • A conducting strip of width l, thickness d • x - current，y - magnetic field •  z- voltage UAA’ B A • Exp. carrier q，force fL= q vB • q > 0 , v - positive x ，fL - positive z • q < 0 , v - opposite x ，fL - positive z direction A’ • for q > 0 ，positive charges pile up on side A， • negative on A’  produce an electric field Et • fe = qEtopposite to fL slow down qEt= qvB •  stop pilingq movesalong x（as without B） •  the Hall potential difference：UAA’ = Et l = vBl

  39. The Hall Constant • I = q n ( vld ) •  v = I/qnld •  UAA’ = IB/qnd • write：UAA’ = K IB/d • proportional toIB/d （macroscopic） • Hall constant：K = 1/qn （microscopic） • determined by q 、n • q > 0 ,K > 0  UAA’ > 0 • q < 0 ,K < 0  UAA’ < 0 • （ A - negative charges，A’ - positive ）

  40. Exercises • p.216 / 5-5- 1, 3, 4, 5, 6

  41. §6. Magnetic Force on a Conductor • 1. Ampere Force • 2. Rectangular Current Loop in a Uniform • Magnetic Field • 3. The Principle of a Galvanometer

  42. B  I dS dl dl and j in same direction j and dS in same direction  I = j  dS = jdS 1. Ampere Force • current  carriers magnetic force  on conductor • electron：f = -ev  B • current：j = -env • force on current element Idl ： • dF = N (-ev  B ) • = n dS dl (-ev  B ) • = dS dl ( j  B ) • = Idl  B • Ampere force：

  43. I I B  n n F4 （ ） B  F2 2. Rectangular Loop in Magnetic Field • Normal vector n and current I • ------ right-hand rule • u①： • d③： • l ②： • r ④： ① ④ l2 （up） l1 ② ③ （down） （ ⊙ ） l1 F1，F3cancel out F2，F4 produce a net torque： T = F2l1sin = IBl2l1sin = ISBsin

  44. T B I pm The Magnetic Dipole Moment • Torque on a current carrying rectangular loop ： • T = ISBsin （ direction：n B） • Definition： • Magnetic Dipole Moment of a • current carrying rectangular loop • pm = ISn • then the torque • T = pmB • （Comparison：in an electric field • p = ql， T = pE）

  45. S B I n Magnetic Moment of Any Loop • Divided into many small rectangular loops • outline ~ the loop，inner lines cancel out • dT = dpmB = IdS nB • all dT in the same direction • T = dT = IdSnB = InBdS • = IS nB = pmB • Definition： • Magnetic Dipole Moment of Any Loop pm = ISn • no matter what shape • （same form as that of a rectangular loop）

  46. B I n Magnetic Dipole Moment of Any Loop • pm making an angle  with B • maximum T for  =  /2 • T = 0 for  = 0 equilibrium, stable lowest energy • T = 0 for  =  equilibrium, unstable highest energy 

  47. N S 3. The Principle of a Galvanometer • n turns： T = nISB • countertorque by springs • T’ = k • when in balance •  =nISB/ k  I • （  =0 for I = 0 ）

  48. Exercises • p.217 / 5-6- 1, 5, 8

  49. I P o a B R §7. Field of a Current Loop • Circular loop of radius R ，current I，on axis • pm = ISnis important • torque exerted by magnetic field • produce magnetic field

  50. Exercises • p.219 / 5-6- 11

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