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 FieldMagnetic Field Chapter 6 Electromagnetic Induction Chapter 7 Magnetic Materials Chapter 8 Alternating Current Chapter 9 Electromagnetic Waves

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

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

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

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

Magnetic Field B F Experiment ： Helmholts coils in a hydrogen bulb ， an electron gun I I M M’ F Conclusion ： moving charge F = q v  B （ Definition of B ） ( Electric Field ： F = qE )  Unit ： Tesla F 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 lines through unit area perpendicular to B equals the magnitude of B

§2. The Law of Biot and Savart 1. The Law of Biot and SavartThe Law of Biot and Savart 2. Magnetic Field of a Long Straight Line CurrentMagnetic Field of a Long Straight Line Current 3. Magnetic Field of a Circular Current LoopMagnetic Field of a Circular Current Loop 4. Magnetic Field on the Axis of a SolenoidMagnetic Field on the Axis of a Solenoid 5. ExamplesExamples

1. The Law of Biot and Savart The field of a current element Idl dB  Idl ， 1 / r 2 ， sin  r ： Idl  P  ： angle between Idl and r Proportionality constant ：  0 / 4  = dBdB IdlIdl r P  Direction ： dB  Idl ， dB  r Integral ： Compare with ：

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

3. Magnetic Field of a Circular Current current I ， radius R ， P on axis, distance a z o R r a dBdB P dldl I   = 90 o cos  = R / r r 2 = R 2 + a 2 component dB || = dBcos  symmetry ， dB  cancel ， B  = 0

4. Field on the Axis of a Solenoid current I, radius R, Length L, n turns per unit length dB at P on axis caused by nIdl （ as circular current ） I P R l dldl L  ctg  = l / R  l = R ctg  dl= - Rd  / sin 2  R 2 + l 2 = R 2 / sin 2 

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

Example （ p.345 / ） Uniform ring with current ， find B at the center. Sol. ： I I  O B C 1 2 I1I1 I2I2  B 1 = B 2 opposite direction  B = 0 Straight lines ： （ circular current ： ） arc 1 ： arc 2 ： parallel ： I 1 R 1 = I 2 R 2

Exercises p.212 / , 8, 12, 13, 16

§3. Magnetic Flux 1. Magnetic FluxMagnetic Flux 2. Magnetic Flux on Closed SurfaceMagnetic Flux on Closed Surface 3. Magnetic Flux through Closed PathMagnetic Flux through Closed Path

F Flux on area element dS d  B = B · dS = B dS cos  F Flux on surface S ( integral ) F if B and dS in same direction (  = 0 ), write dS  = Magnitude of B F Unit ： 1 Web = 1 T · m 2 define number of B lines through dS = B · dS = d  B then line density = 1. Magnetic Flux dSdS B  B ： Flux per unit area perpendicular to B

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

Magnetic Flux on Closed Surface Show ： (2) magnetic field of any currents superposition ： B = B 1 + B 2 + … 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

Turn the normal vector of S 1 opposite, same as that of S 2 then 3. Magnetic Flux through Closed Path Any surfaces bounded by the closed path L have the same flux Show ： L S1S1 S2S2 n n —— called Magnetic Flux through Closed Path L

Exercises p.214 / , 3

§4. Ampere’s Law 1. Ampere’s LawAmpere’s Law 2. Magnetic Field of a Uniform Long CylinderMagnetic Field of a Uniform Long Cylinder 3. Magnetic Field of a Long SolenoidMagnetic Field of a Long Solenoid 4. Magnetic Field of a Toroidal SolenoidMagnetic Field of a Toroidal Solenoid

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

L Encloses a Long Straight Current I Field of a long line current I Field of a long line current I ： I L I dsds dldl dd L B （ direction: tangent ）

L Encloses no Currents Current I is outside L I  L2L2 L1L1

L Encloses Several Currents L encloses several currents Principle of superposition ： B = B 1 + B 2 + …  I is algebraic sum of the currents enclosed by L  direction of I i with direction of L （ integral ）： right hand rule ， take positive sign

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

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

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 B’ B B’’ z’z’ z a b dc direction ： right hand rule

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

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

Exercises p.215 / , 3, 4, 5

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

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 circleq moves in a circle with constant speed Centripetal force ： O R v F m, q= - e  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

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

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

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 F relativity ： v   m   f  varying frequency —— Synchrotrons Cyclotrons F Compare with straight line accelerator  Str. ：  Cyc. ： To gain the same v ， need

F Exp. carrier q ， force f L = q v  B  q > 0, v - positive x ， f L - positive z  q < 0, v - opposite x ， f L - positive z direction A’ 5. The Hall Effect F A conducting strip of width l, thickness d x - current ， y - magnetic field  z - voltage U AA’ x y z I B A l d fLfL fe fe for q > 0 ， positive charges pile up on side A ， negative on A’  produce an electric field E t  f e = qE t opposite to f L  slow down  qE t = qvB  stop piling  q moves along x （ as without B ）  the Hall potential difference ： U AA’ = E t l = vBl

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

Exercises p.216 / , 3, 4, 5, 6

§6. Magnetic Force on a Conductor 1. Ampere ForceAmpere Force 2. Rectangular Current Loop in a UniformRectangular Current Loop in a Uniform Magnetic FieldMagnetic Field 3. The Principle of a GalvanometerThe Principle of a Galvanometer

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 ： I B dldl dSdS  dl and j in same direction j and dS in same direction  I = j  dS = j dS

2. Rectangular Loop in Magnetic Field Normal vector n and current I right-hand rule u ①： d ③： l ②： r ④： I I B n ① ② ③ ④ l2l2 l1l1  （ up ） （ down ） （ ⊙ ）（ ⊙ ） （ ） F 1 ， F 3 cancel out B n l1l1  F2F2 F4F4 F 2 ， F 4 produce a net torque ： T = F 2 l 1 sin  = IBl 2 l 1 sin  = ISBsin 

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 p m = IS n then the torque T = p m  B B pmpm I T （ Comparison ： in an electric field p = ql ， T = p  E ）

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

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

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 ） N S

Exercises p.217 / , 5, 8

§7. Field of a Current Loop Circular loop of radius R ， current I ， on axis o R a P I p m = IS n is important torque exerted by magnetic field produce magnetic field B

Exercises p.219 /