1Wed. Mar. 26, 2008 Physics 208, Lecture 17 1Exam 2 covers Ch. 27-32,Lecture, Discussion, HW, Lab Chapter 27: Electric flux & Gauss’ law Chapter 29: Electric potential & work Chapter 30: Electric potential & field Chapter 28: Current & Conductivity Chapter 31: Circuits Chapter 32: Magnetic fields & forces (exclude 32.6,32.8,32.10)Exam 2 is Wed. Mar. 26, 5:30-7 pm,2103 Ch: Adam(301,310), Eli(302,311), Stephen(303,306),180 Science Hall: Amanda(305,307), Mike(304,309), Ye(308)Wed. Mar. 26, 2008 Physics 208, Lecture 17 2Electric currentproduces magnetic field Current (flow of electric charges )in wire produces magnetic field. That magnetic field aligns compass needleCurrentMagneticfieldWed. Mar. 26, 2008 Physics 208, Lecture 17 3Law of Biot-Savart Each element of currentproduces a contributionto the magnetic field.rθIds! dB =µo4"Ids #ˆ r r2B out of pagedIdBrWed. Mar. 26, 2008 Physics 208, Lecture 17 4Magnetic field fromlong straight wire:Direction What direction isthe magneticfield from aninfinitely-longstraight wire?I ! dr B =µo4"Idr s #ˆ r r2xyWed. Mar. 26, 2008 Physics 208, Lecture 17 5Current dependenceHow does the magnitude ofthe B-field change if thecurrent is doubled?I ! dr B =µo4"Idr s #ˆ r r2xyA) Is halvedB) QuadruplesC) Stays sameD) DoublesE) Is quarteredWed. Mar. 26, 2008 Physics 208, Lecture 17 6Distance dependenceHow does the magnitude ofthe B-field at 2 compareto that at 1?I ! dr B =µo4"Idr s #ˆ r r2xy12A) B2=B1B) B2=2B1C) B2=B1/2D) B2=4B1E) B2=B1/42Wed. Mar. 26, 2008 Physics 208, Lecture 17 7Why? Biot-Savart says Why B(r)∝ 1/r instead of 1/r2 ? ! dr B =µo4"Idr s #ˆ r r2 ! r r Large contribution fromthis current element.Decreases as 1/r2ISmall contribution fromthis current element.~ independent of rWed. Mar. 26, 2008 Physics 208, Lecture 17 8Long straight wire All current elements produce B out of pagexa! r = x2+ a2rθ! dB =µo4"Ids #ˆ r r2=µo4"Ir2sin$=µo4"Ir2ar=µo4"Iax2+ a2( )3 / 2! B =µoIa4"dxx2+ a2( )3 / 2=#$$%µoI2"aAdd them all up:Wed. Mar. 26, 2008 Physics 208, Lecture 17 9Forces between currentsWhich of these pairs of currents will attracteach other?A. AB. A & CC. BD. A & BA B CWed. Mar. 26, 2008 Physics 208, Lecture 17 10Force betweencurrent-carrying wires Attractive for parallelcurrents. Repulsive forantiparallel currentsWed. Mar. 26, 2008 Physics 208, Lecture 17 11Field from a circular loop Each current element produce dB All contributions add as vectors Along axis, allcomponents cancelexcept for x-compWed. Mar. 26, 2008 Physics 208, Lecture 17 12Magnetic field from loopWhich of these graphsbest represents themagnetic field on theaxis of the loop?BzzxyBzBzBzA.B.C.D.zzzz3Wed. Mar. 26, 2008 Physics 208, Lecture 17 13Magnetic field from a current loop One loop:field still loops around the wire. Many loops: same effectWed. Mar. 26, 2008 Physics 208, Lecture 17 14Solenoid electromagnet Sequence ofcurrent loops canproduce strongmagnetic fields. This is anelectromagnetWed. Mar. 26, 2008 Physics 208, Lecture 17 15Comparing Electric, Magnetic Biot-Savart:calculate B-field from current distribution. Resulting B-field is a vector, and… complication: current (source) is a vector! Coulomb:calculate E-field from charge distribution Resulting E is a vectorbut charge (source) is not a vectorWed. Mar. 26, 2008 Physics 208, Lecture 17 16A shortcut: Ampere’s law Integral around closedpath proportional tocurrent passing throughany surface boundedby path. Ampere’s law! B • ds"=µoIclosed pathsurfacebounded bypathI Right-hand ‘rule’: Thumb in direction ofpositive current Curled fingers showdirection integrationWed. Mar. 26, 2008 Physics 208, Lecture 17 17‘Testing’ Ampere’s law Long straight wire , B⊥r! B r( )=µoI2"rr! B • ds"IB(r)B||dspath hasconstant rpath length= 2πr! B • ds"=µoI2#rds"! B • ds"=µoI2#rds"=µoI2#rds"! B • ds"=µoI2#rds"=µoI2#rds"=µoI2#r2#r =µoICircular pathSurface bounded by pathWed. Mar. 26, 2008 Physics 208, Lecture 17 18Using Ampere’s law Could have used Ampere’s law to calculate BrIB(r)! B • ds" = Bds =" B ds" = B2#r =µoI $ B =µoI2#rCircular pathSurface bounded by pathB||dsB constant on pathpath length= 2πr4Wed. Mar. 26, 2008 Physics 208, Lecture 17 19Quick Quiz Suppose the wire has uniform current density. Howdoes the magnetic field change inside the wire?A. Increases with rB. Decreases with rC. Independent of rD. None of the aboverB! B"• ds = 2#rB =µoIcut=µoI#r2#R2$ B r( )=µoI2#R2rWed. Mar. 26, 2008 Physics 208, Lecture 17 20Building a solenoidWed. Mar. 26, 2008 Physics 208, Lecture 17 21Ampere’s law for the
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