Chapter 30: Magnetic Fields Due to Currents

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1 d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.1-1 T) i to ue a lage cuent flowing though a wie. The calculation of the magnetic field poduced by a wie can be vey complicated a it involve vecto multiplication and integation! n the ealy 18 it wa ealized by iot and Savat that the magnetic field due to a conducto caying a cuent could be expeed a: d d 3 The vaiable ae defined in the pictue. θ n thi equation i the pemeability contant: x1-7 T-mA d P i into the page The magnitude of i given by: dinθ d. Ka P13 Sp θ i the angle between and d Some thing to keep in mind about the iot Savat aw: a) The diection of d i pependicula to the vecto d and. b) The magnitude of d vaie a 1. c) The magnitude of d i popotional to the cuent () and the length of the cuent element, d, d) The magnitude of d i popotional to the ine of the angle between the vecto d and and. e) To get the field fom a finite ize wie we mut integate d! 1

2 Diection of Magnetic Field due to Cuent in a Wie The econd ight Hand ule The diection of the magnetic field due to a cuent in a wie can be found uing you ight hand: 1) You thumb point in the diection of the cuent ) The diection that you finge natually cul give the diection of the magnetic field. HW 3- out of the page x into the page Example howing the line of the magnetic field fo a cuent going into the page. The field line fom concentic cicle aound the wie. The diection of the -field at a point i tangent to the cicle at the point of inteet. n thi figue a vecto epeenting the magnetic field i dawn fo two diffeent ditance fom the wie. The elative ize of the vecto give the elative magnitude of the -field at the two point. Anothe view of the elationhip between the diection of the cuent and the magnetic field. HW 3-. Ka P13 Sp

3 Foce on a Wie fom Anothe Wie Quetion: Two wie ae next to each othe a in the figue. Fo both wie the cuent i NTO the page. What i the diection of the magnetic foce on wie due to the magnetic field of wie 1? HW 3- F HW 3- wie 1 F wie 1 wie wie Fit we need to find the diection of: The diection of follow the cuent (poitive chage flow) and i into the pape. The diection of at wie i found uing the nd ight hand ule and i pointed DOWN. The diection of the vecto co poduct i towad wie 1. The foce i alo in thi diection ince we ae auming the cuent i made of poitive chage caie. Theefoe wie i attacted to wie 1. What about the foce on wie 1 due to wie? The magnetic field at wie 1 due to wie i pointed towad the top of the page. Uing the (1 t ) ight hand ule (co poduct) we find that wie 1 i attacted to wie. f the cuent in the wie un in oppoite diection then the wie will epel each othe.. Ka P13 Sp 3

4 . Ka P13 Sp Calculation of the Magnetic Field Due to a Cuent in a wie 3 d P d d θ et conide the imple cae (which i not o imple) of a long (length ) wie caying a contant cuent (). We want to calculate at a point along the middle of the wie a ditance away fom the wie. Since the cuent i contant we can pull it outide of the integal: 3 d Fo thi ituation d i alway into the page, no matte whee we ae along the wie. So, thee i only one component of. Thi implifie thing and we now have: in d θ no longe a vecto equation! We now need to do a bit of tig: θ ) in( in θ θ 1 3 ) ( ) ( d 1 ) ( Fo a vey long wie, >>, thi implifie to: d F 1 The foce between two wie epaated by a ditance d i: i into the page ntegal #19 Appendix E f >> then ( ) 1

5 Ampee aw When we tudied electotatic we found that fo cetain poblem with a high degee of ymmety (e.g. cylinde, phee, plane) we could find the electic field eaie uing Gau aw intead of diect integation. E da q enc ε Ou old fiend, Gau aw A imila ituation exit when the magnetic field i geneated by a cuent. d enc Ou new fiend, Ampee aw ike Gau aw, Ampee law contain an integal with a dot poduct of vecto. t alo involve an integal ove a pecific path. Fo Gau aw the integation wa ove a cloed uface that encloed chage. Fo Ampee aw the integation i ove a cloed loop that encloe cuent. The ight ide of Ampee aw include all cuent encloed by the loop. et do ome example with cuent caying wie to illutate the featue of Ampee aw. A wie with a dot ha cuent coming out of the page, one with an x ha cuent into the page. Ampeeian loop 1 Ampeeian loop 1 The cuent encloed by thee The cuent encloed by the Ampeian loop i 1. Ampeian loop i 1 fo both f the cuent ae equal but the example. diection oppoite then enc.. Ka P13 Sp The cuent encloed by thi Ampeian loop i till 1. Cuent outide the loop do not contibute to enc. 5

6 Ampee awcontinued d enc We now need a ign convention fo the encloed cuent. We ue the convention of the text to define the diection of integation uing the finge of the ight hand fo the loop diection and the thumb to aign the diection of poitive encloed cuent. HW Fig 3-1 Following ou convention, enc 1 -. Since the integal contain the dot poduct between d and we need to define the angle between thee vecto. The vecto d i alway tangent to the cuve and i oiented along the diection of integation, counte clockwie accoding to ou convention. The angle θ i the angle between d and a hown in the figue. Putting it all togethe we have fo thi example: HW Fig 3-11 d enc coθd ( 1 ) Without moe infomation we can not olve fo. All we know i that the integal i equal to ( 1 - ). The cuent 3 doe not contibute to enc ince it i outide the loop. UT thi cuent doe contibute to!. Ka P13 Sp 6

7 Application of Ampee aw Jut like Gau aw thee ae only a few intance whee we can actually olve fo. One uch cae i the magnetic field outide a long taight wie. HW Fig.3-13 Fo a long taight wie the magnetic field i only a function of the ditance fom the wie. We want to exploit thi piece of info to pick ou Ampeian loop. We pick a cicle with the wie at the cente fo ou Ampeian loop becaue ha the ame value at evey point on the cicle. n addition (exta bonu), ince d i tangent to the cicle it i paallel to eveywhee o coθ1. The integal i evaluated aound the cicle (emembe ac length, d θ) d Ampee aw ay: d (co ) d enc d dθ dθ Same eult a fom diect integation!! Example: inide a wie (of adiu ) with unifom cuent (). Since we have a unifom cuent inide the wie enc ( )( ). Since we till have a long taight wie the integal pat of the poblem i identical to the one we jut olved. enc ( ). Ka P13 Sp 7

8 Application of Ampee awcontinued Anothe ituation whee Ampee aw i ueful i to calculate the -field inide a olenoid. A olenoid i jut a hollow cylinde with wie wapped aound it a hown in the figue. n an ideal olenoid thee i a unifom magnetic field inide the coil and zeo magnetic field outide. Fo cuent flowing in the diection hown in the figue the -field point to the ight. HW Fig d b a d c b We can ue Ampee law to find inide the olenoid by beaking the integal into piece. d. Ka P13 Sp d c d a d d The integal fom b to c and d to a ae zeo ince d i pependicula to (coθ). The integal c to d i zeo ince in thi egion. So we ae left with: d b a d b a d h HW Fig ou loop The encloed cuent i jut the cuent in a tun () time the numbe of tun pe length (n) time the length (h): enc nh. So, fo an ideal olenoid i: n 8

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