hsics 1, Fall 1 3 Octobe 1 Toda in hsics 1: finding Foce between paallel cuents Eample calculations of fom the iot- Savat field law Ampèe s Law Eample calculations of fom Ampèe s law Unifom cuents in conductos? Andé-Maie Ampèe, fo whom the law, and the amp, ae named. 3 Octobe 1 hsics 1, Fall 1 1 ief emindes fom the ecent past Foce laws: df d dsinn cuents F sin d nd F Qv point chages Coss poducts: ab ba ab sinn ab if a b ab abn if ab Coss poducts of Catesian unit vectos: d o Qv df o F La finges of ight hand along ac fom d to, pointing to. Thumb points along df. 3 Octobe 1 hsics 1, Fall 1 ief emindes fom the ecent past (continued) iot-savat field law: d d d 4 d 7 1 whee 4 1 T m A Note the was the ight-hand ule can help one wok out the diection of : d d Finges along ac fom d to Thumb along d ; ; thumb finges cul in points along d. diection of d. d 3 Octobe 1 hsics 1, Fall 1 3 (c) Univesit of ocheste 1
hsics 1, Fall 1 3 Octobe 1 Foces between cuents Conside a section of a cuent with length L, paallel to a long wie caing cuent 1 and ling a distance awa. What is the foce on this wie? Let both cuents flow in the diection, let the fist one lie along the ais, and the second one lie a distance awa along the ais. The field at the location of the second wie is unifom, as we found last time: 1 d d 1 d df 3 Octobe 1 hsics 1, Fall 1 4 Foces between cuents (continued) So, fom the iot-savat foce law, F d L 1 d L 1 That is, paallel cuents attact each othe with a magnetic foce; antipaallel cuents epel each othe. 1 d df 3 Octobe 1 hsics 1, Fall 1 5 iot-savat vs. Coulomb The setup, and most of the eecution, of calculations fom the iot-savat field law ae the same as fo E calculations using Coulomb s law. That is, choose an appopiate coodinate sstem, dissect the souce distibution into infinitesimal elements, use the smmet of the souce distibution to simplif the vecto addition as much as possible, and then integate the esulting epession. No new ticks ae involved, and no new complications besides the intusion of coss poducts, and fields that lie sidewas with espect to the distance fom the infinitesimal element. 3 Octobe 1 hsics 1, Fall 1 6 (c) Univesit of ocheste
hsics 1, Fall 1 3 Octobe 1 Eample calculations of A cuent flows along two 9- degee acs with adii a and b, and though the adii connecting them, as shown. Calculate the magnetic field at point, the cente of the acs. A Catesian coodinate sstem with oigin at and aes along the staight segments seems appopiate; we can see a use fo both - and pola coodinates. b a 3 Octobe 1 hsics 1, Fall 1 7 Eample calculations of (continued) The appopiate infinitesimal cuent elements: d 1 bd d d 3 ad 4 - values (note = ): 1 b a 4 3 1 b a 4 3 3 Octobe 1 hsics 1, Fall 1 8 Eample calculations of (continued) Fo the staight segments 1 and 3, d d, 1 and d. d b 4 a 4 d 4 b 3 + 4 d a 3 Octobe 1 hsics 1, Fall 1 9 (c) Univesit of ocheste 3
hsics 1, Fall 1 3 Octobe 1 Eample calculations of (continued) Note, fo the pola unit vectos, etc. So 4b 4a 1 1 8 a b 3 Octobe 1 hsics 1, Fall 1 1 Eample calculations of (continued) A cuent flows as shown in a cicula loop of adius. Calculate the magnetic field a distance above the loop, along its ais. A coodinate sstem with oigin at the cente of the loop and ais pependicula to the loop seems good. nfinitesimal cuent element: d d ( ) d 3 Octobe 1 hsics 1, Fall 1 11 Eample calculations of (continued) d Distance fom element: cos sin So the integal contains d d d ut points in opposite diections fo elements acoss the loop fom one anothe, fo which diffes b. d d d d d d 3 Octobe 1 hsics 1, Fall 1 1 (c) Univesit of ocheste 4
hsics 1, Fall 1 3 Octobe 1 Eample calculations of (continued) d So the components cancel out in pais, and the components add in pais. Keep onl, multipl b, and integate fom to : 3 d 4 d 3 d 3 d d d d 3 Octobe 1 hsics 1, Fall 1 13 d Eample calculations of (continued) The solenoid. Find the magnetic field at point on the ais of a tightl-wound solenoid (helical coil) consisting of n cicula tuns pe unit length wapped aound a clindical tube of adius and caing cuent. Epess ou answe in tems of 1 and. What is the magnetic field on the ais of an infinite solenoid? 1 3 Octobe 1 hsics 1, Fall 1 14 Eample calculations of (continued) 1 Suppose that n is so lage that we can conside the loops in the coil to be displaced infinitesimall; then the numbe of loops in a length d is nd, and nd d 3 3 Octobe 1 hsics 1, Fall 1 15 (c) Univesit of ocheste 5
hsics 1, Fall 1 3 Octobe 1 Eample calculations of (continued) Substitute 1 tan d tan d d d cos d d sin so 1 tan 3 Octobe 1 hsics 1, Fall 1 16 Eample calculations of (continued) 3 nd n d sin d 3 sin n sin d ; n n sind cos cos 1. 1 3 Octobe 1 hsics 1, Fall 1 17 1 Eample calculations of (continued) 1 Fo an infinite solenoid, and 1, so n cos cos n 3 Octobe 1 hsics 1, Fall 1 18 (c) Univesit of ocheste 6
hsics 1, Fall 1 3 Octobe 1 Ampèe s law n 186, Andé-Maie Ampèe (Fance) saw that Østed s esult fo the magnetic field of a long wie could be cast in a diffeent, useful wa. Handwaving deivation follows. (Fo a eal one, click hee, and stat at page 8.) We saw last time that the long wie poduced a magnetic field 3 Octobe 1 hsics 1, Fall 1 19 Ampèe s law (continued) This can be witten as The lines of ae cicles centeed on the wie, in a plane pependicula to the wie. The tem looks like an integal of displacement vectos aound the cicle: d d d d 3 Octobe 1 hsics 1, Fall 1 Ampèe s law (continued) ut is constant in magnitude along the cicles, so it can be taken inside the integal. Futhemoe, is the cuent enclosed b the cicula loop. Thus d encl Ampèe s law This elation, which tuns out to be quite geneal, can be used to find fo cases in which is smmeticall distibuted, in much the same wa that Gauss s law can be used to find E. 3 Octobe 1 hsics 1, Fall 1 1 (c) Univesit of ocheste 7
hsics 1, Fall 1 3 Octobe 1 Unifom clindical cuent Eample. A long staight wie with adius caies a cuent which is unifoml distibuted ove its cosssectional aea. Calculate inside the wie. We alead know that is unifom in magnitude on cicles outside the wie: nside the cuent is still clindicall smmetic, so we suspect will be too. A cicle is a good Ampèean path hee. 3 Octobe 1 hsics 1, Fall 1 Unifom clindical cuent (continued) Define the cuent densit J (cuent pe unit coss-sectional aea of the wie): J A Unifom cuent means that J is unifom acoss the wie s coss section: the total cuent is J, and the cuent enclosed b a cicle of adius coaial with the wie ou Ampèean path is encl J 3 Octobe 1 hsics 1, Fall 1 3 Unifom clindical cuent (continued) Now we appl Ampèe s law, noting again that has unifom magnitude along the cicle: d encl J inceases lineal with adius, stating fom eo, and matches up with the outside value at the wie s suface. 3 Octobe 1 hsics 1, Fall 1 4 (c) Univesit of ocheste 8
hsics 1, Fall 1 3 Octobe 1 Aside: cuent distibution in conductos Note that didn t sa that this unifom cuent is caied in a conducto! aallel cuents attact each othe, as we showed ealie toda. So an effot to make a unifom cuent in a conducting wie will esult in the cuent tending to pile up at the cente: cuent theads attact one anothe, and the chages ae fee within the conducto to move in the diection the e being pulled. The calculation of a self-consistent cuent distibution in a conducting wie is beond the scope of HY 1; ou will lean how to do this in a couse like HY 17. 3 Octobe 1 hsics 1, Fall 1 5 Field in an infinite solenoid, Ampèe s vesion ectangula Ampèean loop, as shown. The smmet of the coil dictates that the field must be along, and must be a lot stonge inside than out, so if the numbe of tuns pe unit length is n, and the cuent is, d enclosed n n Same as befoe. 3 Octobe 1 hsics 1, Fall 1 6 Ampèe vs. Gauss Hee ae the conditions unde which it is pofitable to use Ampèe s law to find, compaed to Gauss s law to find E. Ampèe nfinite linea cuent nfinite plana cuent nfinite clindical cuent, an adial dependence nfinite solenoid Tooid Gauss nfinite linea chage nfinite plana chage nfinite clindical chage, an adial dependence Spheicall smmetic chage, an adial dependence 3 Octobe 1 hsics 1, Fall 1 7 (c) Univesit of ocheste 9