Physics 1, Fall 01 6 Decembe 01 Toay in Physics 1: Examples in eview By class vote: Poblem -40: offcente chage cylines Poblem 8-39: B along axis of spinning, chage isk Poblem 30-74: selfinuctance of a tooi. Double ainbow ove the VLA. If you can explain eveything in this pictue, you will have unestoo all of electicity an magnetism. 6 Decembe 01 Physics 1, Fall 01 1 Poblem -40 A vey long nonconucting cyline of aius 1 is unifomly chage with chage ensity E. It is suoune by a cylinical metal (conucting) tube of inne aius an oute aius 3, which has no net chage. If the axes of the two cylines ae paallel, but isplace fom each othe by a istance, etemine the esulting electic fiel in the egion > 3, whee the aial istance is measue fom the metal cyline s axis. Assume < ( 1 ). 6 Decembe 01 Physics 1, Fall 01 Poblem -40 (continue) The key featues o this poblem ae that (a) it involves a supeposition of two infinite cylinical chage istibutions, so the electic fiel has no component in the iection of the axes, suggesting the use of Gauss s Law; an (b) E = 0 insie the conucting cylinical shell, also suggesting the use of Gauss s Law. E 6 Decembe 01 Physics 1, Fall 01 3 3 1 Gaussian cylines of length h, pepenicula to (c) Univesity of ocheste 1
Physics 1, Fall 01 6 Decembe 01 Poblem -40 (continue) The chage pe unit length of the nonconucting cyline is 1 E 1. Call that on the othe sufaces an 3. Daw a Gaussian cyline, aius an length h, coaxial with the conucting shell an lying within it. E = 0 eveywhee on its suface, except the cicula ens, which ae E(i.e. EA0). E 6 Decembe 01 Physics 1, Fall 01 4 3 1 Gaussian cylines of length h, pepenicula to Poblem -40 (continue) So Gauss s Law fo this suface is 1 1 h EA 0 Qencl 0 0 1 E1 The chage/length is istibute nonunifomly on the inne suface of the shell, in such a way that its E cancels that of the chage o, within the bouns of the shell. E 6 Decembe 01 Physics 1, Fall 01 5 3 1 Gaussian cylines of length h, pepenicula to Poblem -40 (continue) The shell has no net chage, so 3 E 1. This chage must be istibute unifomly on the oute suface. Othewise its E woul not be eo insie the shell, as it must be since E fom the othe two chage istibutions a up to eo within the shell. E 6 Decembe 01 Physics 1, Fall 01 6 3 1 Gaussian cylines of length h, pepenicula to (c) Univesity of ocheste
Physics 1, Fall 01 6 Decembe 01 Poblem -40 (continue) Thus the calculation fo the fiel outsie the shell is familia: EA Qencl 0 E h 3 h 0 E 1 h 0 E E1 ˆ. 0 Note that the answe oes not epen on whee the chage o is, as long as it s insie the shell. E 6 Decembe 01 Physics 1, Fall 01 7 3 1 Gaussian cylines of length h, pepenicula to Poblem 8-39 A nonconucting cicula isk, of aius, caies a unifomly-istibute electic chage Q. The plate is set spinning with angula velocity about an axis pepenicula to the plate though its cente. Detemine (a) its magnetic ipole moment an (b) the magnetic fiel on its axis a istance away fom the cente. (c) Is 0 B 3 fo? I ll o (b) fist. 6 Decembe 01 Physics 1, Fall 01 8 Poblem 8-39 (continue) (b) We fist beak the poblem own into pats we have seen befoe. Consie an infinitesimal annulus, wie at aius. This annulus caies a chage Q A, an the isk is spinning, so the annulus is like a cicula loop of cuent: Chage Q, aius Q I Q Chage ensity Q 6 Decembe 01 Physics 1, Fall 01 9 B (c) Univesity of ocheste 3
Physics 1, Fall 01 6 Decembe 01 Poblem 8-39 (continue) That is a poblem we ve seen befoe: in class on 30 Octobe 01 we showe that the magnetic fiel on the axis of a cicula loop of aius that caies a cuent I is 0 I B ˆ 3 So switch I I, : 3 0 I 0 B ˆ ˆ 3 3 6 Decembe 01 Physics 1, Fall 01 10 B Poblem 8-39 (continue) An so we a the contibutions of all the annuli into which the isk can be ecompose: 3 ˆ 0 B 3 0 3 Substitute: u u As 0, u, so... B 6 Decembe 01 Physics 1, Fall 01 11 Poblem 8-39 (continue) 0 ˆ 0 4 3 3 4 0 u B ˆ 1 1 ˆ 0 u u 4 1 1 0 ˆ 0 1 ˆ 1 1 6 Decembe 01 Physics 1, Fall 01 1 u u (c) Univesity of ocheste 4
Physics 1, Fall 01 6 Decembe 01 Poblem 8-39 (continue) (a) The magnetic ipole moment of the infinitesimal annulus at aius is its cuent times its enclose aea: ˆ ˆ 3 AI so the sum of the moments of all the annuli is just the integal of this: 4 ˆ 3 ˆ 4 0 B 6 Decembe 01 Physics 1, Fall 01 13 Poblem 8-39 (continue) (c) Fo this we nee a powe-seies expansion that moves the poblem past the scope of PHY 1: n n! m 1 x x m! 0 n m! m nn 1 1nx x if x 1. 1 an 1 : Hee we have in min x n 1 1 0 1 1 B 6 Decembe 01 Physics 1, Fall 01 14 Poblem 8-39 (continue) 0 1 1 B ˆ 1 1 4 4 0 4 1 1 1 3 ˆ 1 1 4 8 8 0 ˆ 3 So, yeah, it oes what is suggeste in the book. 6 Decembe 01 Physics 1, Fall 01 15 (c) Univesity of ocheste 5
Physics 1, Fall 01 6 Decembe 01 Poblem 30-74 (a) Show that the self-inuctance L of a tooi of aius 0 containing N loops each of iamete is 0N L if 0. 80 (b) Calculate L fo =.0 cm an 0 = 66 cm. Assume that the fiel insie the tooi is unifom, an that thee ae 550 loops in it. 6 Decembe 01 Physics 1, Fall 01 16 Poblem 30-74 (continue) Let s calculate the fiel insie the tooi fist. The cicula geomety will foce B to be constant in magnitue on cicles an point clockwise, so we use Ampèe s Law an a cicula loop: B 0Iencl B 0NI 0NI B ˆ 0 6 Decembe 01 Physics 1, Fall 01 17 Poblem 30-74 (continue) If 0 then all points within the tooi lie at appoximately the same aius: the fiel is almost unifom. Thus the self-inuctance becomes NB NBA L I I 0N 0N 0 80 as avetise., 0 6 Decembe 01 Physics 1, Fall 01 18 (c) Univesity of ocheste 6
Physics 1, Fall 01 6 Decembe 01 Poblem 30-74 (continue) The aithmetic: 5 L.910 H. 0 6 Decembe 01 Physics 1, Fall 01 19 (c) Univesity of ocheste 7