Chapte 3 Souces of the Magnetic Fie n the ast chapte we stuie the foce exete by a magnetic fie on cuent an chages in motion. We now go on to iscuss how the magnetic fies ae pouce by cuents an chages in motion. The iot-savat aw is anaogous to the expession fo the eectic fie obtaine fom Couomb s aw fo the foce between point chages. Ampee s aw is anaogous to Gauss s aw in eectostatics. t is usefu in etemining the magnetic fie ue to a symmetic cuent istibution. 3. Fie ue to a Long, Staight Wie How a cuent-cay wie pouces a magnetic fie? A cuent in a ong, staight wie pouces a magnetic fie with cicua fie ines --- as may be veifie by spinking ion fiings on a boa noma to the wie.
3. Fie ue to a Long, Staight Wie () iot an Savat foun that the magnetic fie is invesey popotiona to the istance fom the wie. t was ate foun that the cuent is iecty popotiona to the cuent. n S units, we expess these esuts as Whee, cae the pemeabiity constant, is efine to have the vaue 7 4 T m / A 3 3. Fie ue to a Long, Staight Wie () How o we etemine the iection of the fie? The iection of the fie is given by the ight-han ue: When the thumb points aong the cuent, the cue finges inicate the iection of the fie. 4
T +. 3 ut 5 ut.4. 5 ut ut.5 Exampe 3. Two ong staight, paae wies ae 3 cm apat. They cay cuents 3 A an 5 A in opposite iections as shown in Fig. 3.3a. (a) Fin the fie stength at point P. (b) At what point, besies infinity, is the fie stength zeo? Soution: Hint: This is a vecto aition. Diections must be taken into consieation. 5 3. Magnetic Foce etween Paae Wies What is the magnetic foce between two cuent-caying wies? Oeste s emonstation that an eectic cuent exets a foce on a compass neee i not, of couse, poves that thee is an inteaction between two cuents. Ampee emonstate that two cuent-caying wies o in fact exet foces on each othe. 6
3. Magnetic Foce etween Paae Wies () Consie two ong, staight wies that cay cuent an, as shown in Fig. 3.4. F F The foce pe unit ength on eithe wie is the same: F We see that cuents in the same iection attact each othe. Convesey, cuents in opposite iections epe each othe. 7 3.3 iot-savat Law fo a Cuent Eement Having etemine the magnetic fie fo a ong staight wie, iot an Savat next sought a moe genea expession fo the fie ue to an infinitesima ength of any cuent-cay wie. Lapace pointe out to them that the esut fo a ong wie impies that the fie ue to a cuent eement shou epen on the invese squae of the istance. ong cuent-cay wie Long unifom chage ine? k k sinθ kλ E λ E k ˆ 8
3.3 iot-savat Law fo a Cuent Eement () n S units an vecto notation, the iot-savat aw fo the magnetic fie ue to a cuent eement, shown in Fig. 3.5b. is 4 ˆ The magnitue of the fie is sinθ 4 9 ˆ 4 sinθ 4 + / + / / Exampe 3. Fin the fie stength at a istance fom an infinite staight wie that caies a cuent. Soution: tanα cosα sec α α sec α cosα α ( secα) / cosα α
Exampe 3.3 A cicua oop of aius a caies a cuent. Fin the magnetic fie aong the axis of the oop at a istance z fom the cente. Soution: axis axis sinα a 3 4 a Exampe 3.3 ()
The Magnetic Fie of Soenoi 3 Exampe 3.4 A soenoi of ength L an aius a has N tuns of wie an caies a cuent. Fin the fie stength at a point aong the axis. Soution: Sine the soenoi is a seies of cosey packe oops, we may ivie into cuent oops of with z, each of which contains nz tuns, whee nn/l is the numbe of tuns pe unit ength. The cuent within such a oop is (nz). 4
z a tanθ z asec nz nasec ( a a + a tan n cosθθ θ n cosθθ θ n(sinθ sinθ) θθ Exampe 3.4 () A soenoi of ength L an aius a has N tuns of wie an caies a cuent. Fin the fie stength at a point aong the axis. Soution: axis θ ) θθ 3/ nasec θ n(infinite ong soenoi) 5 3.4 Ampee s Law Ampee ha sevea objections to the wok of iot an Savat. Fo exampe, accuacy an assumption. He pusue his own ine of expeimenta an theoetica eseach an obtaine a iffeent eation, now cae Ampee s aw, between a cuent an the magnetic fie it pouces. Athough Ampee s aw can be eive fom the iot-savat expession fo, we wi not o so. nstea, we can make it pausibe by consieing the fie ue to an infinite staight wie. We know that the fie ines ae concentic cices fo a infinite ong, staight cuent-caying wie. () 6
3.4 Ampee s Law () (). We may intepet it as foows: is the ength of a cicua path aoun the wie, is the component of the magnetic fie tangentia to the path, an is the cuent though the aea boune by the path. Ampee geneaize this esut to the paths an wies of any shape. 7 3.4 Ampee s Law () Accoing to Ampee s aw the sum (intega) of this pouct aoun a cose path is given by Whee is the net cuent fowing though the suface encose by the path. The sense (cockwise o countecockwise) in which the intega is to be evauate is given by a ight-han ue: When the thumb of the ight han points aong the cuent, the cue finges inicate the positive sense aong the path. 8
9 Exampe 3.5 ( ) > An infinite staight wie of aius caies a cuent. Fin the magnetic fie at a istance fom the cente of the wie fo (a) >, an (b) <. Assume that the cuent is unifomy istibute acoss the coss section of the wie. Soution: (a) (b) ( ) < Exampe 3.6 An iea infinite soenoi has n tuns pe unit ength an caies a cuent. Fin its magnetic fie. Soution: n nl L ab ab b a a c c b b a + + +
Exampe 3.7 A tooia coi (shape ike a oughnut) is tighty woun with N tuns an caies a cuent. We assume that it has a ectangua coss section, as shown in Fig. 3.8. Fin the fie stength within the tooi. Soution: N N The fie is not unifom; it vaies as /. The tooia fies ae use in eseach on fusion powe. Execises an Pobems Ch.3: Ex. 9, 3, 8, 9 Pob., 3, 5, 9,