How Electric Currents Interact with Magnetic Fields

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1 How Electic Cuents nteact with Magnetic Fields 1 Oested and Long Wies wote these notes to help ou with vaious diectional ules, and the equivalence between the magnetism of magnets and the magnets of electic cuents (called electomagnetism). n 1820, H.C.Oested, in Copenhagen, intending to see if heating of a wie b cuent flow might cause deflection of a neab compass needle, used cuent diven b a batte of twent Cu-Zn voltaic cells. _ batte + A, guide to the ee A ' ' compass base A '' '' choice of leads Figue 1: Oested s expeiment. The compass is fixed in place, but the connections can be changed so that the cuent flows along eithe the uppe path (A-A -A ) o the lowe path (- - ). The compass needle did not deflect towad o awa fom the wie, no did it deflect paallel o antipaallel to the wie. t deflected in the thid diection; Oested concluded that the magnetic field must ciculate aound the wie. When the cuent was evesed, the deflection evesed. We will call the ule giving the diection of ciculation the magnetic field due to the wie Oested s Right-Hand Rule Point the thumb of ou ight hand along the diection of cuent flow. Cul ou finges, which will now be pat of a cicle. (Do this! t is an ode! You finges have to lean this unusual ule as much as does ou conscious bain.) The diection in which ou finges cul gives the diection of ciculation of the magnetic field aound a long cuent-caing wie. ee Fig. 1a. Oested's Right-Hand Rule 1 2 #1 2a #2 Figue 2: Oested s Right-Hand-Rule. The thumb points along the diection of the cuent and the finges ciculate along the diection of the magnetic field. chematic of clockwiseciculating field lines aound a long wie caing cuent into the page. upeposition of the fields of two wies, one into the page and one out of the page. Evewhee along the axis of the wie, the field ciculates about the axis; Fig. 1a gives onl thee of the infinite numbe of cicles associated with the field lines. Fig. 1b gives the field lines fo a 1

2 long wie caing cuent into the page. Check that the have the coect diection of ciculation b pointing the thumb of ou ight hand into o out of the page, and culing ou finges. Example: Oested s ight-hand-ule is of moe than qualitative significance. Fom it we can also aive at some quantitative conclusions. Q: Conside two long wies, caing the same cuent, #1 out of and #2 into the page, and sepaated b a distance 2a, as in Fig. 1c. At a point a distance along thei pependicula bisecto, b smmet the poduce magnetic fields of the same magnitude, sa T. Fo a = 2 cm and = 1 cm, find the net magnetic field at this point. A: The acs in Fig. 1c ae centeed aound wies #1 and #2. Using Oested s ight-hand-ule, wie #1 caing cuent out of the page (#2 caing cuent into the page) poduces a field 1 ( 2 ) that ciculates counteclockwise (clockwise). This detemines the diections of 1 and 2 at the point in question. We then pefom vecto addition to find the total field. Fom Fig. 1c the total field is upwad and of magnitude 2 1 cosθ = 2(0.005)(a/ 2 + a 2 )= T. 2 Ampee and Cuent Loops and Magnets Ampèe s Equivalence Conside a small cuent loop, within a single plane, of cuent and aea A. ee Fig. 3a. Ampèe established that the magnetic dipole moment of this small cuent loop has magnitude = A. (moment of equivalent cuent loop) (11.1) The diection of the equivalent magnet is detemined b what we call Ampèe s Right-hand Rule: Cul the finges of ou ight hand in the diection of the cuent flow; ou thumb then points in the diection of the equivalent magnetic dipole moment. ee Fig. 3b. The magnetic field at the cente of the loop also points in this diection. nea side Ampee's ` Right-Hand Rule Figue 3: Ampee s Right-Hand-Rule fo eplacing a cuent loop b an equivalent magnet, of moment. The finges ciculate along the cuent and the thumb points along. The magnetic dipole moment is nomal to the plane of the loop, and is popotional to the cuent and to the aea A of the loop. f thee ae seven tuns one wa and thee tuns the othe wa, the moment is 7 3 = 4 times lage than foa single tun, and points in the diection of the moment of the majoit (seven) of tuns. ometimes we will emplo magnetic moment, o even dipole moment, in place of magnetic dipole moment. ee Fig. 4a fo the field lines due to a cuent loop; see Fig. 4b fo the field lines of an equivalent magnet. At a distance lage compaed with the loop itself, the field lines of a cuent loop ae indistinguishable fom those fo a long thin magnet, as in Fig. 4c. 2

3 Figue 4: Field-lines of a cuent-loop with pointing to the ight. Field-lines of a ba magnet with pointing to the ight. At lage distances fom both, the field-line pattens ae the same. Example 1: The equivalence of a cuent loop to a magnet enables us to pedict qualitativel what the magnetic field of a cuent loop is like, and how two cuent loops will inteact. We simpl eplace each loop b its equivalent magnet, and then use opposites attact, likes epel. Q: Conside Fig. 5a. Loop, in the cente, is fixed in place, loop A is fee to tanslate, and loop C is fee to otate. How do loops A and C espond? nea 1 A nea 2 3 C nea Figue 5: Cuent loops one could be wapped aound a speake cone! Equivalent magnets. A: Ampèe s ight-hand-ule, Fig. 5a becomes equivalent to Fig. 5b. Then A is epelled b, and C otates clockwise, so that its faces the of. Just as a cuent loop is equivalent to a thin disk-shaped magnet (called a magnetic sheet), so a disk-shaped magnet is equivalent to a cuent loop. Application: The a Magnet and ts Equivalent Cuent. Fig. 6a depicts a ba magnet of magnetic moment, length l, and unifom coss-sectional aea A. The equivalence of a magnet to a cuent loop enables us to eplace the magnet b an equivalent clindical cuent sheet ciculating aound that coss-section. Just as the thin magnetic disks of Fig. 5b ae equivalent to the cuent loops of Fig. 5a, hee the long magnet of Fig. 6a is equivalent to a sheet of cuent with cuent pe unit length K ciculating as indicated in Fig. 6a. K M magnet magnetization M, cuent/length K, magnetic moment A solenoid tuns/length n =nl tuns =A aea A length l Figue 6: Magnet with moment pointing upwad. An equivalent cuent/(vetical length) K = M is also shown. Equivalent solenoid with n tuns/length and cuent, so K = n. Fo length l the cuent is = Kl, so b (11.1) the magnet has magnetic moment = A = (Kl)A = KV, whee V = Al is the volume of the magnet. ecause the magnetization is the dipole moment pe unit volume, o M = /V, fo the cuent sheet C M = V = KV V = K. (11.2) A chaacteistic magnetization fo a pemanent magnet is M 10 6 A/m. (11.2), the coesponding cuent pe unit length is thus K 10 6 A/m, coesponding to a wie of diamete 1mm 3

4 (so n = 10 3 /m) caing = 1000 A, wapped aound a clinde. o wonde magnets ae such poweful souces of magnetic field! This geomet can be poduced b using a long, tightl-wound coil, o solenoid, of coss-section A and length l, with n tuns pe unit length, caing cuent. ee Fig. 6b. This has = nl tuns, each contibuting to the magnetic moment, so = (nl)a. The magnetization of this solenoid satisfies M = V = (nl)a = n. (11.3) Al Thus, when viewed fom the magnetic pole viewpoint, b the magnets chapte the magnet has chage densit σ m = ±M and magnetic pole stength q m = ±σ m A = ±MA on its poles, and when viewed fom this chapte s equivalent cuent viewpoint, the magnet has cuent densit K = M ciculating about its axis. The diection of the cuent is detemined b Ampèe s ight-hand-ule. The uface Cuent is eal, and is the sum of electon cuents fom all of the atoms. Macoscopic cuents deca with time due to electical esistance, but atomic cuents can flow aound an atom without decaing. Magnetic Moment of a Cuent-caing Paallelogam Let us appl Ampèe s Equivalence to a paallelogam cicuit, whee the cuent successivel goes along side a and then along side b. ee Fig. 7a. The aea of the paallelogam is given b A = a b sinθ ab = a b, (11.4) whee θ ab is the angle between a and b in thei common plane, and a b is the vecto coss-poduct. Fig. 7b gives the diection of the magnetic moment (the thumb) using Ampee s Right-Hand Rule. a = a x b b ab a a x b b ight-hand Figue 7: Cuent loop and its magnetic moment. Vecto-Poduct Right-Hand-Rule. Equivalence of Oested s and Ampèe s Right-Hand Rules (RHR s) Fig. 8a-b illustates that Oested s and Ampee s Rules give the same qualitative diections fo the magnetic field of a long wie. z x L 1 Figue 8: Long wie (Oested) and cuent loop (Ampee) that has pat of the wie along one am. Long wie (Oested) and an altenative cuent loop (Ampee) that has pat of the wie along one am. The field line pattens ae the same in each case. Fig. 9a-b illustates that Oested s and Ampee s Rules give the same qualitative diections fo the magnetic field of a cuent loop. 4 z x L 2

5 . Figue 9: Cicula loop (Ampee) and its field-line pattens, with a shaded close-up of a staight pat pat of the loop (Oested). quae loop (Ampee) and its field-line pattens, with a shaded close-up of pat of the loop Oested). haded close-up appopiate to a small pat of eithe the cicula loop o the squae loop, showing the field lines due to that pat of the loop. 3 ome Consequences of Ampèe s Equivalence.. Quantitativel, Ampèe s Equivalence states that, in an extenal field, the cuent loop behaves like a pemanent magnet of moment. This has a numbe of impotant consequences. Toque on a Cuent Loop As fo a pemanent magnet, the toque τ on a cuent loop is given b τ =, τ = sinθ,. (11.8) One application of this equation is to the toque that tuns the needle of a galvanomete. Hee due to a pemanent magnet, and = A being due to tuns of wie caing cuent and having coss-sectional aea A. Application: The lip Ring and Motos. Eq. (11.8) applies to otating motos, which use Ampèe s 1832 invention of the slip-ing. Fig. 10a gives a schematic, whee a pemanent magnet poduces ext, and a batte poduces a cuent that passes though two half-ings sepaated b two gaps and goes to a small otatable solenoid. f the connection of the windings to the batte wee fixed, then the solenoid would oscillate about ext, just like a pemanent magnet. Howeve, when is neal aligned with ext, the angula momentum of the solenoid takes the sliding contacts past the gaps to switch the connections. Then the cuent eveses diection, as do and the toque. Hence the moto keeps otating in the same diection. ee Fig. 10b. ext Figue 10: chematic of a slip-ing. A batte dives cuent though the cicuit, which includes multiple loops of wie attached to a oto in sliding contact with the fixed stato. A fixed extenal field ext points downwad. The toque τ on the oto as a function of oientation of the stato. The evesal of the connections at the gaps causes the toque to alwas cause otation in the same diection. Equivalent Magnets Fig. 11a-c gives some examples of situations whee one can think of a cuent loop as a magnet. Example a: n Fig. 11a we ma eplace the cuent loop b a magnet to see that the toque will 5

6 z x A l q m q m A l Figue 11: A cuent loop in an extenal field. A squat solenoid. At a distance it can be appoximated as a dipole. A long thin solenoid. ea (but not too nea) each end it can be teated as a sum of monopoles. tun the loop clockwise as viewed fom above. Example b: n Fig. 11b the squat solenoid can be thought of as a dipole. We eplace the solenoid a magnet of the same shape to help us visualize the magnetic field lines it poduces. Let the solenoid have n tuns pe unit length, length l, and unifom coss-section A, each tun caing cuent, so K = n. eithe (11.2) o (11.3) it has magnetic moment = MV = KV = (n)(al). At a distance l along its axis, the magnetic field satisfies the dipole law = = 2k m 3, no matte what the shape of the magnet o its equivalent cicuit. ee Fig. 11b. Example c: n Fig. 11c the tip of a long thin solenoid is like a monopole. Conside a magnet and equivalent cicuit that is long and thin, as in Fig. 11c, so that its adius a l. Fo lage distances fom eithe tip of the solenoid, such that a l, b Ampèe s Equivalence the tips can be teated as monopoles q m = ±/l = ±na with field stength = k m q m / 2. Pof.P.Helle (andeis) has successfull wound small solenoids with a ve high and unifom winding densit n, and measued thei -fields. He finds expeimentall that the invese squae law is well-satisfied, both on and off the axis. Fig. 12 shows the obital motion and magnetic motion of ciculating chages, both positive and negative. Thei angula momenta have the same diection but thei magnetic moments ae opposite because the ae oppositel chaged. L q>0 L q<0 q,m v q,m v Figue 12: Obital motion and magnetic moment of a positive chage q > 0. Obital motion and magnetic moment of a negative chage q < 0. 6

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