The Curlmeter [AUTHOR(S)] [SUBHEAD] BUILDING BLOCKS | Undergraduate Research Projects Experiments with toroids by Armian Hanelli, Cyrus Northern1Virginia Comm 111Equation Chapter 1 Section 111Equation 1 [SECTION] Chapter 1 Section [SECTION] [AUTHOR(S)] Building Blocks Building Blocks [BODY] The Magnetic James Clerk Maxwell, th by Armian TK [HEAD]Hanelli, Cyrus Hossainian, [HEAD] specifically, for donut-sha Northern Virginia Community College, Annandale The Curlmeter The Curlmeter In 1861 he wrote, “Let th [BODY] covered wire. It may be s [SUBHEAD] [SUBHEAD] effect will that— ofm a James Clerk Maxwell, the father of electrodynamics, had athe fondness for be donuts Experiments with toroids objects Experiments toroids specifically, for donut-shaped wrappedwith in wire. Maxwell’s wire-covered r properties anduniformly rich physw [AUTHOR(S)] In 1861 he wrote, “Let there be a [AUTHOR(S)] circular ring of uniform section, lapped covered wire. It may be shown that if an electric current is passed through thiscalled wire multipole moment the by effect will be that ofCyrus a magnet bent round till its two poles are in contact.” simplest of multipolar cu Armian Hanelli, Hossainian, by Armian TKHanelli, Cyrus Hossainian, TK Northern Virginia Community Northern College, Annandale Virginia Community College, Annandale Maxwell’s wire-covered ring, or solenoid, exhibits aStudying number Colof interesting MEMBERS OF THE SPS CHAPTER attoroidal the Northern Virginia Community toroids has yiel properties and rich physics. The current flowingHossainian. in its windings is characterized by a [BODY] [BODY] lege (NVCC) in Annandale, VA. Third from left: Cyrus To the rightlike miniatur can behave multipole moment called a toroidal dipole, or an anapole, moment. Toroidal current by Armian Hanelli and Cyrus Hossainian electromagnetic of him: Walerian Majewski. Seventh from left: Armian Hanelli. Image courtesy of media a James Maxwell, the father James of electrodynamics, Clerk Maxwell, the hadfather a fondness of electrodynamic for donuts simplest of Clerk multipolar currents that produce only finite-range magnetic fields. Northern Virginia Community College, Annandale NVCC. specifically, for donut-shapedspecifically, objects wrapped for donut-shaped in wire. objects wrapped in w We investigated the simp Studying toroids has yielded new insights into fundamental“curlmeter” physics. Atoms, for exam using a magn In 1861 he wrote, “Let there be In a1861 circular he wrote, ring of“Let uniform theresection, be a circular lapped ring uniform of un can behave like miniature toroids. Toroids are also finding applications in covered wire. It may be shown covered that if an wire. electric It may current be shown is passed that if through an electric thiscu w electromagnetic hot in thermonuclear reactors. Toroidal moment the effect willmedia be thatand of aconfining magnet the effect bentplasma round will betill that its of two a magnet poles are bent in contact.” round till its James Clerk Maxwell, the father of electrodynamics, had a We Maxwell’s investigated the simplestring, toroidal multipole: the dipole. Weaor began byofof making ae Just as loops wire are wire-covered or Maxwell’s toroidal wire-covered solenoid, exhibits ring, number toroidal solenoid, interestin fondness for donuts—more specifically, for donut-shaped “curlmeter” using magnetic model, a ring of 12rich neodymium bemagnets. usedcurrent curlmeters to properties andarich physics. The properties current and flowing in physics. its windings The isascharacterized flowing in Just as loops of wire are used as Gaussmeters to measure objects wrapped in wire. amoment. vectordipole, characterizing multipole moment called a toroidal multipole dipole, moment or an called anapole, aistoroidal Toroidal or an curr an magnetic fields, toroidscurrents may be used as to detect theproduce In 1861 he wrote, “Let there be a circular ring of uniform section, Toroidal moment calculated as a fields. vector simplest of multipolar simplest that produce of curlmeters multipolar only finite-range currents that magnetic only pr fi curl of magnetic fields. The curl of a magnetic field B isofa Bvector lapped uniformly with covered wire. It may be shown that if an elecin the Cartesian coo Just as loops of wire are used as Gaussmeters to measure magnetic fields, toroids the has amount of circulation, orinto vorticity, in thenew Bphysics. field. tric current is passed through this wire. . . . the effect will be that of a characterizing Studying toroids yielded new Studying insights toroids has fundamental yielded insights Atoms, into fund for be used as curlmeters to detect the curl of magnetic fields. The curl of a magnetic ¶ inare also ¶ f can behave like toroids. can behave Toroids like areminiature also finding toroids. applications Toroids It is calculated as miniature a vector product of the differential operator magnet bent round till its two poles are in contact.” ˆ ˆ Ñ = x + y + is a vector characterizing the amount of circulation, or vorticity, in the B field. It is ¶ ¶ x y in electromagnetic media and confining electromagnetic hot plasma media in thermonuclear and confining reactors. hot plasma acting onas the components in the Cartesian coordinate system. Maxwell's wire-covered ring or its equivalent, a magnet bent into calculated a vector productofofBthe differential operator > acting on the compon Ñ a loop, exhibits a number of interesting properties and rich physics. of B in the Cartesian coordinate system. We investigated the simplest We toroidal investigated multipole: the the simplest dipole. toroidal We began multipole: by making the (1) The current flowing in its windings is characterized by a multipole “curlmeter”¶using ¶a magnetic “curlmeter” model, a ring using of 12 a neodymium magnetic model, magnets. a ring of 12 n ¶ (1) moment called a toroidal dipole, or an anapole, moment. Toroidal Ñ = xˆ + yˆ + zˆ ¶x ¶y ¶z Toroidal moment Toroidal moment current is the simplest of multipolar currents that produce only finiterange magnetic fields. Just as loops of wire are usedJust as Gaussmeters as loops of wire to measure are used magnetic as Gaussmeters fields, toro to Current flowing in a coil or on the surface of a magnetic torus Studying toroids has yielded new insights into fundamental be used as curlmeters to detect be the used curl asof curlmeters magnetic to fields. detect The thecurl curlofofa magnet magne directed along the axis of symmetry generates a characterizing toroidal moment physics. Atoms, for example, can behave like miniature toroids. m is a vector thetis amount a vector of circulation, characterizing or vorticity, the amount in the of B circulation field. It i running through object’s donut hole. This moment m is of Toroids are also finding applications in electromagnetic media and calculated as athe vector product calculated of the differential as a vector operator product > acting the differential on the com op Ñ system. analogous the magnetic dipole moment produced by current of B in thetoCartesian coordinate of Bsystem. in the Cartesian coordinate confining hot plasma in thermonuclear reactors. (1) flowing through a loop of wire and interacting with B via a torque (1) We investigated the simplest toroidal multipole: the dipole. We ¶ ¶ ¶ ¶ ¶ ¶ ˆ ˆ ˆ ˆ x B. τ = tm × ( Ñ x= B) began by making a “curlmeter” using a magnetic model, a ring of x with + yenergy + z U = -tm · Ñ = x By+ analogy yˆ + zˆ with the ¶x ¶y ¶z ¶x ¶y ¶z magnetic dipole moment, the toroidal moment of a thin toroid with 12 circumferentially magnetized neodymium magnets. Curlmeter EXPERIMENTS WITH TOROIDS TOROIDAL MOMENT LEFT: We passed a linear wire through the hole of our suspended magnetic model to measure its dipole moment. ABOVE: A magnetization M creates a surface current density J and magnetic field B in a torus, illustrated here. Images courtesy of NVCC. 10 Fall 2014 / The SPS Observer xˆ = B+intaA, yˆbe equal Ñ + zˆ toA tand be equal B toint=tmshould where a int are, the torus’s areas of the torus’s =B aA,respectively, where A andareas a are,ofrespectively, ¶x ¶y ¶z m and of the crosshole section “donut” and of its the“limb.” cross section of its “limb.” toroidal moment tm directed along the axis of symmetry running through the object’s donut hole. This moment is analogous to the magnetic dipole moment produced by ic toroid,toroidal withmagnetic approximately circumferential magnetization M, has a toroidal Our toroid, with approximately circumferential magnetization a toroidal moment tm directed along the axis of symmetry running throughM, thehas object’s current flowing through a loop of wire. An external magnetic field B can exert a torque τ ent tm that can hole. be written directly inisbe terms of M: dipole moment that can written directly terms ofdipole M: moment produced by donut Thistmmoment analogous to the in magnetic = tm × B on the toroid and give it the potential energy Um = -m × B. By analogy with the (2) (2) current flowing through a loop of wire.be Anequal external B can a torque τ internal magnetic field Bthe should to tmagnetic = BintaA,field where A exert of its magnetic field B on the magnetic dipole moment of our torus int toroidal moment of amthin toroid magnetic dipole moment, with internal magnetic field 1 1 toroid tand ×B on the and give it the potential energy U mdV m = -m × B. By analogy with the r ´ (= M t = r ´ M dV ) a are, respectively, the torus’s hole and ofareas the of became much larger than the torque on the toroidal moment from Bint should equal( to)tm =areas BintaA,ofwhere A and“donut” a are, respectively, the torus’s m be 2 2 moment, the toroidal moment of a thin toroid with internal magnetic field magnetic dipole “donut” andofofits the“limb.” cross section of its “limb.” cross hole section curl B. Bint should be equal to tm = BintaA, where A and a are, respectively, areas of the torus’s Our magnetic toroid, approximately In a future experiment we will try to detect fields that, accord“donut” hole and of the crosswith section of its “limb.”circumferential magOur magnetic approximately circumferential magnetization a toroidal be written M, has netization M,toroid, has awith toroidal dipole moment tm that can ing to Nobel laureate Vitaly Ginzburg, appear outside of a toroidal dipole moment tm that can be written directly in terms of M: directly in terms ofwith M and the positioncircumferential r inside the toroid: immersed in an electromagnetic medium. (No such fields Our magnetic toroid, approximately magnetization M, hasdipole a toroidal (2) dipole moment tm that can be written directly in terms of M: appear in a vacuum.) We also plan to rotate our magnetic to1 ssumingEquation then that we almost and an small (as compared with (as compared with deal with almost ideal and small dVthat t m2Assuming = deal r ´with ( Mthen )an (2)we ideal extent roidfrom around f the external field B) toroid, its interaction with the of B means, from theof B means, the of 12 the external field B) toroid, itscurl interaction with the curl the its diameter and observe the toroid’s magnetic field tm = r ´with dV ( M )an . (2) (normally axwell lLaw, an interaction current having current density Ampere-–Maxwell lLaw, anexternal interaction withi,an external current i, having current densitylocked inside the toroid) escaping outside, creating an 2 eraction J, with the rate of change therate electric field E.of the electric field E. and/or interaction withofthe of change electric-dipole-type toroidal antenna. // ò ò ò ò Assuming then(3)that we deal with an almost ideal and small This research was funded by a 2013 Sigma Pi Sigma (3) (as compared with then the extent the external field B) toroid, Undergraduate Research Award. Equation with an almost ideal andits small (as compared with ¶E 2Assuming ¶that E weofdeal B = m0 J +the m0eextent B the = m0 J + m0e 0field B) toroid, its interaction with the curl of B means, from the 0 Ñ ´of interaction withexternal thethen curl that of means, fromanthe Ampere–Maxwell Law, ¶t 2Assuming ¶t Bwe Equation deal with almost ideal and small (as compared with Ampere-–Maxwell lLaw, an interaction with an external current i, having current density anextent interaction an external current having current density J, B means, the of the with external field B) toroid, itsi, interaction with the curl of from the J, and/or 3interaction with the rate of change of the electric field E. Equation and/or interaction withan the rate of change the electric field E. Ampere-–Maxwell lLaw, interaction with anofexternal current i, having current density uation Eq. 3 isthat a differential form of athe integral law we see in our textbooks. Note equation Eq. 3 is differential form theelectric integral lawE. we see in our textbooks. J, and/or interaction with the rate of change ofofthe field Get Money for (3) ¶E (3) Ñ´B = m J + m e (3) 0 0 0 Chapter Research! ¶¶Et ¶t SPS chapters are eligible for up to $2,000 in funding for research projects through the SPS Chapter ReEquation 3 Notethat that Eq. 3 isEq. a differential form ofform the of integral law we see ourin oursearch Award (formerly the Sigma Pi Sigma UnderNote equation 3 is a differential the integral law weinsee textbooks. Equation 3 textbooks. graduate Research Award). Applications are due Note that equation Eq. 3 is a differential form of the integral law we see in our textbooks. November 15th. For details see www.spsnational.org/ (4) (4) programs/awards/research.htm. 1 ¶ 1 ¶ ds = m0i +(4) E s=m i+ B × ddA E × dA Ñ ´ B = m0 J + m0e 0 º c 2 ¶t ò c 2 ¶t ò 0 CORRECTION OUR EXPERIMENTS “Get Inspired!” on page 17 of the Spring 2014 issue of The SPS Observer referred incorrectly to physics demos put on by “Juanita Community College.” The actual name of the school is Juniata College, and it is a bachelor’s granting institution. We also incorrectly stated that their most popular demo includes smashing a Equation 4Our experiments We paramaterized the torque resulting from the interaction of the cement block on the chest of a math professor—we should have current density in i with the toroid as τ = t x i. It rotated the toroid's said a physics professor! We apologize for these errors. See beTo axis create a curlmeter capable interacting with the conduction current density in to align with the wire’s of current. low for the demo in action as college president Jim Troha whacks equation Eq. 3, we ran a linear wire with current i through the hole of our toroid. The To find t, “the effective” toroidal moment, we used the method physics torque resulting from the interaction of the current and the toroid, τ = t x i, pushed the professor and SPS advisor Jim Borgardt. // that Gauss ago to make the first measurement of toroid’s axis toused align180 withyears the wire’s current. the Earth’s magnetic field: we measured the frequency f of oscillaTotions find t, toroidal moment, we used method of“the our effective” freely suspended toroid under the the influence ofthat curlGauss B. used 180 years ago to make the first measurement of the Earth’s magnetic field: we measured the From thef torsional equation motion of our toroid weFrom foundthe f (Eq. 5) equation of frequency of oscillations of ouroffreely suspended toroid. torsional 2 , as a function of i, we the found torus’ fmoment inertia, 1.01 × 10 motion of our toroid (equationofEq. 5) asI a= function of–3i, kg themtorus’ moment of inertia, I = suspension 1.01 × 10–3 kg m2, and the suspension and the wire’s torsional constant k:wire’s torsional constant k. (5) (5) f 2 = t / 4p 2 I i + k / 4p 2 I (4) 1 ¶ = m0i + 2capable E × dA To create of (4) interacting with the conduction ò B ×adscurlmeter c1 ¶¶t ò current B density in0i Eq. 3, weEran a linear wire with current i × s = + × d d m A 2 ò c ¶toroid, tò through the hole of our creating a non-zero curl B there. ( ) ( ) /4π2I)i + k/(4π2I)We were ultimately able to extract the crucial measurement for our device, the ultimately empirical toroidal t =crucial 1.20 ×measurement 10–5 Nm/A. Wfor ire currents that did not We were able to moment, extract the pass through the donut hole in the torus exerted no torque, in –5 agreement with the theory. our device, the empirical toroidal moment, t = 1.20 × 10 Nm/A. Wire currents that did not pass through the donut hole in the We then tried to replicate our findings with electromagnetic toroids connected to dc or ac torus exerted no torque, agreement with the theory; from the voltage. This proved difficultinbecause the single layer of windings around the toroid perspective ofaour toroid, their fields werea curl-less. created not only toroidal moment but also net magnetic dipole moment. As the current the linear wire threaded through thewith toroid’s holetoroids increased, Weinthen tried to replicate our findings electric con-the effect of its JUNIATA magnetic field on the magnetic dipole of our torus became much larger torque COLLEGE PRESIDENT JIM TROHA puts the faculty nected to dc or ac voltage. This proved difficult because the singlethan the onlayer the toroidal moment: oscillations of the toroid became unobservable as the torus’ in line axis by hitting physics professor James Borgardt with a sledge of windings around the toroid created not only a toroidal moaligned with the magnetic field of the wire. hammer as SPS students "Danger" Dave Milligan (right) and ment, but also a net magnetic dipole moment. As the current in the Caitlin Everhart (left) look on. Photo by Rick Hamilton. wire threaded we through toroid’sfields holethat, increased, thetoeffect Inlinear a future experiment will trythe to detect according Nobel laureate Vitaly Ginzburg, appear outside of a toroidal dipole immersed in an electromagnetic medium. (No such fields appear in a vacuum.) We also plan to rotate our magnetic toroid around its diameter and observe the toroid’s magnetic field (normally locked inside the toroid) escaping outside, creating an electric-dipole-type toroidal antenna. The SPS Observer / Fall 2014 11

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