Motion through a velocity selector analyzed using a galilean transformation

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1 Motion through a veloit seletor analzed using a galilean transformation Doug Bradle-Huthison Phsis Department, Sinlair Communit College, 444 W. 3 rd St., Daton, OH douglas.bradle-hut@sinlair.edu (Reeived 30 Ma 04, aepted November 014). Abstrat The motion of a harged partile through a veloit seletor is an eample of a general lass of motions through rossed eletri and magneti fields. In this paper partiular attention is paid to the relationship between observables of this tpe of motion measured in two referene frames onneted through a galilean transformation: the lab frame and a frame traveling at speed E/B relative to the lab ( seletor frame ). The eletromagneti field onl has a magneti omponent in the seletor frame, whih makes solution of the equations of motion easier, and also affets onserved quantities suh as energ and angular momentum. The eletri field in this ontet breaks the rotational smmetr in one frame affeting angular momentum onservation onsistent with Noether s theorem. Analzing the motions in this manner is a wa to introdue onepts from relativit into disussions of eletromagnetism. Kewords: Eletromagnetism, Phsis Eduation. Resumen El movimiento de una partíula argada a través de un seletor de veloidad es un ejemplo de una lase general de los movimientos a través de ampos elétrios magnétios ruzados. En este doumento se presta espeial atenión a la relaión entre los observables de este tipo de movimiento, medido en dos maros de referenia onetados a través de una transformaión de Galileo: el sistema del laboratorio un maro que viaja a la veloidad de E/B relativo al laboratorio ("maro seletor"). El ampo eletromagnétio sólo tiene un omponente magnétio en el maro seletor, lo que hae que la soluión de las euaiones de movimiento sea más fáil; también afeta a antidades onservadas omo la energía el momento angular. El ampo elétrio en este onteto rompe la simetría de rotaión en una trama que afeta a la onservaión del momento angular onsistente on el teorema de Noether. El análisis de los movimientos de esta manera es una forma de introduir los oneptos de la relatividad en las disusiones sobre el eletromagnetismo. Palabras lave: Eletromagnetismo, Eduation en Físia. PACS: Fk, g ISSN I. INTRODUCTION A ommon eample of the pratial uses of magneti fields is the veloit seletor; a devie whih is usuall a omponent of a mass spetrometer. Mass spetrometers are instruments found in man pure and applied siene laboratories. When treated in phsis tetbooks [1, ] the veloit seletor is an eample of how a ombination of eletri and magneti interations an be used to steer partiles that differ from a partiular veloit (the seletor veloit ) awa from the aperture of the spetrometer, thus reduing the number of variables that the trajetor of the partile depends on within the spetrometer to one: the mass. The motion of a harged partile through a veloit seletor is part of a more general lass of motions; namel motions through rossed eletri and magneti fields. In an eduational setting it an be used to distinguish between the two interations, one of whih is learl dependent on veloit and the other whih is independent of this variable. The ombination of these two interations is usuall referred to as the Lorentz fore: F qe qv B, (1) whih desribes the interation of a partile with harge q moving with veloit v interating with fields E and In the ases onsidered here we are interested in field onfigurations suh as that desribed in Figure 1, where a uniform eletri field is direted parallel to the ais desribed b the vetor E Eˆ and a uniform magneti field is parallel to the z ais, and desribed b the vetor B Bˆ z. Tetbook disussions are mostl foused on the motion at the seletor veloit, but eamining the other trajetories an shed light on the nature of the two distint interations that olletivel are thought of as eletromagnetism. Lat. Am. J. Phs. Edu. Vol. 8, No. 4, De

2 Doug Bradle-Huthison In this paper we eamine those trajetories b using a galilean transformation to what we will all the seletor frame, namel a referene frame moving at the seletor veloit relative to the lab. An observer in this frame will not be able to detet an eletri field, onl a magneti field will be present in an operational sense. The partile trajetories in this frame are all irular (or helial) and thus solving the equations of motion in this referene frame is muh easier than it would be in the lab frame. The form of Equation b suggests that a hange of variable, that an be viewed as a galilean transformation, will simplif the mathematis, namel: whih is onsistent with: v v v s, v s t, (3a) (3b) where v s E B is the magnitude of the seletor veloit. Partiles that enter the field region with veloit v s v sˆ will follow straight line (inertial) paths at onstant speed. Appling Equations 3 to Equations, we find: m dv qv B, (4a) FIGURE 1. The oordinate sstem and field diretions. An inverse transformation an then be used to determine trajetories in the lab frame. This is of some pratial mathematial use to be sure, but what is more interesting is the disappearane, if ou will, of the eletri field. This turns out to be a speial ase of the more general Lorentz transformations, and an be used to illustrate the relationship between eletri and magneti fields. In addition, onserved quantities differ from frame to frame. Mehanial energ is onserved in both frames, but the form differs. Angular momentum onservation, on the other hand, is diretl affeted b the presene (or absene) of an eletri field whih breaks the rotational smmetr onsistent with Noether s theorem [3]. m dv qv (4b) Note that in this referene frame a harged test partile at rest will remain at rest. There is no eletri field. Onl a veloit dependent interation will affet the test partile, namel a magneti fore. This result is onsistent with the Lorentz transformation [4], for an observer moving along the diretion relative to the unprimed frame: E (E vb z ), (5) but must be viewed as a v approimation as the Lorentz transformations also predit that the magneti field will be different in this frame: B z (B z v E ). In the primed (seletor) frame the veloit and position vetors onsistent with Equations 4 are: v v os( t )ˆ v sin( t )ˆ, (6a) II. TRAJECTORIES Aording to Equation 1 the equations of motion for a harged partile of mass m and harge q moving in the - plane through the field onfiguration desribed in Figure 1 are given b: m dv qv (a) Lat. Am. J. Phs. Edu. Vol. 8, No. 4, De m dv qe qv (b) r r sin( t )ˆ r (os( t ) C)ˆ, (6b) Where qb m, and r qb. If we hoose 0 and C=-1 Equations 6 desribe the trajetor of a positivel harged partile that enters the field region moving parallel to the ais in the positive diretion, and passing through the origin at t=0. This partile will eeute half of a irular trajetor with lokwise rotation. If the field etends into quadrants and 3 (or somehow just 3) the irle will be ompleted onfined to quadrants 4 and 3 in the - plane. With suitable

3 adjustments to the free parameters and the sign of the frequen, Equations 6 an desribe motions for positive and negative partiles (all irular) passing through the origin at t=0 parallel to the ais. For eample, setting C=+1, and hoosing a negative frequen, will desribe the ounter lokwise motion of a negativel harged partile, that passes through the origin along the ais, with a positive veloit. Inverting the transformation desribed b Equations 3 we an determine solutions in the lab frame: Motion through a veloit seletor analzed using a galilean transformation partile, both observers measure the same vetor, thus the net fore is the same in both frames. It is the relationship between the veloit and aeleration vetors that is different for the two observers. This affets the resulting motions, and the salar quantities that are onserved. In this frame: v ( v v s os( t ) v s )ˆ v v s sin( t )ˆ, r (r sin( t ) v s t)ˆ r (os( t ) C)ˆ, r v v s. (7a) (7b) In Figure we show several trajetories of positivel harged partiles with positive initial veloit as viewed in the lab frame. With suffiientl high initial veloit (relative to the seletor veloit) the trajetories are loids, but at lower veloities the redue to nearl sinusoidal shapes reduing further to a straight line motion at the seletor veloit. III. CONSERVATION LAWS A primar differene between an eletri field and a magneti field is that a magneti field is inapable of doing mehanial work. The distintion between the two interations in this regard is evident here, espeiall if we ompare the two referene frames. In the primed, or seletor frame, the net fore on the partile takes the form: F qv B and the fore is alwas orthogonal to the veloit. As a result, no work is done and the kineti energ of the partile is onserved. This is not true in the lab frame where an eletri field is present and the net fore takes the form of Equation 1. In this frame, the quantit (to within an arbitrar onstant) 1 mv qe is onserved. In both FIGURE. Two trajetories for a singl ionized positive ion (m=16 amu) entering the field region along the positive ais at t=0. Top: v 0 =.5v s. Bottom: v 0 =1.5v s. B=0.1 T, E=00 V/m. We an also understand this using a lagrangian/ hamiltonian approah. Equation an be derived from the lagrangian: 1 L 1 qbv qe, ases we an refer to the onserved quantit as the whih an be written in the form: mehanial energ, but in the primed frame we would desribe the quantit b H mv. The galilean L 1 mv 1 mv d (qb) qbv qe. transformation preserves the energ onservation priniple as it applies to mehanial energ, but the form is different. Sine Hamilton s priniple is invariant to a total time The primar priniple, that the quantit is onserved, is derivative added to the lagrangian: retained. It is interesting to note that, the net fore vetor is L 1 invariant under the transformation desribed. The galilean mv 1 mv qbv qe, (8) transformation does not hange the aeleration of the Lat. Am. J. Phs. Edu. Vol. 8, No. 4, De

4 Doug Bradle-Huthison is equivalent in that it will generate the same equations of motion. From Equation 8 using the generalized momenta p qb and p, we an generate the hamiltonian: H 1 1 qe, whih is a onstant of the motion in the lab frame and an also be epressed in terms of the momenta and oordinates alone. But Equation 8 an also be written in the form: L 1 mv 1 mv qb(v v s ) whih suggests a transformation to the primed frame. Sine galilean transformations do not affet time derivatives of veloities, we an add primes to the two kineti energ terms, and write out a lagrangian (in the same form) that will generate the equations of motion in the primed frame: L 1 mv 1 mv qbv. (9) The generalized momenta also have the same form in this frame: p qb, p. But now, ombining these momenta with Equation 9 we generate the hamiltonian: 1 1 H, d r qv B 0, (10) due to the entripetal nature of the fore. In the lab frame the same result will be arrived at even though this observer epliitl takes into aount the effet of the eletri field: r qv B r qe. (11) To see this note that in eah frame the radial vetor is the differene between the position vetor of the partile and the position of the enter of the irle, both relative to the respetive origin: r r r 0 and r r r 0. However, displaements are invariant under a galilean transformation so r r. Also, v s E B ˆ, whih means E v s If we make this substitution into Equation 11 reognizing that v v v s, we find: d r qv B, so, the angular momentum about the ais of one of the irular orbits is onserved in both frames. Following a moving ais, not surprisingl is equivalent to transforming to a moving frame of referene. But what if eah observer determines the angular momentum relative to the origin of their respetive oordinate sstems? In the primed and unprimed frames respetivel we have: d r qv B, (1) r qv B r qe, (13) where the absene of a potential energ term is a diret onsequene of there being no eletri field in this frame. This quantit is a onstant of the motion in the primed and the position vetors are related b the transformation frame. In eah frame the hamiltonian does not depend equation r r v s t. Using this relationship in Equation 13 epliitl on time and is therefore a onstant of the motion. and the fat that E v Noether s theorem [4] states that translational s B we find: invariene of the lagrangian leads to onservation of linear momentum. The omponent of the generalized linear momentum (whih inludes an eletromagneti term) is d qev tˆ z. (14) onserved in both frames, as the eletri field does not break the translational smmetr of the sstem in that This is atuall a somewhat general result, independent of diretion: in eah frame the lagrangian is li in the where on the ais, the ais of rotation is hosen. For oordinate. eample, if both observers shift the ais along their We annot sa the same for rotational smmetr as the respetive aes to r, the primed observer eletri field does pla the role of disrupter, in this regard, will find that the angular momentum is onstant as in the lab frame. This affets angular momentum as we desribed b Equation 10. In the lab frame this quantit will disuss below. hange at a rate given b Equation 14 namel, Angular momentum is another quantit assoiated with a onservation law. In the primed frame it is fairl eas to qe v tˆ z. This result an be understood in light of see that an observer using an ais of rotation that passes Noether s theorem. The eletri field breaks the rotational through the enter of one of the irular orbits, will find smmetr about the z ais in the lab frame, and angular that the partile s angular momentum is onserved, namel: momentum is not onserved. In the primed frame, where Lat. Am. J. Phs. Edu. Vol. 8, No. 4, De

5 there is no eletri field, an observer an hoose an ais of rotation about whih angular momentum is onserved. The smmetr differenes are apparent even if we hoose the origin as the ais. Sine r r r ˆ, and onsidering Equation 10, we an alulate the derivative of the angular momentum in the primed frame about the origin: d r qv Bˆz, for whih the time average over a le vanishes, due to the sinusoidal harater of the veloit. This is not the ase in the lab frame due to the epliit time dependene of Equation 14. And in the primed frame, it is possible through a fied origin shift (0,0) (0, r ) to onvert this vanishing time average to a vanishing instantaneous value. IV. DISCUSSION Eletri and magneti interations are distint but intimatel related. Like other measurable quantities the must be defined operationall, or in terms of how the are measured. This is where the idea of a test harge or in the ase of magnetism, test urrent, omes into pla. To an observer in the lab frame of referene, a harged partile travelling along the ais, at the seletor veloit has zero eletromagneti fore ating on it; an observer moving at the seletor veloit draws the same onlusion but the partile is at rest in this frame. The lab frame observer upon viewing partile motions with varing veloities will onlude that both eletri and magneti interations are taking plae and these interations are simpl balaning at Motion through a veloit seletor analzed using a galilean transformation the seletor veloit. For the observer in the seletor frame, no measurements will reveal the presene of an eletromagneti interation that is not veloit dependent; hene, this observer annot detet an eletri field. In an operational sense this field is not present. If we were to argue that it is present but undetetable, we would be arbitraril singling out the lab frame as some speial referene frame. Doing so, would violate the basi priniples of relativit (either galilean or einsteinian). Thinking relativel is hard for students, as the idea of absolute motion is embedded into our ommon sense notions about the world. But the fat that magneti interations depend on veloit and eletri interations do not, is a hint that referene frames must pla a role in eletromagnetism. The field onept is also hard to grasp, and ma seem like mere mathematial onveniene when introdued in the ontet of eletrostatis. Eamples like the one analzed here, where both interations are in pla and fluid, demonstrates the full power of Mawell s theor. REFERENCES [1] Serwa, R. W. and Jewett, J. D., Phsis for sientists and engineers with modern phsis, 9th Ed. (Brooks/Cole, Boston, 014). [] Young, H. D. and Friedman, R. A., Universit phsis with modern phsis, 13th Ed. (Pearson, San Franiso, CA, 014). [3] Deslodge, E. A. and Karh, R. I., Noether s theorem in lassial mehanis, Am. J. Phs. 45, (1977). [4] Fenman, R. P., Leighton, R. and Sands, M., The Fenman letures on phsis. Vol., (Addison Wesle, Reading: USA, 1963). Lat. Am. J. Phs. Edu. Vol. 8, No. 4, De