INFLUENCE OF INITIAL VELOCITY ON TRAJECTORIES OF A CHARGED PARTICLE IN UNIFORM PARALLEL ELECTRIC AND MAGNETIC FIELDS

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1 The Online Jounal of Science and Technolog - Apil 18 Volume 8, Issue INFLUENCE OF INITIAL VELOCITY ON TRAJECTORIES OF A CHARGED PARTICLE IN UNIFORM PARALLEL ELECTRIC AND MAGNETIC FIELDS Siti Nuul KHOTIMAH, Spaisoma VIRIDI, WIDAYANI, and KHAIRURRIJAL Depatment of Phsics, Facult of Mathematics and Natual Sciences, Institut Teknologi Bandung, Indonesia Coesponding Autho: nuul@fi.itb.ac.id Abstact: An eploato stud on the tajecto of a chaged paticle moing in paallel unifom electic and magnetic fields has been caied out whee electomagnetic adiation of an acceleated paticle is not consideed. A geneal solution fo the paticle motion equation is deied analticall using a simple method b appling the second Newton s law to the Loent foce acting on the chaged paticle. The tajecto is a cicula heli with time-dependent pitch. Specific solutions ae obtained b aing the initial paticle elocit in the absence of electic field. The esult shows two basic pattens of tajectoies: cicula and cicula heli with constant pitch. Paametes such as adius and helical pitch fo cicula heli tajecto as well as adius and its cente position fo cicle tajecto of ccloton motion can be obtained. Kewods: Tajecto, Heli, Pitch, Electic field, Magnetic field, Initial elocit Intoduction The motion of a chaged paticle in unifom electic and magnetic fields has been discussed in man undegaduate phsics tetbooks, especiall fo a staight line tajecto of a chaged paticle with eo esultant of magnetic and electic foces in a elocit selecto (Sewa, 1996; Cutnell and Johnson, 1; Hallida et al., 11; Benson, 1996). The othe fequent topic discussed in the tetbooks is a cicula tajecto of a chaged paticle moing in a plane at ight angles to a unifom magnetic field (Sewa, 1996; Cutnell and Johnson, 1; Hallida et al., 11; Benson, 1996; Giffiths, 1989) and if the paticle stats with an additional elocit paallel to magnetic field then it moes in a heli (Sewa, 1996; Hallida et al., 11; Giffiths, 1989). Spial paths of chaged paticles occu in a magnetic bottle as a plasma confinement (Sewa, 1996; Hallida et al., 11; Benson, 1996). Photodetachment of H in electic and magnetic fields has been conducted theoeticall b seeal authos, such as in cossed electic and magnetic fields (Petes and Delos, 199), in cossed electic and magnetic fields nea a metal suface (Wang, 1), in cossed gadient electic and magnetic fields (Wang and Cheng, 16), and in paallel electic and magnetic fields nea metal suface (Tang et.al., 16) in which classical motion equations of the detached electon ae deied fom the Hamiltonian goening the electon. Using simple method, motion equations of a chaged paticle in unifom cossed electic and magnetic fields ae obtained b appling the second Newton s law to the Loent foce acting on the paticle (Khotimah et al., 17). This wok eploes the motions of a chaged paticle within unifom paallel electic and magnetic fields. Analtical fomula is deied fom a simple method (Khotimah et al., 17) to obtain the motion equations and the tajectoies which ae influenced b initial elocit of the paticle and the magnitude of electic field. Theo In this wok, it is consideed a chaged paticle q with mass m moes with elocit ( iˆ ˆj ) unifom electic field E E k and unifom magnetic field B B k fom initial position iˆ ˆj and initial elocit iˆ ˆj as shown in Figue 1. The fields affect the ˆ paticle motion though the Loent foce ( F ) (Sewa, 1996; Cutnell and Johnson, 1; Hallida et al., 11; Benson, 1996; Giffiths, 1989). ˆ in Copight The Online Jounal of Science and Technolog 89

2 The Online Jounal of Science and Technolog - Apil 18 Volume 8, Issue Figue 1. Schematic plot of a chaged paticle with initial conditions iˆ ˆj and iˆ ˆj in unifom paallel magnetic and electic fields along the -ais ( iˆ ˆj ) B F qe k ˆ q Appling the second Newton s law on the chaged paticle acted b the Loent foce is m a iˆ a ˆj a q B iˆ q B ˆj qe ˆ () ( ) ( ) ( ) ( )k Theefoe, thee diffeential equations ae obtained. d d d d d qe ω, ω, and () dt dt dt dt dt m ω is the ccloton fequenc, at which the paticle would obit in the absence of electic field (Sewa, 1996; Cutnell and Johnson, 1; Hallida et al., 11; Benson, 1996; Giffiths, 1989). qb m ω () The fist two of equations () ae coupled diffeential equations. B diffeentiating the fist of equations () with d espect to t and using the second equations () to eliminate, we obtain a linea thid ode diffeential dt equation with constant coefficients, d d ω dt dt The solution fo ( t) is obiousl obtained. ( t) C1 cosω t C sinωt C (5) Inseting solution (5) into the fist of equations (), we obtain d and then it gies the solution fo ( t ). dt ( t) C cosω t C1 sinωt C (6) The thid of equation () is fo motion unde constant acceleation so that the position of paticle in -ais as a function of time is 1 qe ( t) C5 C6 t t (7) m Equations (5), (6), and (7) ae a complete set of solutions fo paticle motion. The initial condition of the chaged paticle is specified fist. It stats at position moing with initial elocit iˆ ˆj ( ), ( ), ( ), and ( ) Thus, the equations of motion fo the paticle ae iˆ ˆj. These si conditions, i.e. ( ) (1) and, ( ),, detemine the constants C 1, C, C, C, C 5, and C 6. Copight The Online Jounal of Science and Technolog 9

3 The Online Jounal of Science and Technolog - Apil 18 Volume 8, Issue ( t) cosωt sinωt ω ω ω ( t) sinωt cosωt ω ω ω 1 qe ( t) t t m (8) (9) (1) Equations (8) and (9) ae sinusoidal functions and the define the tajecto of the paticle in - plane. ( t) (11) ω ω ω ω Equation (11) is a fomula fo a cicle in - plane of adius R R (1) ω ω which is centeed at point C,. ω ω Theefoe, the geneal tajecto of the paticle is a cicula heli with adius R and pitch p. π m qe π m p t (1) q B m q B The pitch is along -ais and is a function of time. Results and Discussion In this stud, an electon ( m kg, q C ) is used as an eample of a chaged paticle to show the paticle motions. The electon moes in unifom magnetic field B k ˆ T and unifom electic field E 1 N/C. The magnetic foce poides the centipetal acceleation in the - plane and the electic field gies constant acceleation in the -ais. Fo eample, if the electon stats fom initial position ( ˆ i ˆj 6 ) m with initial elocit ( iˆ ˆ j. 1 ) m/s then the equations (), (8) to (1) gie (in SI units): qb ω (1) m ( t).85 cos( t).85sin( t). 85 ( t).85sin( t).85cos( t). 85 (15) (16) 1 7 ( t) 6 1 t.51 1 t (17) The tajecto of the paticle in - plane and the helical pitch ae (18) ( ) ( ) ( ) p( t) t (19) Copight The Online Jounal of Science and Technolog 91

4 The Online Jounal of Science and Technolog - Apil 18 Volume 8, Issue Figue. The motion of an electon fom ( ˆ i ˆj 6 ) m with ( ˆ ˆ. 1 ˆ i j k)m/s in B k ˆ T and E 1 N/C. (a) -component, (b) -component, (c) -component, (d) - plane, and (e) -D space Figue shows the paticle motion in -component, -component, -component, - plane, and cicula heli tajecto with its pitch lineal changes with time. It can be seen cleal that the motion in -component (Figue (a)) and -component (Figue (b)) ae sinusoid. Accoding to equation (11), pojection of the tajecto in - plane poduces a cicle with adius R of. m whose cente at (.85m,.85m) as shown in Figue (d). The paticle motion in -component has a constant acceleation of m/s due to electic foce and in this eample it stats fom 6 m as pesented in Figue (c). The tajecto of the paticle is a tenuous cicula heli as shown in Figue (e). In this wok, the influence of initial elocit iˆ ˆ j is studied. The and -component of initial elocit ( and ) influences the adius R of the cicula heli accoding to equation (1). Meanwhile, the -component of initial elocit ( ) influences the diection of helical motion. Figue shows the influence of 1 m/s,, and 1 m/s in ( iˆ ˆ j k ˆ ) on the tajecto. The tajectoies ae tenuous heli with thei pitch changes with time due to constant acceleation in the -component b the electic field as witten in equation (1). The diection of helical moement in Figue (c) is initiall towad -negatie and then finall to the -positie as pesented in Figue (d). In the absence of electic field ( E ), the motion of the paticle is influenced b the magnetic field alone so that the tajecto depends on the initial elocit of the paticle as shown in Figue. If is constant then the tajecto is a cicula heli with constant pitch (Figue a). This cicula heli has adius R of. m and constant pitch of 7.15m. Howee, if then the tajecto is a cicle in - plane (Figue b). The cicula tajecto has the same adius R of. m. The cente of cicula tajecto is at point C (.85,.85, 6) m. This cicula motion is usuall onl chaacteied b the adius of cuatue in man phsics tetbooks (Sewa, 1996; Cutnell and Johnson, 1; Hallida et al., 11; Benson, 1996; Giffiths, 1989). Copight The Online Jounal of Science and Technolog 9

5 The Online Jounal of Science and Technolog - Apil 18 Volume 8, Issue Figue. Cicula helical tajectoies of an electon in B k ˆ T and E 1 N/C. ˆ i ˆj 6 m with ( ˆ. 1 i 5. 1 ˆj ˆ)m/s : (a) It stats fom ( ) 1, (b) 5 k, (c) 1 and (d) the coesponding -component Figue. Helical and cicula tajectoies of an electon in ˆ i ˆj 6. 1 It stats fom ( ) (a) 1, (b) B k ˆ T and E. m with ( ˆ i 5. 1 ˆj ˆ)m/s : 5 k and (c) the coesponding -component The influences of initial elocit on cicula tajecto of an electon in - plane stating fom initial position ˆ i ˆj within B k ˆ T and E ae shown in Figue 5. The influence of initial ( ) n ˆj m/s; n speed is shown in Figue 5a b taking fou alues, i.e ,,,. and it shows longe adius of the cicula tajecto fo geate initial speed with R n.1 m; n 1,,,. The influence of initial elocit diection with the same speed is also pesented in Figue 5b using initial speed π 5. 1 m/s and initial angle θi i ; i,1,,...7. with espect to ais which esulted 8 Copight The Online Jounal of Science and Technolog 9

6 The Online Jounal of Science and Technolog - Apil 18 Volume 8, Issue diffeent positions of the cente same adii R.85 m. π π C.85sin i,.85cos i,m,1,, i but with the B 1. 1 ˆ T Figue 5. Cicula tajectoies of an electon in and E. ˆ i ˆj with diffeent initial elocities in - plane: It stats fom oigin ( ) ˆ m/s; (a) influence of initial speed n.5 1 j n 1,,,. (b) influence of initial diection (angle) using speed 7 k π 5. 1 m/s : θ i i ; i,1,, Conclusions This wok has descibed the possibilit tajectoies of a chaged paticle in unifom electic field E Ek ˆ paallel to magnetic field B Bk ˆ influenced b initial elocit. The tajecto is cicula heli with its pitch is a function of time if the paticle has elocit component on the - plane ( iˆ ˆ j, iˆ ˆ j, iˆ o ˆ j ), cicula heli with constant pitch if the paticle has elocit component in -ais ( iˆ ˆ j, iˆ, ˆ j ) and E, and cicle if the paticle moes onl in - plane ( iˆ ˆ j, iˆ, ˆ j ) and E. The magnitude of initial elocit influences the adius and the diection of the initial elocit influences position of the cente. Acknowledgement Authos would like to thank Pogam Riset Inoasi ITB in ea 17 with contact numbe 17/I1.C1/PL/17 fo suppoting this wok. Copight The Online Jounal of Science and Technolog 9

7 The Online Jounal of Science and Technolog - Apil 18 Volume 8, Issue Refeences Benson, H. (1996). Uniesit Phsics, eised edition, John Wile & Sons, Inc., Singapoe. Cutnell, J.D. and Johnson, K.W. (1). Intoduction to Phsics, Intenational Student Vesion, 9 th Ed., John Wile & Sons, Inc., Singapoe. Giffiths, D.J. (1989). Intoduction to Electodnamics, nd Ed., Pentice Hall: New Jese. Hallida, D., Resnick, R. and Walke, J. (11). Pinciples of Phsics, Intenational Student Vesion, 9 th Ed., John Wile& Sons, Inc, Singapoe. Khotimah, S.N., Viidi, S. and Widaani (17). Influence of Initial Velocit on Tajectoies of a Chaged Paticle in Unifom Cossed Electic and Magnetic Fields, Euopean Jounal of Phsics, 8():5 Petes, A.D. and Delos, J.B. (199). Photodetachment coss section of H in cossed electic and magnetic fields. I. Closed-obit theo, Phsical Reiew A, 7(), -5. Sewa, R.A. (1996). Phsics fo Scientists and Enginees with Moden Phsics, th Ed., Saundes College publishing, Philadelphia. Tang, T.T., Zhu, Z.L., Yao, J.G., Kong, D.B. and Yin, D.F. (16). Photodetachment of H in paallel electic and magnetic fields nea metal suface, Indian Jounal of Phsics, 9(7), Wang, D. (1). Photodetachment of H ion in pependicula electic and magnetic fields nea a metal suface, Canadian Jounal of Phsics,, 9(1), Wang, D. and Cheng, S. (16). Photodetachment of H ion in cossed gadient electic and magnetic fields, Pamana Jounal of Phsics, 87:75 Copight The Online Jounal of Science and Technolog 95