Method of Loclistion nd Controlled Ejection of Swrms of Likely Chrged Prticles I. N. Tukev July 3, 17 Astrct This work considers Coulom forces cting on chrged point prticle locted etween the two coxil, likely chrged rings lying t some distnce etween one nother. Conditions under which the chrged prticle, while eing ffected y Coulom repulsion forces of chrges of the rings, loclises in the spce etween the rings nd moves long closed trjectory round the xis of chrged rings, not leving the spce etween them, were determined. Also, conditions t which controlled ejection of the chrged prticle from its loclistion zone or the cpility to control the kinetic energy nd direction of ejection of the prticle is possile, were determined. Bsed on the otined results, we conclude tht the considered method of loclistion nd controlled ejection of chrged prticles is pplicle oth in experiments on the nucler synthesis in swrms of loclised positively chrged prticles nd for the formtion of ems of likely chrged prticles with the given velocity of motion reltively to the chrged rings. Keywords: Coulom forces, electric chrge, nucler synthesis, prticle em. Figure 1 illustrtes n interction etween prticle nd two rings hving like electric chrges chrges of the prticle nd the rings re of the sme sign. By ring we men solid torus gel-shped mnifold. The following definitions re used t the figure: m is n electriclly chrged prticle; P is P electriclly chrged ring; P is P electriclly chrged ring; F is surfce on which unit vectors of forces cting on the prticle from the side of the P ring re lying; F is surfce on which unit vectors of forces cting on the prticle from the side of the P ring re lying. Let us ssume tht the prticle is point prticle nd forces cting from the side of rings on the point prticle re coming from the circles formed y centres of infinite set of cross-sections of the rings. The rings hve the sme circulr cross-sections Figure. The plnes t which the circles of rings re lying re prllel. The xes of the X nd Y coordinte system in which the interction is reviewed re lying in plne which is prllel to those of circles of the rings nd is locted in the middle etween the plnes e-mil: in-t@rmler.ru
Figure 1: Interction of chrged prticle nd two chrged rings. Figure : 1 is circulr cross-section of the ring; is centre of the circulr cross-section of the ring; 3 is circle formed y centres of infinite set of cross-sections of the ring the circle of the ring; 4 is the centre of the circle of the ring; 5 is rdius of the cross-section of the ring.
of circles of the rings. Projections of centres of circles on the X, Y plne coincide. The positive xis Z is coming from the projection point of centres of circles of the rings to the X, Y plne nd directed towrd the P ring. Coordintes of the centre of the circle of the P ring re,, h the coordinte definition order is x, y, z. Coordintes of the centre of the circle of the P ring re,, h. Coordintes of the prticle loction re x, y, z. In order to determine the eqution of motion of the chrged prticle under the ction of Coulom forces of chrges of the rings, we will use the following vriles nd constnts: m is mss of the prticle; R = x + y + z is the rdius vector of the prticle position; r = x+ y is vector prllel to the X, Y plne, linking the Z xis nd the prticle position point; q is n electric chrge of the prticle; Q is n electric chrge distriuted uniformly in the P ring; Q is n electric chrge distriuted uniformly in the P ring; R is vector coming from the centre of the circle of the P ring to the point on this circle; R is vector coming from the centre of the circle of the P ring to the point on this circle; R m is vector issued from the point of the circle of the P ring to which the R vector comes to, to the point of loction of the prticle; R m is vector issued from the point of the circle of the P ring to which the R vector comes to, to the point of loction of the prticle; γ is n ngle etween the vectors r nd R equl to the ngle etween the vectors r nd R ; ω is mgnitude of n ngulr rottion velocity of the r vector. In order to consider forces cting on the prticle from the side of two rings s those tht re coming from the circles formed y centres of infinite set of cross-sections of the rings, the following conditions should e stisfied: R m >> r s nd R m >> r s where r s is the rdius of cross-sections of the rings Figure. Let us write down the eqution of motion of the prticle s n equlity of the force cting on the prticle to the sum of Coulom forces pplied to the prticle from the side of ll electric chrges of the rings: m d R dt = qq π π R m R 3 m dγ + qq π π R m R 3 m dγ, 1 where: R m = r + z h ẑ R, R m = r + z + h ẑ R, 3 R m = R + z h + r rr cos γ, 4 R m = R + z + h + r rr cos γ. 5 In the 1, the integrtion is performed with respect to the γ vrile, with constnts R, R, R, h, nd vriles R m nd R m. 3
Let us determine the projection of the force 1 to the r vector: d R m dt r = qq π Rm r dγ + qq π Rm r dγ. 6 π Rm 3 π Rm 3 Let us determine the projection of the force 1 to the z vector: d R m dt z = qq π Rm z dγ + qq π Rm z dγ. 7 π Rm 3 π Rm 3 After the trnsformtion of equtions 6 nd 7, tking into ccount - 5, we otin: m d r dt mω r = qq π r R cos γ π R + z h + r rr cos γ dγ+ 3/ + qq π π r R cos γ R + z + h + r rr cos γ dγ, 8 3/ m d z dt = qq π z h π R + z h + r rr cos γ dγ+ 3/ + qq π π We introduce nd determine the following functions: z + h R + z + h + r rr cos γ dγ. 9 3/ r = R + z h + r, r = R + z + h + r, 1 s = r r, s = r r, k = R r, k = R r, l = z h r, l = z + h r. 11 Using 1 nd 11, we trnsform 8 nd 9: m d r dt mω r = q π Q s k cos γ π r 1 s k cos γ + Q s k cos γ dγ, 1 3/ r 1 s k cos γ 3/ m d z dt = q π Q π r l 1 s k cos γ 3/ + Q r As the following conditions re stisfied: we will hve from 1 nd 11: s k = l 1 s k cos γ 3/ dγ. 13 h, z ±h, 14 rr R + z h + r < 1, s k = 4 rr R + z + < 1. 15 h + r
Therefore, for 1 nd 13, under the conditions of 14, we cn use the expnsion into Mclurin series: 1 1 = n + 1! n 3/ n n!, < 1. 16 n= Applying the 16, we will otin from the 1: m d r dt mω r = q π q π n= n= n + 1! n n! n + 1! n n! Q r Using the vlues of definite integrls: Q r s n k n+1 s n+1 k n + Q r π s n+1 k n cos n γdγ + Q π s n r k n+1 cos n+1 γdγ. 17 π cos n γdγ = π n! n n!, π cos n+1 γdγ =, n =, 1,...,, 18 we integrte nd trnsform the right prt of the 17: m d r dt mω r = q q n=1 n= 4n + 1! Q 4n n! n! r 4n! Q 4n n 1! n! r s n 1 k n s n+1 k n Then, pplying 16 nd 18, we will hve from the 13: m d z dt = q n= Under the following conditions: we rewrite 19 nd s follows: m F 4n + 1! Q l 4n n! n! s n r k n + Q r + Q r + Q l r s n+1 k n s n 1 k n. 19 s n k n. Q = Q = Q, R = R = R, F = qq, 1 R d r dt ω r = n=1 m d z F dt = n= 4n + 1!R 4n n! n! 4n!R 4n n 1! n! n= 4n + 1!R 4n n! n! s n 1 r s n+1 k n r l s n k n r Let us introduce nd determine dimensionless functions: k n + sn 1 r + l s n + sn+1 r k n r kn k n,. 3 r = r R, z = z R, h = h R, 4 5
r = r R = 1 + Then nd 3 will look s follows: F r = 4n + 1! r n+1 n= 4n n! n! z h + r, r = r R = 1 + z + h + r, 5 F r = m d r F dt ω r, Fz = m d z F dt. 6 F z = 1 r 4n+3 n= + 1 r 4n+3 4n + 1! r n 4n n! n! n=1 4n! r n 1 4n n 1! n! z h + r 4n+3 1 r 4n+1 + 1 r 4n+1. 7 z + h r 4n+3. 8 As follows from 5, 6 nd 8, if z =, then d z/dt = s well. Thus, t z = nd dz/dt = the prticle will move within the X, Y plne. Let us determine the projection of the force 1 on vector which is norml to the r vector lso: d R m dt r = qq π Rm r dγ + qq π Rm r dγ, 9 π Rm 3 π Rm 3 nd let us consider the 9 under the conditions of 1, t z = nd t dz/dt = : m d r ω r dt Tking into ccount tht: π = qq π integrting the 3, we will get: π R sin γ dγ. 3 R + h + r 3/ rr cos γ R sin γ dγ =, 31 R + h + r 3/ rr cos γ z =, dz dt =, mr ω = Const. 3 Grphing the dependence of dimensionless functions F r nd F z on the vlues of r, t vrious vlues of z nd h, using the equtions 7, 8, 3, nd grphing the functions similr to dimensionless those F r nd F z otined from the equtions 19 nd t Q Q llows for the following conclusions: 1. Under the condition of h < 1/, there re the vlues of vriles r, z, dr/dt, dz/dt, ω tht determine initil conditions of prticle motion t which the prticle cn e loclised in the spce etween the chrged rings.. There re conditions of vrition of the distnce etween the rings nd conditions of vrition of chrges of the rings, oth overll nd sectorl i.e., chrges of certin segments of rings, t which loclised prticles will e ejected from the loclistion zone nd will ccelerte under the ction of Coulom repulsion forces of the rings long certin directions nd to certin kinetic energies. 6
Figure 3: Zones of loclistion nd ejection of chrged prticle during its interction with the two chrged rings. Figure 3 demonstrtes two surfces mrked trnsprent lue nd yellow tht split the spce etween the rings into three res. The surfces hlved y the X, Z plne re shown. In the re etween the surfces two forces determined y the equtions of 7 nd 8 ct on the prticle under the condition of h =.5: the force prllel to the Z xis is directed towrd the X, Y plne while the force prllel to the r vector is directed to the Z xis. The mgnitude of force prllel to the r vector nd cting on the prticle t the trnsprent lue surfce equls to zero. The mgnitude of force prllel to the Z xis nd cting on the prticle t the yellow surfce equls to zero. In the re eyond the trnsprent lue surfce the force prllel to the r vector nd cting on the prticle is directed wy from the Z xis. In the re eyond the yellow surfce the force prllel to the Z xis nd cting on the prticle is directed wy from the X, Y plne. Therefore, in the re etween the surfces there will exist some certin set of trjectories not contcting to the surfces; while moving long them, the prticle will e loclised. This set includes circulr trjectory in the X, Y plne t the following conditions: h =.5, z =, dz dt =, ω = F mr, ω = ω, r.556446..., 33 ω ω n=1 4n! 4n n 1! n! r n 1 + h + r n+1/ n= 4n + 1! 4n n! n! r n 1 + h + r n+3/. If the prticle enters the re eyond the yellow surfce see Figure 3 while the distnce etween the rings is chnging or the chrges of the rings re chnging, it is repelled from the zone etween the rings with the force directed prllel to the Z xis wy from the X, Y plne. If the prticle enters the re eyond the trnsprent lue surfce see Figure 3 while the distnce etween the rings is chnging or the chrges of the rings re 7
chnging, it is repelled from the zone etween the rings with the force directed prllel to the r vector wy from the Z xis. As clcultions show, the loclistion of prticles in the inner spce etween the rings is possile only for prticles tht hve strictly determined initil conditions of their motion reltively to the chrged rings. Prticles unit vectors of which velocities re lying within the X, Y plne nd hving definite initil vlues of moments of moment reltively to the Z xis nd definite initil vlues of their rdil velocities reltively to the Z xis will overcome the repulsion of rings nd will concentrte long certin circulr trjectories etween the rings. These initil conditions of prticles motion reltively to the rings re determined s follows: Bsed on the lw of conservtion of the sum of kinetic nd potentil energies of the prticle during its motion within the X, Y plne z =, dz/dt =, we will otin: m dr = E J U r, h, 34 dt mr E = m dr + J + U r dt mr, h, J = mrω, 35 where: r is mgnitude of rdius vector of initil position of the prticle in the coordinte system where the interction is considered, dr /dt is n initil rdil velocity of the prticle, ω is mgnitude of n initil ngulr velocity of the prticle. Using the definition of potentil [1], pplying 1 nd 11, we find the potentil energy of the prticle in the system of two chrged rings: U r, z, h = q π π Under the conditions: we will hve from the 36: U Q r 1 1 s k cos γ 1/ + Q r 1 1 s k cos γ 1/ dγ. 36 Q = Q = Q, R = R = R, z =, 37 s = 1 + h 1/ + r, k = r/s, 38 r, h = qq π πr s From the vlues of functions 38 we will get: 1 1 k cos γ 1/ dγ, 39 h, k < 1. 4 Therefore, under the conditions of 4, we cn represent the integrted function in the 39 s n infinite series: U r, h = qq πr s n!k n n= n n! 8 π cos n γdγ. 41
Using the 18, we integrte the 41: U r, h = qq R s n= 4n!k n 4n n! n!. 4 We sustitute s nd k with their determintions 38 nd finlly otin: U r, h = qq R n= 4n! 4n n! n! r n 1 + h + r n+1/. 43 Then we will determine four dimensionless functions: Ă = mr3 d r E R J, B =, C =, 4qQ dt qq 4mqQ R Ŭ = n= Using 44, we rewrite the 34: 4n! 4n n! n! r n 1 + h + r n+1/. 44 Ă = B C Ŭ. 45 r Let us find susequently the first, the second nd the third prtil derivtives of Ă with respect to r nd let us determine the three functions of r nd h: Ă r = C r 3 Ă r C = 6 r 4 Ŭ r. 46 Ŭ r. 47 3 Ă 4 C = 3 Ŭ r 3 r 5 r. 48 3 From the functions 45-47 we form system of three lgeric equtions reltively to the unknowns B, C, r depending on the vlue of h: C 1. B = r + Ŭ,. r C 3 Ŭ = From 48 nd 49 we otin n inequlity: r, 3. 3 r Ŭ r = Ŭ r. 49 1 r Ŭ r 3 Ŭ r 3, 5 for the purpose of determintion of the h vlues t which oth the system of equtions 49 nd the inequlity 5 re true. The system of equtions 49 nd the inequlity 5 determine conditions under which the Ă function 45 hs n inflection point t which the vlue of the function equls to zero. As lso follows from these conditions, the 9
function 46 hs n extremum t the inflection point of the function 45; t the point of this extremum the function 46 lso equls to zero. The function 45 is dimensionless function of the squred rdil velocity of the prticle. The function 46 is dimensionless function of rdil ccelertion of the prticle. Therefore, the rdil velocity nd the rdil ccelertion of the prticle t the inflection point of the Ă function 45 will equl to zero. The prticle hving the initil conditions of its motion s determined from 49 nd 5, with the negtive vlue of rdil velocity will overcome the repulsion of the rings, nd the trjectory of its motion will grdully trnsform to circulr with certin constnt vlues of r nd ω. Prticles which initil conditions of motion do not comply the conditions of loclistion will e ejected from the system of rings to the infinity. The dynmics of the prticle moving from the infinity towrd the rings nd which trjectory trnsforms to the circulr s determined y the conditions of 33 is demonstrted t Figure 4. The grphs of the following three functions re drwn there: - red curve determines the mgnitude of the dimensionless rdil velocity of the prticle depending on the distnce to the origin of the coordinte system the Ă1/ function otined from the 45; - lue curve determines the dimensionless rdil ccelertion of the prticle depending on the distnce to the origin of the coordinte system the Ă/ r function 46; - yellow curve determines the direction of the dimensionless force cting on the prticle prllel to the Z xis the negtive vlues of the function men tht forces re pressing the prticle to the X, Y plne depending on the distnce to the origin of the coordinte system, with z =.1 the F z function 8. F r Ro,z Ro,h Ro 1..8.6.4..5 1. 1.5. r Ro Figure 4: The dynmics of loclistion of chrged prticle during its interction with the two chrged rings. The results of the study regrding trjectories of prticles motion performed y using the numeric methods demonstrte tht there is theoreticl possiility of creting the conditions for the nucler synthesis in swrms of positively chrged prticles loclised using the method s determined hereinove, nd tht there is theoreticl possiility of formtion of ems of likely chrged prticles with the given kinetic energy nd with the given direction of their motion reltively to the system of chrged rings. If conclusions formulted in Modified Coulom Forces nd the Point Prticles Sttes Theory [] concerning the existence of the proton nd the electron condenstes re 1
correct, then the system of the two chrged rings will help crete the conditions for the formtion nd loclistion of volumes of oth proton nd electron condenstes. Loclised volumes of condenstes could possily e used either s men for jet-propelled motion or s tool for the destruction of spce ojects steroids nd comets potentilly thretening Erth or for chnging their motion trjectories. References [1] Jckson, J.D. Clssicl Electrodynmics // John Wiley & Sons, Inc., New York, 196 [] Tukev, I.N. Modified Coulom Forces nd the Point Prticles Sttes Theory// http://vixr.org/pdf/177.7v1.pdf 11