ON A SIMPLE SYSTEM OF CHARGED PARTICLES IN MILNE'S KINEMATICAL THEORY

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1 ON A SIMPLE SYSTEM OF CHARGED PARTICLES IN MILNE'S KINEMATICAL THEORY BY D. N. MOGHE (Research Scholar, Bombay Uni,~er.dty) Received March 9, 1939 (Communicatcd by Dr. G. S. Mahajnni) 7. Introduction IN a recent paper Milne 1 has defined mass, energy, force, work, etc., according to bis kinematical theory. Ir the system of particles in his kinematical theory ate supposed to be charged particles then some interesting results ate obtained, it being assumed that the charge associated with the particles is the same for each particle. The value of the charge is e. The results obtained give ah interpretation of the principal equations of the electro-magnetic field in terms of the kinematical theory of relativity. Let us suppose that Mis the momentum of any particle belonging to the kinematical system under consideration ; then where p = mu, etc., and V = (u, v, w). M = (p, q, r), (1) Therefore, d (mu) = ex (2) dt is an equation of motion of any charged particle of the system. If, as a general case, these particles are supposed to collide with each other, then the loss of momentum due to the collision is proportional to the velocity of the particles and the equation (2) can be written in the general forro, viz., d Cm-)=~X-,,, (3) dt In Part I of this paper we shm1 obtain the principal equations of electro-magnetism taking (2) as the equation of motion of any charged particle of the system ; and in Part II these very equations will be deduced, taking into account the collisions of these charged particles. Milne defines force by the relation d (V ~), (4)* 31

2 32 D, N. Moghe where F is the Newtonian measure of force, mis the Newtonian measure P of the mass of an individual partiele, and V-- 7 is the velocity of the P constrained particle relative to its surroundings. -/ may be regarded as the eosmic velocity. 2. Part I : Non-colliding Particles In the Maxwell-Lorentz theory of eleetro-magnetism, the electric force (X, Y, Z) and the magnetic force (a, /~, ~) are determined by the equations, viz., and ~x ~t' etc. (5) ~H ~G etc., (6) = ~y ~z' where (--F, -G, --I-I, ~91 is the electro-magnetic four rector denoting the electro-magnetic potential. Ir should be pointed out here that the electro-magnetie equations used in this paper ate expressed in terms of Heaviside-Lorentz units. If, during the motion, the mass m of each partiele remains constant, then (2) can be written as du m 2~ = ex. (7) As the particles ate moving under the influenee of the substratum, the x x-component of the veloeity of an individual particle will be u -- ] instead of u. Therefore, we have from (4) and (7) Fa-m 2/} ~( u-- ~) =ex (8) of, in vectorial form (8) can be expressed as F-m~ ~( V-- ~) ---- e (X, Y, Z) (9) where V = (u, v, w) and P = (x, y; z). electric force we have e div (X, Y, Z) =m~91 ~ +~ + bz Taking the divergence of the Milne's equations for the motion of a system of particles in free space ate given by the single vector relation, viz., dv (P -- Vt) Y =.X G (91 (llp

3 Simple Syslem of Charged Particles in Milne's Kiuematical Theory 33 in the usual notation. For the fundamental system of particles G (r + 1--~0, X-->t ~, Y--~I, and Z-.t so that (11) becomes dv P -- Vt d t--r---. (12) If we regard the system of charged particles as the fundamental system then (10) reduces to m mv (13) e div (X, Y, Z) = 7 div V = div -}-. Accordingly, we have mv ep = div --U' (14) where p is the density of electric charge. From (13) follows mv e(x,y,z) =-~. (15) II the influence of the substratum is negligible the average acceleration V g of an individual particle is 7" The relation (15) can now be expressed as where m' m g (x, v, z) = ~'g, (16) The Maxwellian equations of the electro-magnetic field are: da bz by dt -- ay ~z' etc., (17) and dx ~r ~fl d-[ + ~~" = ~y ~z' etc., (18) where (ax, ay, a.) is the density of the electric current ; and as the current produced in the case under consideration is due only to charged particles in motion, we have From the equations (17) and (15) we deduce (ex, ay, a,) =p (u, v, w) ----pv. (19) db mv -- d-7 = curl -~, (20) where B = (a, ti, ~). Also from equations (18), (15) and (19) we get the following relation to determine the magnetic vector B, viz., A3 mp curl B =pv -- ep (21) p

4 34 D. N Moghe The usual equation of continuity, viz., takes the forro ~ +~ ~P = o (22) m~ (d~vv-~) 247 =o (%), If p is not a function of position, gradp =0; relation then p is given by the p- 1 et ~ = 2--m + A, (24) where Ais a constant of integration. Ir follows from (24) that p decreases with time and that p--~0 as t--+~. Also when t-~ 0, p--+p0, Po being the initial density of charge the constant of integration Ais, therefore, equal to p0-1 The mechanical stress (P, Q, R) in space due to the electro-magnetic field is given by P = px q- yay --/]%, etc. (25) Using (15) we obtain The rateat which work is done by the mechanical force is dw _ pmv 2 ( mv~ (27) d--t =- Xax + x[a r -l- Z~: et -- pv. -~ j and the Poynting flux ~ = (rrx, ~r, %), where 7rx = fiz- yy, etc., is expressed as [B.V]. (28) rr=}7 Ir should be remembered that we ate using the acceleration formula for the fundamental system as given by (12) for which d F-m ~ (V Pi) ---~ O. (29) This simple value of the aeceleration den0ting a limiting state-of the substratum is to be used only after the necessary operations have been performed. For instance, dv P -- Vt e(x,u =0 for di -- F- ' (30) therefore, ep = 0 But the correet result is given by (14). (al)

5 Simple System of Charged Particles in Milne's Kinematical Theory Part II: Colliding Partieles Itis quite probable that the charged particles belonging to the kinematical system under consideration may have frequent collisions with each other. Let us assume, therefore, that the mass of each individual particle remains constant during the motion before and after the collisions. Hence, the equation corresponding to (7) can be written as du m -)~ = ex --/zu, (32) so that the equations of motion for this case can be expressed by a single rector relation, viz., m--~ ~V(v -- ~)=~(~~~/-~(v ~) ~~~/ The divergence of the electric force would give the density of the electrie charge as div [(m +.t) V] 3~. ep (34) [. t J t ' but the of continuity (22) determines p more directly through a differential equation of the first order ir we put grad p = 0 (i.e., pis a function of t only), viz., dp- I 3mp- I et (35) dt + t (m + tlt) = m + t~t which integrates into p-1 =(t +q [ A+-74em log (m +t~t)]+me/3t, z+ 4- e t [1 + 7m/t m~/t,~t 2 + 4m3/l~3t3]. (36) t* A being a constant of integration, p-+0 when t-+ oo. From equations (17) and (18) we obtain the following formulm to determine the magnetic force B, viz., db (mv ~ y_a) P ---~ =curl --~ +,V 1 = V -- 7' (37) and mp curl B =pv -- et 3. (38) The mechan cal stress brought {nto play by this system of charged particles in motion is given by (P,Q,R) =p (mv ~ +-~- ~V~ + [B.V]) (39)

6 - - c- 36 D.N. Moghe and the work performed thereby is -~dw =PL[mV~et +-e91 (Z -- ty) J per unir time. (40) ~ Lastly, the flow of energy per unir area of the Poynting flux ~r can be expressed as ~r = m +e~./zt [B.V] --e91161 [B. PI. (41) REFERENCE.~ 1 Milne, E. A., " On the Foundat, ions of Dynamics," Proe. Roy. Soc., 154 A, z Asa general case, let us take the force as given by F = --MV(~ -- 1)/Z+u189 ~/y d [Mg Y89 X/Z] where M = m~89 g = -- (P -- VZ/Y)/X, then the equation of mo~ion for the system of charged particies (which cannot be called a simple system) is givcn after considerable reduction by e(x,y,z) = --.~I(O + 1)[g +V (~-- 1)/Z]. The fundamental system being given by G + 1 ~ 0, we have e ~ 0. This gives F ~ 0 for G + I - 0 which leads to relations of the forro gimen by (4:) and (12). a Here X = i~ pg. ~, Y = 1 -- V ~, ~ andz =t-- P ~ V are nob to be confused with the cnmponents of electric force denoted by ~he same letters. 4 Here F is the eleetric force (X, Y, Z). a Here as in referente (3) Y a nd Z ate not to be confused with the component.s of electric force.