Moon_RetPotentialClass_Relative

Physics Essays volmne 7, number 1, 1994 The Electrodynamics of Gauss, Neumann, and Hertz Parry Moon, Domina Eberle Spencer, Arian S. Mirchandaney, Uma Y. Shama, and Philip J. Mann Abstract This paper is based on the fundamental criteria for the electrodynamic equation suggested by Gauss. It considers the definitions of the electric field proposed by Neumann and by Hertz. The classical formulation utilizing Neumann's definition and Einstein's postulate on the velocity of light does not satisfy the criteria suggested by Gauss. A new electrodynamic equation is proposed, utilizing the Hertzian definition of the electric field and the universal time postulate on the velocity of light, which does satisfy the Gaussian criteria. Key words: electrodynamics, force, moving charges, charges, potentials, postulates, velocity of light 1. INTRODUCTION Gauss ~) suggested in 1835 that electromagnetic theory should be derived from a single electrodynamic equation for the force between moving charges which should be a function of the relative velocity of the charges, and in 1845 added the condi- tion c2> that it should include retardation. Gauss wrote to his friend Weber about his ideas and even proposed an equation <3~ that can be derived from a scalar potential, is a function of position and relative velocity, and describes AmpSre's equation for the force between current elements. But the equation suggested by Gauss did not include retardation. Gauss was not satisfied with his own formulation. Following his motto pauca sed matura, Gauss never published his equation. It appears only in letters and in his collected works. Gauss's younger friend and collaborator, Weber, generalized Gauss's equation to include an acceleration term and published ~4) it in 1848. This equation can also be derived from a scalar potential which is a function of position and relative velocity, but does not include retardation. However, this equation is still seriously considered by a number of scientists including Wesley, c5) Pappas, c6) Phippsf ) Graneau, ~a) and Assisfl ) The idea that an electromagnetic field should be derived from both a scalar and a vector potential was introduced by Neumann c1~ in 1845 and has been utilized in Maxwellian can be expressed in the form (r)R = c(t -- r), (4) where (r)R is distance (in meters), t is time (in seconds) of reception, and z is time (in seconds) of emission. All distances are measured in a nonrotating coordinate system. Then for any of these postulates on the velocity of light, differentiation of 28 Parry Moon, Domina Eberle Spencer, Arjan S. Mirehandaney, Urea Y. Sharaa, Philip J. Mann Eqs. (4) with respect to t gives at )~ d( r )R[d~] (5b) _ _ =c 1 - _ _ dt dt " and differentiation with respect to rectangular space coordinates in the nonrotating coordinate system gives = - -C .~ [ ~ yzt yzt " ~ xzl XZt [ a(r)R Or xyt The scalar potentials considered in this paper will have the form r = a (7) 47re0(r) R ' and the vector potentials will have the form A = KQw(z(t),t) (r)R (8) where Q is the charge (in coulombs) whose motion produces the electromagnetic field, (r)R is the distance (in meters) from the charge to the point at which the fields F, E, and B are defined, e0 is the permittivity of free space (in farads/meter), K is an arbitrary constant, and W is an arbitrary vector function of r and t. 3. DERIVATIVES OF THE POTENTIALS In order to determine general expressions for the fields E, B, and F, it is necessary to differentiate the scalar and vector potentials. Differentiating Eq. (7), grad r = - Q (grad (r)R) , . (9) 47r%((r)R) 2 The space derivative of the vector potential that is needed is found by evaluating the curl of the vector potential by differ- entiating Eq. (8): (10) Partial and total derivatives of the vector potential are also needed. Differentiating Eq. (8) it is found that I cW 11 IO'r~ ~ 1 lOW ] (-~t)xyz =KQ L-((r)R) 2 , - t "---t )xyz)+(r)----Rt O'---T )xyz and (lla) dt ((r)R) 2 d-7 (r)R d'---T " 4. GENERAL FIELDS Substitution of Eq. (10) into Eq. expression for the magnetic field: (2) gives a general B- m KQ W x(grad(r).) ,+ KQ [ OW ] X (grad(r),),. ((r)R) 2 ~ _ -~-- - ' (12) For all possible choices of W and (r)R, there is an inverse square field that is proportional to W and also an inverse first power field if W changes with time. Depending on whether we employ the Neumann or the Hertzian definitions of the electric field, we have by substitution of Eqs. (9) and (11) into Eq. (3), E N = Q "(grad (r)R) t ((r)R) 2 47re0 E H = Q - (g rad ( r )R) t ((r)R) 2 47re0 _ KQ dW (13b) -3T " To find the general expressions for the force per unit charge F, substitute Eqs. (9), (10), and (11) into Eq. (1). Then for the Neumann definition 29 The Electrodynamics of Gauss, Neumann, and Hertz o r ra r'R t cWI1--tO } } ((r)R) 2 L 47re 0 --~ x,z + Kw x (W x (grad (r)R),) l 1] (14a) while for the Hertzian definition FH = ((r)R)2Q [(grad47r%(r)R)t + KcW(1 - -d-Tdr) + Kw x (W X (grad (r),),)] +KQ(r)R [ _ dW+w x d t c ((-~-rW) x (g radt (r)g)t)] " (14b) Both force fields contain an inverse square term and an inverse first power term. The expressions depend on arbitrary choices of grad (r)R, Or/Ot and dr/dt, and W. 5. POSTULATES ON THE VELOCITY OF L IGHT The two postulates on the velocity of light that have survived the binary star test (~6~ in Euclidean space and explain the Michelson-Gale experiment (~7~ are Postulates I* and III*. Postulate I*: The velocity of light in free space is a constant c irrespective of the velocity of source or receiver in any coordinate system which is not in rotation. This is equivalent to Einstein's 1907 postulate.