A GENERAL FORM FOR THE ELECTRIC FIELD LINES EQUATION CONCERNING AN AXIALLY SYMMETRIC CONTINUOUS CHARGE DISTRIBUTION

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1 A GENEAL FOM FO THE ELECTIC FIELD LINES EQUATION CONCENING AN AXIALLY SYMMETIC CONTINUOUS CHAGE DISTIBUTION BY MUGU B. ăuţ Abstract..By using an unexpected approach it results a general for for the electric field lines equation. It is a general forula, a derivative-integral equation structured as a ulti-pole expansion series. By solving this equation we can find the electric field lines expressions for any type of an axially syetric ultipole continuous electric charge distributions we interested in, without the need to take again the calculus fro the beginning for each case particularly, for instance as in discrete charge distribution case. Key words: electric field lines equation; ulti-pole expansion series; axially syetric continuous electric charge distribution. 1. Introduction Fro an axially syetric agnetic ulti-pole of arbitrary degree n, (Jackson, 1975), we can derive the exact equation for the field lines, (Jeffreys, 1988). The ethod presented in (Jeffreys, 1988) deals with spherical haronics in the ost general way. Consequently the equation for the field lines is the expression of a general case. Another two exact equations for the field lines are given in (Willis & Gardiner, 1988). The equations are for two special agnetic ulti-poles of arbitrary degree with no axial syetry. These cases ay be classified as either syetric or anti-syetric sectorial ulti-poles. By using the above considerations the ai of this paper is to find a general for for an exact equation for the field lines of an electric ulti-pole with axial syetry.. Theory Let s consider now a continuous electrostatic charge distribution within a spatial volue. We ust evaluate the electric potential in a point P outside the distribution, as we can see in figure below:

2 z Charge eleent d r θ y x V Fig.1 The electric field lines equation is the well known expression: (1) E dl 0 By assuing that we have a charge distribution with an axial syetry with respect to z axis, we can explicit the length eleent and the electric field as: () dl d u d u and: V 1 V (3) E V u u The cross product (1) leads after an eleentary calculus to the well known field lines equation written in polar coordinates: d V V (4) d 0 For a continuous charge distribution the electric potential V can be expanded as a Legendre series, according to (Eyges, 1980): V P rr, cos Consequently the potential derivatives fro equation (3) can be written as: V P cos rr and: V P cos rr By introducing these results within equation (3) and considering the property: 3 3 P cos rr P cos rr

3 the electric field lines equation can be expressed as: d 1 1 (5) P cos cos 0 1 d P 0 0 This is a general expression for the electric field lines equation under continuous charge distribution hypothesis. At first sight it exhibits a coplicate for which requires for solving a derivative-integral equation ethod. Despite this appearance the solutions can be obtained in a siple and direct anner, as its show in the following exaples. It is useful for our calculations to consider the odrigues representation of Legendre polynoials: 1 d (6) P cos cos 1! d cos Under these circustances equation (5) becae ore explicit and siple. The derivative with respect to θ of expression (6): P cos d 1 (7) cos 1 cos sin! d cos leads to an iportant observation that we can ake the derivatives with respect to cosine before we ake the integration, and thus the equation (5) becae only an integral equation, ore sipler to solve. It is obvious that the case =0 doesn t exist because the derivatives (7) don t exist. More interesting is the dipole case: 1 By taking into account the expressions (6) and (7), the equation (5) can be written as: d 1 1 d 0 cos 1 cos sin d cos 1 d d cos d cos After trivial siplification and obvious derivatives we obtain the equation: d sin cos d 0 which can be directly integrated as: (8) Csin and it is the well-known expression, in polar coordinates, of the field lines for an electric dipole. The atheatical treatent of the case is the sae as the previous case. We obtain the equation:

4 d 1 d 1 [(cos 1) cos sin ] 3 d cos 3 1 d d (cos 1) 0 4 d cos fro which is deduced the ost siplest for: d 3cos 1 (9) d sin cos Finally, after integrating equation (9), we are obtaining the following relation: (10) k sin cos which is the well-known expression of the field lines for an electric 4-pole. Equation (5) is the direct consequence of the equation (3). If the electric field couldn t be an expression of a scalar potential, then all the above atheatical stateent has no basis. The agnetic analog for V doesn t support sources. Subsequently the agnetic analog for equation (3) can be written only with the vector potential A. The vector potential is defined in ters of current density. Under axial syetry and continuous distribution of current density hypothesis, A can also be expanded in Legendre series. But copared with the electric field this is the only siilarity. The agnetic field lines equation appears in a double cross-product for. The solutions of this equation are ore coplicate than equation (5), (see (Jeffreys, 1988)). 3. Conclusions The ai of this paper is to deduce a new for for the electric field lines equation. We obtain a general forula, a derivative-integral equation structured as a ulti-pole expansion series. The equation has exact solutions corresponding to an axially syetric electric ulti-pole continuous charge distribution, without the need to consider special assuptions for 0. Equation (5) can be the starting point of the entire section., because is valid in entioned approxiations, without the need to deduce it fro equation (1) for each case fro the beginning, for instance as in discrete charge distribution case. EFEENCES Eyges L., The Classical Electroagnetic Field, Addison-Wesley, Mass. 197, reprinted by Dover(1980). Jackson J. D., Classical Electrodynaics, Wiley, New York, 137, (1975). Jeffreys B., Derivation of the equation for the field lines of an axis syetric ultipole, Geophy. J. International, 9(), (1988).

5 Willis D. M. and Gardiner A.., Equations for the field lines of a sectorial ulti-pole, Geophy. J. International, 95(3), (1988).