University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-1967 Conductive Sphere on a Charged Conductive Plane Birney Robert Fish University of Tennessee, Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Physics Commons Recommended Citation Fish, Birney Robert, "Conductive Sphere on a Charged Conductive Plane. " Master's Thesis, University of Tennessee, 1967. https://trace.tennessee.edu/utk_gradthes/3705 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Birney Robert Fish entitled "Conductive Sphere on a Charged Conductive Plane." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Physics. Robert J. Lovell, Major Professor We have read this thesis and recommend its acceptance: R. H. Ritchie, Isabel H. Tipton Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Augutt 4, 1967 To the Graduate Counelh I am submitting herewith a thesfs written by Birney Robert Fish entitled "Conductive Sphere on a Charged Conductive Plane." I recommend that ft be accepted for nine quarter houn of credit In partial fulfillment of the requirements for the degree of Master of Scfence, with a maJor In PhysiCJ. We have read thIs thesis and recommend its acceptance: rZtl.� Acoepted for the Council: Graduate Studies and Research CONDUCTIVE SPHERE ON A CHAR GED CONDUCTIVE PLANE A Thesis Presented to the Graduate Council of The University of Tennessee In Partial Fulfillment of the Requirements for the Degree Master of Science by Birney Robert Fish August 1967 ACKNOWLEDGMENTS The experimentalwork described in this study was based on ideas developed by Mr. F. G. Karioris, Associate Professorof Physics, Marquette University. Substantially all of the laboratorywork was done by Mr. Karioris with the assistanceof /v\r. J. L. Thompson, Health Physics Technology Section, Health Physics Division, Oak Ridge National Laboratort, and with only minor assistance by the author. Their aid has been indispensible in this effort and is gratefullyac knowledged. The author is indebted to Miss Blanche Reed, Health Physics Technology Section, for her excellent work in preparing this manuscript under a most demanding schedule. Thanks alsoar e due to the author•s graduate committee, Dr. R. J. Lovell, Dr. Isabel H. Tipton, and Dr. R. H. Ritchie, forth eir thoughtful review of the material presented in this ·reportand forth eir many va luable suggestions. * Operated by Union Carbide Corporation for the United States Atomic Energy Commission. ii 73061.0 TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION ... Statement of the Problem . ....... Scientific Justification .........•............•.•. Technological Justification . 2 Plan of Attack . .. 4 Units . .. 5 II. ESTIMATION OF CHARGE, FORCE AND FIELD ..•......... 6 Estimate of Charge . • . • . • . 6 Estimate of Normal Force • . • . 9 Estimate of Field Strength . • . 13 Summary of Estimates • . • . • . • . 14 Ill. IMAGE CHARGE APPROACH ...................... 17 Background . 17 Basic Principles . • . • . • 18 Application to. Sphere on a Plane ...... ·. • . 22 IV. LAPLACE EQUATION IN BISPHERICAL COORDINATES.. .... The Laplace Equation . '19 Bispherica l Coordinates . 31 Solution for the Field . • . 39 A . • . 41 symptotic Solution forthe Fo rce . iii CHAPTER PAGE V. EXPERIMENTAL STUDIES . • 45 Introduction . 45 Charging of a Sphere on a Plane . 45 Coulometric Ana log Study of Surface Charge Density . 54 VI. SUMMARY AND.CONC LUSION . 65 Summary . _. 65 Discussion . 65 Conclusion . .. 66 BIBL IOGRAPHY. • . 68 iv L 1ST OF FIGURES FIGURE PAGE I. Two Contacting Spheres Showing the Electrical Images in the System of Planes Formed as the Electrical Inversion of the Spheres aboutthe ir Point of Contact. 7 2. Image Charge in a Plane under the Influence of a Po int Charge Outside the Plane . 19 3. Image Charge in a Spherica l Conductor under the Influence of a Po int Charge Outside the Sphere . 21 4. Three-D imensional Image Charge Distributions in the Plane and in the Sphere . • . 25 5. Resu lts of Graphic Solution fo r the Surface Charge Distribution on the First and Second Three-Dimensional Images in the Sphere, as Compared with the Starting Distribution of Charge . 27 6. Sphere and Plane in the Bisphericol Coordinate System . 32 7. The Problem of Two Charged Spheres of Equal Radii con be Related to the Problem of a Conductive Sphere on a Charged Plane . 42 8. Experimenta l Setup fo r Measurement of Charge Acquired by Sphere Contacting a Charged Plane . 47 v FIGURE PA GE 9. Strobe Photograph of 50 Micron Diameter Gold Sphere Oscillating (vertically in silicone oil) between Gold Electrodes held at Constant Potential . • • . 48 10. Diagram of Charged Capacitor used to Determine Velocity of a Go ld Sphere as a Function of Applied Field • . • . • . • 50 II. Charge Acquired by a 50 Micron Diameter Gold Sphere Contacting a Charged Go ld Electrode • . • . • . 53 12. Coordinates fo r a Hemisphere on a Plane . 59 13. Segments of Copper Plated on a Hemisphere in Contact with a Plane Electrode . 61 14. Distribution of Electrodeposited Copper on a Hemisphere in Contact with a Plane Cathode in a Plating Ta nk . 62 15. Distribution of Electrodeposited Copper on a Plane in the Vicinity of a Hemisphere ..................... 63 16. Distribution of Copper Deposited on a Sphere in Contact with a Plane Cathode in a Plating Tank . 64 vi ABSTRACT The electrostatic problem of a conductive sphere on a charged conductive plane has been solved using theoretical and experimenta l methods. From elementary electrostatics it is seen that there will be a uniform electric field 2 E0 vo lts/meter normal to a semi -infinite plane bearing <70cou lombs/meter • Based on an image charge approach, it was determined that a sphere of radius 3 2 R meters will acquire a tota l equ ilibrium charge q = � 'IT R a0 coulombs at contact with the plane, and the uniform field will be perturbed, resulting in a field strength E = 4.5 E0 vo lts/meter at the top of the sphere. An approxi- motion, valid to within less than about I per cent, for the repulsive force acting on the sphere was obtained by extrapolating to contact, series solutions of the Laplace equation for finite separation between sphere and plane: the -10 2 forceat contact is estimated to be F = 1.537 x lo kR E newtons; where z � . k is the dielectric constant of the surrounding medium. The charge distribution on the sphere was determined experimenta lly by plating copper on a sphere - plane cathode and, within about 10 per cent, is given by the empirical expression 2 4 a(9) = (0.8696 cos ;+ 3.6304 cos �) <70; where 9 is the angle of co latitude in spherical polar coordinates referred to the axis of contact and concentric with the sphere. vii CHAPTER I INTRODUCTION I. STATEMENT OF THE PROBLEM If a conductive sphere is placed in contact with a semi -infinite, conductive plane, the configuration constitutes a three�imensional equipotential surface. Assuming an isolated plane to be electrically charged to a known surface charge density (a0) originally, it is desired to determine the effect produced by placing a sphere on the plane. Specifically, the purpose of th is investigation is to deter­ mine the forceexerted on the sphere norma I to the plane, the to ta l charge acquired by the sphere and the perturbation in the electrica l fie ld due to its presence on the plane. II. SCIENTIFIC JUSTIFICAT ION Whereas there are classical so lutions which are applicable to the problems of a th in disk or af a hemisphere on a plane [e.g., Jeans (1941) ], no such so lution exists fo r the sphere on a plane. Historica lly, the solutions to electrostatic problems involving three-dimensiona l bodies, such as twofinit e spheres or a sphere and a plane, have been approached by the method of images described by Maxwell(l892). We ber(l965), says, fo r example, 11The electrostatic field of twofinite conducting spheres can be described only by an infinite sequence of images. 11 This restriction is especia lly damaging in that the infinite series that result from the method of 2 images are.very unwieldy and have been made to converge in only a few comparatively simple cases. In a recent paper, Taylor (1966) , pointed out that, despite the fa ct the first observations th at I ight objects lying on a su rface will jump to a piece of rubbed amber were recorded about2500 years ago, there rema in areas of work on electro­ statics not yet completely so lved. He went on to indicate that the distribution of charge on axisymmetric bodies is known fo r only a few cases: Taylor (1964) , had earlier pub ished a derivation for the surface charge density over a spheroid of I _ finite eccentricity. Ill. TECHNOLOGICAL JUSTIFICATION Beyond the esthetic challenge presented by the electro static problem of a perfect conducting sphere on a charged sem i-infinite plane, there are many areas of technology in which a so lution to this problem would be, at least, usefu l and, in some cases, vital. The· various mechanisms of adhesion of particles to surfaces are impbrtant in understanding the behavior of airborne particles after they come in contact with a so lid surface.