are x. r at .0 t... .‘ u. .3 . . r... . "2w ,. A\ Q .1 ~\ 0 3 H. r.\.u w“ . .3“ 0“... .0 .v. - . . 5;. w.” ,.. w...” an 5.3 ”W ”Lu ....2.\ via {L3 8.... J «.92.... Pan? n: .32: O W -l S T“: t b 2 . ._ x .. r. .5... .l‘ \ ‘hu ‘ .. . no... u I 5 fig \f.~ “.2 a v o m GOJV 0 ~— 5-} 5 s 34 fl \ 6" q _ A A... l... a o a»: ' \z: {z a u (5 “'- \ : ”nu—o ". J -.. e. .WW . .. f. .3 .. . C G .._..... ,. . ax“ a.a m.. .2” . 3d. It». . “1.2 h o . a: .a [\‘k .c i - I., O 0-& u 1.8 (K: K. 7.: (A. g! j :____2:13:31_::__:_____:,__ mmm IIIIIIIIIIIIIIIIIIIIIIIIIIIIII 01687 5878 THEORY OF QUASI-STATIC FLOW OF CHARGE ON SLIGHTLX CONDUCTING BODIES by Tin Oo Hlaing AN ABSTRACT Submitted to the College of Science and Arts Michigan State University of Agriculture and Applied Science in partial fullfilment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1959 ABSTRACT This paper describes the motion of electric charge when placed on the surfaces of very slightly conducting bodies. It is assumed that no electric charge is initially present inside any conductor which is under consideration. For this reason it follows, from a wellknown case, that there will never be any charge observed inside the con— ductor at any later time. A general method is developed for finding the charge density at any time on the surface of a conducting body in terms of the initial charge den- sity on its surface. This general method is used to find the charge density at any time for the following cases involving specific geometries and initial charge distri- butions: (1) A slightly conducting cylinder with arbitrary initial two-dimensional surface charge density. (2) Two slightly conducting coaxial cylinders with arbitrary initial two-dimensional surface charge density. (3) A single cylinder conducting on its surface only, with arbitrary initial two-dimensional surface charge density. (4) A similar single cylinder with a line charge on its airface initially. (5) A slightly conducting sphere with arbitrary initial two-dimensional surface charge density. (6) A slightly conducting Sphere with a poxnt charge on its s.rface initially. THIDRX OF QHIBI-BTAIIC FLOW OF CHARGE ON SLIGHTLX CONDUCTING BODIES by Tin Oo Hlaing A THESIS Submitted to the College of Science and Arts Michigan State University of Agriculture and Applied Science in partial fullfilment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1959 .4 -,7.,lf5‘1\ (7 g 3/0 ACKNOIIEDGIENT The author wishes to express his'sincere thanks to Dr. Iilliam G. Hammerle for the guidance and encouragement received from him throughout the course of this work. Thanks are also due to the members of the examination committee: Dr. Sherwood K. Haynes, Dr. Donald J. lontgoaery and Dr. Richard Schlegel of the Physics Department; and Dr. John G. Hocking of the Mathematics Department. II III IV VII TABLE 0! CONTENTS ImmUCTIONOCOOOOOOOOOO.0......OOOOOOOOOOOOO Gm TEORYOOOOIOOOOOOOOOOOOOOOOOOOOOOOOOO T'O-DIIENSIONAL CHARGE DISTRIBUTION ON A cwDUCTI'G CWEOOOOOOOOOOOOOOOCOO0.0.0... TWO-DHBNSIONAL CHARGE DISTRIBUTION ON TWO CONDUCTING COAXIAL CYLINDERS............. T'O-DI'IENSIONAL CHARGE DISTRIBUTIW ON A SINGLE CYLINDER WITH SURFACE CONDUCTIVITY... . A. Arbitrary Charge Distribution on a single cylinderOOOOOOOOOOOOOOOOOOOOOOOOOOO B. Line Charge on.a Single Cylinder.......... TWO-DUBNSIONAI. CHARGE DISTRIBUTION ON A CONDUCTING SWOOOOOOOOOOO0.000000000000000 A. Arbitrary Charge Distribution on a Sphere. B. Point Charge on a Sphere.................. WY...OOOOOOOOOOOOOOOOOOOO.OOOOOOOOOOOOOO APPRNDIXES APPENDIX A................................... APPENDIX B................................... APPENDIX C................................... APPENDIX D................................... APPENDIX.E................................... APPENDIX P................................... 12 21 21 24 3O 30 33 39 42 45 46 47 49 50 I INTRODUCT ION If a substance is rubbed against another substance,it generally becomes either positively or negatively charged. Among others, Hersh, Sherman and Montgomery1 have measured the amount of charge produced when a filament of a given material is rubbed against another of the same or different material under controlled mechanical and ambient conditions. They used metals as well as non-metals. Thus, electric charge may be generated on the surface of very slightly conducting non-metallic substance. Once the charge is generated, it is interesting to know how it will move from its initial position into the final equi- librium distribution which will be reached after a very long time. It is discussed in this paper how the electric charge initially placed on the surfaces of slightly con- ducting bodies will behave as time progresses. In Section II, it is shown that there will never be any charge inside a conductor, if no charge is initially present inside it. Although the charge may flow through the interior of the body from one point of the surface to another, measurable amounts of charge are found only on the surface. In the same section, the general method for finding the surface charge density at any time is developed, provided that the initial surface charge density is known. 1. Hershfé'mnsmmanfim. and Montgomery,D.J., "Textile Research Journal, 2&, 426 (1954) In Section III, an arbitrary two-dimensional distri- bution of charge is placed on a slightly conducting cylin- der. The expression for the surface charge density at any time is calculated, and is expressed in closed form. In Section IV, an arbitrary two-dimensional distri- bution of charge is placed on the outer surface of two conducting coaxial cylinders. Two general expressions for the surface charge density at a later time are calculated, one for the surface of the outer cylinder, and the other for that of the inner one. In Section V A., an arbitrary two-dimensional distri- bution of charge is placed on the surface of a cylinder which is conducting only within a very thin layer on the surface. This may be considered as a special case of the one discussed in Section IV. The expressions for the sur- face charge density at any time are calculated. In Section V B., the same cylinder is taken but the initial charge placed on the surface is a line charge. The expression for total surface charge density at any time is calculated and expressed in closed form. In Section VI A., an arbitrary two-dimensional distri- bution of charge is placed on the surface of a slightly conducting sphere. An infinite series for the surface charge density is obtained. In Section VI B., the same sphere is taken but the initial charge placed on its sur- face is a point charge. The expression for the surface charge density at a later time is presented in a form suitable for numerical computations. II GENERAL THEORY Electrodynamic problems can be solved using Maxwell's equations. They are a set of differential equations which are written in the rationalized m.k.s. system of units as follows: a?" “33 ' (21) _. at .. ' e. H . 3 +31) .. 31: (2.2) v3 = 0 _. .. (2.3) v.3 - f (2.4) If the medium is isotropic and homogeneous, KE ‘ -/‘*H 6‘ (2.5) (2.6) Cu out A E. (2.7) From the conservation of charge, the relationship between jandfig A A VJ +§f = O (2.8) g 31: where = electric field intensity; 3 = magnetic induc- .8 tion; H = magnetic field intensityz 3 = current density; 5 2: electric displacement; v? volume charge density; K = dielectric constant; A = magnetic inductive capaci- ty; 5' = electric conductivity. 2. Pipes, Louis A.: ”Applied Mathematics for Engineers and Physicists,” McGraw-Hill Book Company,Inc.,New fork, 1946.1). 364. 3.8tratton, Julius A.: “Electromagnetic Theory,” HcGraw- Hill Book Company,Inc.,New York,19M.P. 5. 4 In all the cases considered in this paper, known elec- tric charge is initially placed on various conducting bodies. If 6‘ , the electric conductivity, is very small, the charge flows very slowly from place to place. .We may, at any instant, consider that the terms involving partial deriva- tives with respect to time vanish in the equations, and also say that 3 , the current density, is Very small. Then, as a zeroth order approximation, ‘ j” o (2.9) O 95;,” (2.10) d A and D and E are approximately independent of time. By using (2.1) and considering the fact that.E is in- dependent of time, the following relations are obtained: B” O (2.11) “ ‘3 O (2.12) thus the magnetic field is negligible, and _' . Thus an electric potential V exists such that E 8 “av (2.14) P where Vzv -.- - T}; (2.15) To this approximation, we have a purely electrostatic problem, that is, the electric potential and field at any instant are the same as would exist were the charge distri- bution at that same instant not changing with time. As the next order of approximation, this electric 5 potential and field may be put back into Ohm's law (2.7) to find a nan-zero current density: 3 6.? a ~C6V. (2.16) Then gag , the rate of change of charge density with time, is obtained from the expression for the conservation of charge (2 8g: 3 '6? V1 e“ 513- . s V . (2.17) But by (2.15). this gives % .-6:—f' (2e18) 3 K The above differential equation (2.18) can be solved t immediately to give ..1% 0 g {R 3c (2.19) where '1- ' g and is known as the tine of relaxation for the conducting material and {’(o) is the value of ‘3 at tgo, that is, it is the original charge density at the same point in space. Two cases can be considered: . (a) If 29(0)“), the original charge at every point of the conducting material decays exponentially. It is clearly seen that the time of relaxation is independent of the size and the shape of the conductor. For example, the conductor may be either spherical or cylindrical in shape and the radius may be either large or small. (b) If ?(O)c.o, then f g 0 , that is, if no charge is present inside a conductor initially, there will never be any charge in the interior at a later time. I.See reference 3.9.15. 6 In all the problems that are considered in this paper, the original charge distribution is confined to the sur- faces of the conducting objects, and thus only case (b) is applicable. The charge density on the surface of the conducting material is defined as charge per unit area and is denoted by (ed) at any unot +0, andOJLO) at t=0. If bad) is known, the potential \/ can be obtained by solving Laplace's equation subject to the proper conditions at the boundary between the various dielectrics. Laplace's equation is written as 6V 8 0 ‘ (2.20) At the boundary, the boundary conditions or? V, a V; (2.21) and (D1,). 49“);- (ACE) (2.22) where V, and V; are the potentials in the two mediums, (1),). anathnlare the normal coaponents cf the electric displace- ments in the two mediums. A ‘When \I is known,\J can be obtained by using (2.14) and (2.7). .r By charge conservation, the relationship between.I and no is such thag " .a .4 3,... -3,“ . 332d) (.1...) d 4 where ‘1 is the normal, 3, and J: are the current densities in the two mediums evaluated at the surface(see Fig.1). 3.5 reference 3.13.164. 6.3ee reference 3,p.483. O” 1. Fig. 1. If 00(i) is known, equation (2.20) can be solved subject to the boundary conditions (2.21) and (2.22), to give V in terms of (0d) . But V in turn gives 3 by equations (2.14) and (2.7). Equation (2.23) then gives .333 , the time rate of change of charge density, in terms of J ,and thus in terms of “((1) . We may con- clude, therefore, that the above set of equations is equivalent to a first order partial differential equation for the charge distribution and can be solved for 03(t) if LOCO) is known. Probably this solution for God.) is satisfactory 1f the relaxation time T in (2.19) is much larger than the time required for light to cross the con- ducting objects. It is not possible to solve these equations in general. Instead, they will be solved for ODCt) in a number of examples involving specific geometries and initial charge distributions. III TWO-DWI“ GiARGI DISTRIBUTION ON A CONDUCTII} W In this section, the investigation is done on an yinfinitely long dielectric cy- linder (Fig.2) whose radius is (K, . Throughout the cylinder, the dielectric constant is K'L and the conductivity is g" , ' where 61 is very small. ' Outside the cylinder, the di- elecgic constant is K0 and the conductivity is zero. The Fig.2. radial'distance from the axis of the cylinder to any point is denoted by l‘ . 9 is the angle between T‘ and the x axis. The surface charge density to is considered to be arbitrary at time t =' 0, except that it is independent of Z and is a function of I 0 only. We will, furthermore, restrict unto be symmetric about the x axis, so that (no) . (ad-0). (3.1) To find the surface charge densitytodkt a later time {*n let us expand it in the orm LOG.) .{ P9) Cosme (3.2) where ESL)“ given by “‘0 Rm- : (3.3) ' A‘. SowC‘l)‘. 11.0 The potentials about the surface - either outside or inside of the. cylinder - satisfy Laplaoe's‘ equation (2.20). Therefore VL , the potential inside the cylinder, is obtained by solving the Laplace's equation, using the boundary conditions (2.21) and (2.22). It can be written a3 cc 'V° 1n “CosmG F¢h+ Co L - ' 71R“-.(K(*KD" ‘ (3.4) ‘1') where Co is an arbitrary constant. it I1: K , by using (2. 23), (2.14) and (2.7) from the previous chapter, we have 103$) .- “0%) at r,& (3.5) = _ 5‘; EKG.) Coshfi ( 6 “a. Ki. + K. N . 3° ) 0n differentiating both sides of (3.2) partially with respect to t, we have 3—95)) aziP-m C°“‘ 9 (3.7) For (3.7) to be equal0 to (3.6), it is necessary that - . (i) — a 43’— ‘0" {Lin )= K. “‘0 (3.8) all: 0 “30 m solving (3.8) for Est), we have 9.5.0) e’t’t - 93'), W‘” ‘ (3.9) (O) I ‘h a. ) 7.3:; Appendix A. P. 10 1': (K1*Ke)/5‘ where ' (3.10) After substituting the value of £3)er (3.9) in (3.2), the magnitude of charge per area of the surface at any time is J; wc*)=z?(°300w1ee "5 + ECO). (3.11) '11-( But the sum can be evaluated in closed form because the exponential is the same for each term in the sum, and aficiasne . 03(0) ._ B0,). (3.12) Thus (3.11)‘ can now be written as -1: .t “(£) 3 WCO)€ m+ ECO (hu- 2 ’1], (3.13) where (.305) is the surface charge density at any time t and 00(0) is the arbitrary charge density on the surface at t=o. Equation (3.13) can be interpreted in the following way: it t = o, tact) . ”(0) , because étéis unity. is time progresses, the initial distributionw(o)decreases to (zero exponentially, with a time constant or time of relaxation "C given by (3.10). In its place arises a charge distri- bution F003)“ - e't‘t] which is independent of 6 and everywhere. the sane on the cylinder. After an infinite tine, the charge distribution is everywhere given by 8(a), the average value of the charge density initially placed on the surface. from (3.3), it can be expressed as Pco) =[m(°)]¢:1J-S:m(°)d° (3. 14) This result (3. 13), incidentty, is correct whether or not the symmetry condition (3.1) - via 00(9) 3 w(—O)- is 11 satisfied. The relaxation time Tie given by(k[ 44(0)“: . It is independent ofR, the radius of the cylinder, and depends only on Kt' K0 and €1- . Thus, the relaxation time is the same for all cylinders, large or small, made from the same material. IV TWO—DIMENSIONAL CHARGE DISTRIBUTION ON TWO CONDUCTING COAXIAL CYLINDERS In this section, the Z investigation is done on two infinitely long conducting KMO’. st3 coaxial cylinders (Fig. 3). K The radius of the inner V201 R. :. cylinder is RV and that of y the outer cylinder is K1. The 4 \9/ >~ conductivities are 6'; for the inner cylinder, 6‘1 for the ”3 outer cylinder, and zero outside the cylinders. The meanings of Fig.3. 9 (and Y’ are same as in . Section III. The problem in this section is to find the surface charge density at a later time, if the charge density at t=0 is given, Initially, the charge distributions on both the surfaces are arbitrary except that they will be assum- ed independent of Z and a function of 6 only. As before, let us expand the charge density at t=0 on the surface of the inner cylinder in the form Q I w’co) . e2.) cane (4.1) 1a where a“) is expressedoas , 3.31.2.) comm 1),. . 3J0.) a. O . (4.2) '5 *gdwEAO Mao 13 The charge density at t=0 on the surface of the outer cylinder can be expanded in the form oo'(o') 320930) f°5m0 (4.3) z... (2.2.. a... a. m. {5.30) = on . (4.4) . VT: ). “"0 “L9 1 ”L 0 At time'): :f o , the charge densities can be expressed as similar series: (59.)- -Z?—. (L) cos-he “,0? (4.5) a V w' (L) = E P ct) Cosme (4.6) 11:0 The only case considered here is with.£0GDidentically zero and therefore from (4.2), EEO): 0 for all n. It is impractical to put electric charge on the surface of the inner cylinder at time t=0 if the cylinders are solid. A general solution of the Laplace's equation in cylin- 8 drical co-ordinates “is V .._._ alnr+ :T‘Ya 005M9+ 1. Say. 1.9) "m (4.7) +21%“, Cosva .1} SMMO) C where \/ isvxhg potential. ' V = V (T39) (4.8) because it is independent of 2. Let us denote the potentials as V‘ for the inner cylin- der, V1 for the outer cylinder, and V3 for the medium outside the cylinders. §.See reference 2,p.407. 14 They are of the form v=“AW C ‘ 1;.“ 'V\ COsfie‘ + O (4.9) m a .‘Y‘ ‘ V1 =28f‘cosme + 'DMT‘Cosnga-EO +FOLMY‘ (4.10) mag! Mt" _ V3 3* :Gfi“c““9 +1401“? (4.11) where A:*,' a“. CO . D,“ . E0 . $0 , q“ and “O are constants whose values will be determined later. The terms contain- ing gm 9 do not appear in any equation because the poten- tial is symmetric about the X axis. The terms containing Tr“ do not appear in (4.9) because the potential is finite at 7‘: 0 . The terms containing Yr do not appear in (4.11) because potential cannot go to infinity as T‘ -) 00 more rapidly than lnT‘ . In (4.10), both 7:“ and Y‘“" can appear because 7‘ = O and T‘ a on are excluded from the re- sion in which V1 is applicable. There are two boundary conditions (2.21) and (2.22) to be satisfied at each boundary. AtT‘-K‘, V\ = V). (4.12) and 3V1) , 3V1 .- t», (‘t (4.13) _K1(5; “Ring? 12R, ) Similarly at T‘ = R1 , V], =-. v3 (4.14) and II AV. a mi) 1 K1('§—T‘)T=Q,_ (4'15) The values of V\ and Vlfrom (4.9) and (4.10) are substitu- " “(gs—Elm + ted into (4.12) and (4.13). Expressions (4.12) and (4.13) 15 are evaluated at T‘ = R‘ . The values of V1 and V3 from (4.10) and (4.11) are substituted into (4.14) and (4.15). Expressions (4.14) and (4.15) are evaluated at r a R, Thus the following four equations are obtained from (4.12),(4.13),(4.14) and (4.15) together with (4.5) and (4,6) . EAhfi: C0!» 118 + Co a: 28." RT Cos 716 +212" QTCos 116 M: “31 1‘12. +EO + Fin R, (4.16) “’K (23“ an R‘Tc“ 7‘9 +2210“) R‘mCOS V19 4E9 2' 111 K1 + K IiiA'nRT Cos Nae-1120?“) Cos 'ne (4'17) ZBR‘: Cosme +2133: cos-we .15.”. “R. ‘13) “‘1: - (4. 8) == “Ea-“R,“Gos‘ne + “chat 1 -K AZQg'DRr-J cos-m + 1), K‘VA \<( 8146-603 110 1- ‘“a‘ “"11 ZD('“)R:\ 603 v.9 +fé>i1 = Z P” 16-3005 716 (4.19) On.“ equating the coefficients.“ of 1‘6 C05 “9 in the above four equations, we have A R1115 QithZM= o (4.20) k (Mn TA-vK'DyR. —KMB R .._P,t') (4.40) / / ck _ -G i (if; ) ' 151%) . (4.41) 'l I AR“) = 53 31- Rfi) ‘ (4 2+2) a? R1 R2 ' where 0‘..[n-RI2-'(2n- -ZIK)(6“C)+R:R;313(K+K3(6*G)]/l4_(443) "‘0‘ OH]: «[§-1R.: (1K1 6 — lKgyl/Ldeno‘ (4.44) Wkd(K-K13@a)-R?R2W\ 3 - (“H + ‘11) " Ed“ 'JX‘») + “‘11!” (4 53) " TL u ' From equation (4.49). the value of 1‘ is J;=-(.u—>)J’. x”- (4 54) Because there are two values for )‘ (4. 52) and (4. 53). the general solutions (of the differential equations) are as follows: I ->t ->t PM"j’-(le“f":-i°z (4.55) at +>~ ~7‘t -H; ‘ a: —(_u__. ch “’ GEL—”>0“ (4. 56) “n. where C‘ and :1 are constants whose values are obtained ’ II at from the requirement that, at t=0, anaemia“ . Eng) . After evaluating 0‘ and C1 , (4.55) and (4.56) become fad) 9550“) j‘>u)(€::' PM) (4.57) p (4): [5”..co>[ (ELM)? +(d\_.___ Hz) and] (4. 58) On solving (4.141) for N .P/(i), it is found that .t P?” ‘ F’s/”me G k . o (4.59) 20 / because RC0) = 0 ” Also, on solving (4.42) for PECE),it is found that POW») = Po'co) (4.60) On substituting (4.57) and (4.59) into (4.5), and (4.58) and (4.60) into (4.6), we have w’=c{) :§(0)(;———:'_‘;' (e‘m’ie "at" °)(1as‘n6 (4. 61) 'E t at. N g. )1 '9‘ afl-n‘4. 62) (SM) 44”}?ch + )e 4;: )e ] ‘th 1 These equations (4.61) and (4.62) complete the general solution for with) and OO'C‘L) , the charge densities at a later time on the inner and outer surfaces respectively. For a particular known (3(0) , the initial charge distri- bution on the surface, Fu26)can be found from (4.4). Then 0‘1” , 04” , (’3‘ and in can be found from (4.43),(4.44), (4.45) and (4.46) respectively..'>‘l and )1 are obtained from (4.52) and (4.53). These results are then put into (4.61) and (4.62) to give the charge distribution at any later time t.. The series involved in (4.61) and (4.62) cannot be summed because the relaxation times of the terms in the series are not all the same as they depend on n. No general conclusions are drawn from these expressions in this section. In Section V, however, we will solve a more specific problem for the surface charge density using the expressions obtained in this section. V'TWO-DIMENSIONAL CHMRGE DISTRIBUTION ON A SINGLE CYLINDER WITH SURFACE CONDUCTIVITI 0 a- 1:. e ' : 91- 09 '2 : ‘9'. ' 4-dr The case considered in this'section is that of an infinitely long cylinder which is not conducting in its interior, but only conducting slightly in a very thin layer on its surface. The expressions obtained in the previous section can be used to solve this problem direct- ly. The only changes necessary are the following: (a) It is assumed that the outer cylinder of the two conducting coaxial cylinders, discussed in the previous section, is very thin, such that RI ___,_ R20 .. ‘0) (5.1) where R‘ is the radius of the inner cylinder, R1 is the radius of the outer cylinder, and ‘6 is very small($<<|). ’ Then all the expressions involving 6 can be expanded in powers of ‘3 , and the [)1 and higher terms can be ne- glected. For example I = '+6+l32+"'#|+5 (5.2) «-5 ‘ Also, la; 4- 55 2:. RIM] ,4; ED +33] (5.3) 0.455 ‘ (QAMXC-fia) c 4 4‘3 (11-6%" ’4‘? 070’ + (b/c - “3%)6 (5.4) (b) It is assumed that the dielectric constants of the two cylinders have the same value and the inner cylinder is not conducting. 22 (Thus _ when the equations of the previous section are rewritten after substituting (5.1),(5.5) and (5.5a) in appropriate places, we have: From (4.43) _ -2 .. - g. n- 1 s/ From (4.44) -1. gen = ERA«whey—inmiflmm. (5.7) From (4.45) -u . ‘2' 7‘ §R1()-7‘6X‘2K56;3§/)4eno\ (5.8) From (4.46) -1. «22 = g R, 0+ $)(1K‘6‘1)§/ld¢.m, (5.9) Here Home! = ~R?(‘+$)C9K\K3+ 9k?) ' (5.10) By using (4.43).(4.1+4).(4.45).(4.45) and putting K,-.-.- K and 6" 3 O , the values of )\’5 in (4.52) and (4.53) ‘2. become >1 3 Meg’a (Kfika) >\ a 6—} + ‘3 ‘V‘GNXKSB K, kKKHka) (5.12) >~,-7~. k|+k3 _ (5.13) A-N . K +\< (5.14) ‘Séfti >;_ 1:: ('L \<.:, K: 3i)()-+ ‘5j) (5.15) On substituting (5.15), (5.12), (5.11) into (4.57) and (5.14).(5.13) (5.12) (5 11) into (4 58). we have -ncbt ROE) = PTSO)[kK' +K K3)(l + §)e—-L_ K.+K3 kl Cat K5t ”57+“;‘afiq (I + ($\)(BT (r “5) --K'K"*K3 3' ‘) _Mth A ears.— ()1): C0 ”"3 F“ P )[Kjkf 3_eit+ st' ‘f’ K) e x, +‘Egéiérg3 :l ().rz) KL-+i< ‘ 3 respectively, we have Then substituting (5.16) and (5.17) into (#. 5) and (4.6) A) g) ~“* 111.1%” +‘n61:35t _( K. )(Hg 5 at 177.7) C05716(S.18) ki-1 K3 . -mett K. oo'ct)= gahzfitwik: k 3 T"; “g _%t max + K‘ e 1(K.+k33 005.719 (9.19) ifi+K3 ' 24 Expressions (5.18) and (5.19) can be rewritten as . - a» ' k“, 1.1a) =_E_1_<1+‘r>)ZP.sov at; co: m -flgdfiefi ZéoaquszS-zm K|+K3 'HM k'+k3 “a! v " 1: i ”c “c m 1: £45” a 915.21) b) C") = RC0) .1, 3 @1110) o.‘ o:- +__1_ c Emmi («M H“: .. k1“; *1" where t _6,_5 (“‘ka 391:1(3/ — K: k K Cl’1 " e 0‘3 = e '< H 3) (5.22) The expressions (5.20) and (5.21) are the general results for any two-dimensional charge distribution. qut)1s the charge density on the surface of the inner cylinder at any 11 time and.)0 (£3 is that on the outer cylinder. B, Line Charge on a Single Cylinder The case considered in Section V A is a general case, .where the original charge placed on the cylinder may be any arbitrary two-dimensional distribution. A special case is considered in his section, where the original charge placed on the cylinder is a line charge. In this case, If E%Co) in.(5.20) and (5.21) becomesg u /1iR,_ “’17” P593 = . (5.23) A flux, 11: o where i) is the charge per unit length of the original 0 line charge. The value of Pom) in (5.21) can be obtained a from (5.23). On substituting the value of PM“) from (5.23) into (3.20) and (5.21), we have 9. See Appendix B. 25 ‘ O . (Jo/(f) C-a-Ki (l +6)"f% 20.? COS'ne a ' 1 ' K. ('1 3 iii-ti. (Browne (5 24) - EfiK, +3 e .‘R‘ 71-11 . k . w’({)= SEQ-(f fik'z-izzarcbs 710 O 1| 1 (5 25) ‘ a. h: K1 9.9% f... 20.; CO! 716 O ‘ ‘ K‘+k3 ‘Rxm‘. The series Q Cos 119. can be summed in a closed form on)“ . 1-0: __._ ‘ ZCL COS 7'9 =[2<|_QQCO&O+}) 3} 1Ie octein two orders of magnitude for relaxation times to give in (5.24) and (5.25): The ones appearing in a. and Q; are of the order of some dielectric constant K divided byég, while the last two terms are multiplied by an exponential which has a relaxation time of K'IG‘ . If 9(< \ and the surface layer is very thin, the relaxation associated with a w: of the order of’Kfllégwill occur much more rapidly than the others. Experimentally, one can probably observe only the slower relaxations associated with a,‘ and a: , because the other relaxation will have already taken place. Also, if 6 is small, it is not possible to distinguish experimentally between charges placed on the surfaces of the inner and outer cylinders. mathematically we may there- fore restrict our solution to determining 00(t) , the sum of («{Ct)nndw'(t) for the case when 6‘1 is very large, 6 is very small, while 616 has some finite non~zero value. The factor 6‘). I} will be designated by G, , which may be called the surface conductivity. From the definition of 10. See Tppendix C. 26 ' . I § , it may be seen that G'- g , where 6‘ is R the conductivity of the surface layer of thickness T , while R is the radius of the cylinder. Thus after substituting O fer 3 ; for 6"1 ; 6", for 6 6‘1 ; and‘Ia.(|:-1(:M?)- fiferia (308710 into (5. 21+) and (5.25), we have $1.. (.01 at L (5.26) I II sit) a and) +u><£)=\:10 laCeSO+T)—_ TR After substituting the value of a, in (5.26), it can be rewritten as 1* 1 E '1 + QCE : ( “— —— (5.27) ) (I- lfiusé+ 211) HR where K . 't "" 5.17.3 (5.28) When both the numerator and the denominator of the right hand side of expression (5.27) are multiplied by e l'f, , we have t' 1: / -/ 03(1):) - e '1. _ c 't L (5.29) -QCOSB 4.5%: 13R; OI" _ . 1: amt) .. ( Sm). fir. j: (5.30) CoskT/rt~¢o$0 11R; where wfl) is the total charge density or the total charge per unit area on the surface at any time. In Fig.4, a set of curves is plotted with 003-) as ordinate and 9 , the angular position, as abscissa for various values of t/ 't . The graph is normalized 27 (by setting f -..-.- 1‘31 R1 ) so that the charge distribution at infinite time is unity everywhere on the cylinder. The charge density at any time m(‘L)for any ‘f‘ can be obtained by multiplying he graph by ‘P/‘l‘fl R1 . The values of +75: used are 0.2, 0.5, 1, 2 and 5. From the curves the following conclusions may be drawn: For the intermediate times between‘t’ -.-.. O and {-1.03 , it is found that, at 9.0 (the original position of the line charge), the chargeeg 5 decays as time progresses. At places where ‘9 is smafleti] the charge density increases atofirst and falls off later. ' At places where '9 is largeg>ihi charge density increases monotonely with time. The flow of charge ceases when the surface charge density is the same everywhere. Thus the line charge placed on the cylinder at t=O spreads out until the charge density is everywhere the same at t e as . TA 3 c ‘F is? 1:“ “as.“ as 6 as degrees 1!— oa. the as tau! tat. 1 tunes 0 10.0149 4.0839 2.1639 1.3130 1.0136 6 '7.8633 - - - - 10 5.7025 - - - ~ 15 3.7140 3.2226 - - - 30 1.3063 1.9920 1.7356 1.2523 1.0117 60 0.3870 0.8303 1.1266 1.1118 1.0067 90 0.1973 0.4621 0.7616 0.9640 0.9999 120 . 0.1324 0.3202 0.5752: 0.8509 0.9932 150 0.1067 0.2614 0.4878 0.7837 0.9884 180 0.0996 0.2449 0.4621 0.7616 0.9866 .11 1 141111112111 11111 11w - 11- 11-111 1.- 1 1- 1 1 1 1 - 111111-111 1141111111-”,11111-11 -11.- . M . .1 H. “11-h m- .. . . . - .. i. a -. .sm 4 . . .- 1 1.111 .- 1 1-...-1 .- 1 1. 1 1 1 111 1111. .1- 1 , - .. 11 H rh— - r1 mar . 1 . - )- h 1 . 1. - 111.11.15.14- efi -. .. . . . . g . u . W a H a ...b- . 1:“ . . . w . . . ..3 4 _ . . - 11 1 . 11 1 . 1 1111 .- 1»! -1. 1- 1 1 1 . t b .. .. 1. -2 w . .... . _ _ . T11 1111-1111- 1‘ 1 1"8-51111 Ile 111.1 1‘11 .11 it'll}- 1 11.111111113111111'111 11.). 1".-.) 1... 1111 1110.911 1 IV... . . n ) _ . . . . a a . .. e . .- a . . . . a W H S . . . _ . 11H 111%] 191 -1- .111- a . a 1 111 1 11” 1 f A. 19.11 1 15.1.. v 1 .1 11 1 11 1.11 11.111 . L1 1111 .- . . . .w . . ._ . e C i a . m . ... mi 1 . . ~ 1 h .11. H . . . . . m _ 1 1.--...... -.....11. - . -. - -- -_ - a _ . . a _ H . 1. m .11.” G1 .11-m.kw .W o 1 w n m . M _ . a ._ 2 .- re--.- - . . - - U m a d .m M m a . m -- . 1.. ~.. - 1.~. o. . - n . . ._ . H a h m- a . . ~ .11.- - - . .a ._ _. -- 1 .. 1- _ n _. .1. c . . m . . - . .- . em 11. . H -.----..---1.-1..T1e1..1 - 1..- : --.-1-1111.- -1. 11 -1 1- --1 . . . w a . a a Tm...- . _ . . . .1111.1~.11...111m11111 TW-«111 -.. 1 _ 1 1 w 1 - a . . H . _ . .. 2a. .- e a . ’ q a . 4 . m . u . M . r . _ . . . . 111-.- 18......- , . 4 11d. _. n . . . . . _ a m .I a M . . . -. .111. - < . a m use m i 1y1.11114.-1 - Iii-M1 w 1 1 -. - - “.1421.- . p . . . y . _ _ «.-.-1.111171 11 .- . . , -. . M w w . ~ . . . . . a .. ... ... E1111 \ n 1.. _ 9 _ 8 7 6 5 1b. 3 H 2 s1. 0 1 i ii. . - . . M. a n . . wl ~ . o . v e . . O I I '1. .1111 ‘ 1N1 ‘1 § .1 ‘91 . u u 1.1 .. . e a . . 1‘ 1.0? ..l. I 0-0.1lm-1.101'11‘.d'.|.._-.. '11...- no. 1" O . ‘ M . . . a 9 ~. ' 1 h.--.I.—.r—b- O 1. ,_.._..-L. ‘ "- - -e -. . . . Q o -e‘ 01.. 11.1.09 P 5.0 UZKNJJL .5.) ....) bt‘l oilflu ZHCFF‘.° \( .9 1.-—4-.. - . . _ l ‘ 1 1 .‘ ..._ a- _‘._...1. - --.i 1 "4--.... - 1...- . .. VI TWO-DIMENSIONAL CHARGE DISTRIBUTION ON A CONDUCTING SPHERE A. Arbitrary Charge Distribution on a Sphere In this section, the inves- tigation is done on a conduc- ting sphere. The radius or . the sphere is F{ . Inside the sphere, the dielectric oon-~ stam- is Ki. and the conduc- tivity is 6‘; . Outside the sphere the dielctric constant is K0 and the conductivity is zero. Radial distance from the center of the sphere to any point is denoted by Y' and 9- is the angle between F15'5° 7‘ and the Z axis(See Fig.5). 9 is sometimes known as the co-latitude angle. The charge density at time t=0 is a two-dimensional dis- tribution and arbitrary except for the fact that it is cy- lindrically symmetric. We will denote it as 00(0) . It can be expanded as 00(0) = i3?) PMCCos-O) 71.0 where R‘CCosO) are Legendre Polynomials and flu) is ex- pressed 11111 ‘ gnu) . $1 goo“) KCCO$9331719¢9 (6,19,) 11.See reference2.13.418 and also Appendix E. 31 oo(d)is considered to be cylindrically s rmetric and inde- pendent of (.6 It has already being proved that if there is no charge in the interior of a conductor at tzo, there will never be any chzge in the interior at a later time. Although charge flows in the interior, it appears only on the surface. The problem is to study the way the charge is distri- buted on the surface of the sphere at a later time if the original charge on it is.of the form given by (6.1). The charge density wetht any time' t, can be expanded as 013(1) =2 915+) 9.19039) (6 2) M20 then the potential inside the sphere é? = Z P.,S‘1)V‘P “(60563 115' Sew-«mm ‘5'” The above expression (6. 3) is the particular solution of Laplace's equation in spherical polar coordinates when the boundary conditions are satisfied. Using (2.23), the time rate of change of charge density is 341.3(1) _:__1/~> at ‘ " 6‘ .R 1... _6 2351311 Pccom ’ 4 0")(Ko'1ki) +ko (O ) But using (6.2), the time rate of )change of charge density is 3 00d?) :2% Est) P.n (COSO) . M804 ~ (605) 12.8ee Appendix D. 32 Since (6.4) equals (6.5). the coefficients of EqCCosakan be equated. Thus 492543 g ,_ 61-“ a) . (6.6) at ‘h Ko+ i+ke The expression (6.6) can be so ved for ehcbto give 61‘"): (6.7) E543): Page TICK m+k3+k where @0319 the value ofPMC‘E)at t:_—O. '.."hen (6.7) is sub- stituted into (6.2), we have, in. r V b)(‘t)= :PC032 Pccose) (6°Q) where cufi¥)is the charge péfi'unit' area a.t any time a.a Vt“ = MCK°+K13+E° (6.9) 5?“ The expression (6.8) is the general result for any two-dimensional charge distribution initially placed on the sphere. For any givenw(°),§‘£o)can be found by using (6.1a). Then if fin“) is substituted into (6.8), the surface charge density at a later time can be found. The value of ”En can be obtained by using (6.9). It is interesting to note that the values of Ty. are independent of the radius R although they depend on n; Because the relaxation times of the terms of the series are not all the same, it is not possible to sum the series in expression (6.8). Therefore no conclusion will be drawn from this expression at present. However,(6.8) will be used in Section VI 9, where we will solve for the surface charge density in a specific example. _1_3. Point Charge on 'a Sphere As a specific example of the results of VI A, the case of a conducting sphere with a point charge on it is con- sidered in this section. Let a point charge of total charge ‘1, be placed on the sphere ate :0 instead of the arbitrary charge distribution. The coefficients of P340563 in (6.1) is then3 PCO)__. (3% +0 6‘] 41"? The above value of P100) is substituted into (6.8) and the general expression for 006:) is now met)":- 1 (1714»09 t"""13-1'sP P140639) (6 10) ‘1‘R n: o ' From the above equation it is seen that at t:O, 09(0) _ L i171 +1) P “(0036) 4-“R‘ 71:0 (6011) and at t: on, cocoa) .-_ 3L... 4‘R1 (6.12) The original charge is a point charge whose magnitude is Co“ and the area of the sphere is [(11 R1 . It shows that the to- tal charge present at the begining is present at the end. It also shows that the point charge placed on the sphere at the begining is uniformly distributed throughout the surface of the sphere after a very long time. 1 3 . See Appendix E . Ill! '1 .II I. I, 34 To determine how the charge is distributed at times 04t< co (.6 10) can be rewritten as cod): 4113' ZC1m+0P (Cos 9)e:/ “to 61:, ~ +41qu~zz.13) RW‘O where rim z “(0* “DIG: (6.14) and 6(0), the Dirac delta function, from (6.11), is equal to Zia-1'1“)?“(cogg) in this case. The first term of the right hand side of equation (6.13) means that the point charge decreases exponentially with relaxation time Too . Also we can write Zczn.\)[é' "In i- In]? (Case): Za4(fi)h e P010961) ”‘1 +ZLML1 “XY.)+°’~CESis the surface charge density at any time. The series 2K“ Pccos931s a relatively small fraction of the entire expression for NC‘L). For t/vt of the order of unity, it is about 1% ofuoci). Also, the series converges r pidly, with the first ten terms constituting about 99% of the sum. Thus, in numerical computation, limiting the series to ten terns gives w(‘t)to an accuracy of about 0.175. For sample calculation, he value of °(.is chosen to be i, that 111,1 3:? 3:13 :3 degzges, ‘hhf' fipgaL ”9&9 '0 15 1.28495 1.51 1.13802 1. 30- 0.87202 1.06329 1.04998 1.00085 60 0.64848 0.88423 0.99088 1.00017 90 0.56623 0.81263 0.96541 ' 0.99979 120 0.52555 0.77514 0.95139 0.99953 150 0.50545 0.75845 0.94403 0.99939 180 0. 49929 0. 75284 0.94172 0. 99936 -It In. (,0 (L) in this table is 2C1m+018tm __ e lP‘hQOSO) 1n.° TABLE III. TABLE FOR 52.9.0039) = 211 ' “-1 degéeeg t4§;?’ :Egzl tigfiL §£§izo 30 40.00885 -0.00337 0.00336 0.00031 60 -0.00348 0.00010 0.00130 0.00012 90 0.00097 0.00035 '-0.00037 -0.00003 120 0.00428 0.00016 -0.00162 -0.00016 150 0.00631 0.00247 -0.00236 -0.00023 180 0.00695 ' 0.00275 -0.00260 -0.00024 _ a o u ... .. .‘ .. .3 z. 0” I“ .-.: "g Dbhloll Us... :"I' 1 hr U1—JJ.’ ... \HMHKHI) VII SUMMARY . (1) A general approximate method for solving Maxwell's equations is given for the case when the conductivities of all materials present are very small. This method of solu- tion permits the charge distribution at any time to be cal- culated if the charge distribution at any preceeding time is known. (2) To the above approximation, it is proved that if no charge is present initially inside a conductor, there will never be any charge inside it at a later time. If charge is originally present inside the conductor, the charge at every point will decay exponentially with a re- laxation time equals to 5, , where K is the dielectric constant and 6‘ is the cgnductivity. This relaxation time is independent of the size and the shape of the conductor. (3) It is fctmd that if any two-dimensional distribu- tion of charge w(c)is placed on the surface of a slightly conducting cylinder, the surface charge'density at any later time is given by acct): 00(0):" ... fibihert’m] (3.13), where 04(0) is the arbitrary two-dimensional charge density at t-_-.-o, a (o) is the average value of the original distribu- tion of charge wee). and '1. , the time of relaxation, is given by M. , where K1 is the dielectric constant in- side the cyslcinder, K0 is the dielectric constant outside the cylinder and 61 is the conductivity of the cylinder. The relaxation time is independent ofthe size of. the cylinder.’ 40 (4) For two conducting coaxial cylinders with an arbi- trary two-dimensional distribution of charge initially on the surface of the outer cylinder, the general formula for wet)“, the charge density at' any time, is given in equations (4.61) and (4.62) as infinite Fourier series. (5) For a single cylinder conducting only within a very thin layer on its surface and with a line charge on it initially, it is found that the surface charge density at any time t can be expressed as . t .. 1° 3...». ”c i) CDC‘E) - {Li—R1 Goth tl't __ 6°39 (5.30) where‘flét)is the total charge density residing on the surfaces of the canducting layer at any time t,'f is the line charge per unit length initially placed on the cylinder, R1 is the radius of the cylinder, andIr- K,'+ K3 , where ‘TFT' K‘is the dielectric constant of the cylinder, K3 is the dielectric constant of the outside medium. 6" is the product of the bulk conductivity of the conducting layer and its thickness divided by the radius of the cylinder. A set of curves are plotted in Fig. 4 for the variation of charge density w(t)with time t and 9- , angular position measured from the line charge. (6) For a conducting sphere, with a two-dimensional charge distribution on its surface, Doc-E) , the charge density at any time, is given by on _‘t '1’. 00G.) -.—. ZR“) 9 5.320» b) (6.8) ‘V\:o 41 where E‘Lo) are the expansion coefficients of (0C0) in Legendre PolynomialsRWCCeso) . I“: “1093+ thKJ, where K0 and kg are the dielectric constants of the mediums outside and inside the sphere, and (Q, is the conductivity of the sphere. (7) For a conducting sphere with a point charge on its surface initially, the surface charge density at any later time is given by equation (6.18). A set of curves are plot- ted in Fig.6 for the variation of charge densityccC€3with time t and é} , angular position measured from the ori-' ginal position of the point charge. It is found that the rate of relaxation of the charge density is independent of the size of the sphere. 42 APPENDIX A IEHEREIALLAEQEEQJLfiflldflDEB The problem is to find the potentials inside and out-' side a cylinder with a two-dimensional distribution of V charge on its surface. The charge density on the surface will be denoted by (10(6). It is an arbitrary function of 9 only, and can be expanded as Me) = Zg‘cos 719 (4.1) The The radius of the cylinder is R and the dielectric con- stant of the cylinder is FKL . The dielectric constant of the medium outside the cylinder is K0 and 1" is the radial distance from the axis of the cylinder to any point. The potentials V" inside the cylinder and V0 out- side the cylinder are particular solutions of Laplace's equation in cylindrical coordinates. They can be written in the form15 0 TI Vi, = ZBJ’ Co: 110 + C. (4.2) 11:1 . A. V0 =28 £05110 + Roi—war ( 3) 11-1 1"" A“ . B“ , A0 and Co are constants whose values are to be determined so that the boundary conditions are satisfied. ' The first boundary condition states that V; 1- Vo at 15.8mythe, William R.: "Static and Dynamic Electricity," McGraw-Hill Book Compeny,Inc..New York,1939,p.67. 43 10 , R , therefore, _ (AA) éflm‘los no + co = Act” R * gA—T—M ‘2’ “9 . n m R g A . (4.5) Also, an - “I K AotmR'r- Co (A.6) The second boundary condition states that at T’ .. R ki(%’f)rak- “(25% “JR. 0) therefore , on (dish—n We... 4. ) - “(n-W . as) =4 igloo: ‘ne “'7’ The K 81113M K A ‘n = “'53) . . * as.— E Also, A0: -323 (11.9) and ' Kc Co -_-, —_2_gln K . (4.10) Using (A.5)(.and (A.8), the values of A.“ and B.“ can be computed A.“ “(P11 R“+‘)/T\(Ki+ K.) (A.11) B“ a B., /T1R “(K-L + K.) (A.12) m substituting the values of the arbitrary constants into (A.2) and (AJ), we have Th, . .— 92-01“ R (111.13) on R. KY1“ V0 3: —:n(Kl-+K) COS T16 11-1 ' o 1’ (A.14) 45 Let us again expandathe charge density 000).. in (Ad): wm- 29. Co: a. (m 'n-e To find 3, , both sides of (A.1) are mltiplied by Cos‘» do and integrated fr” 6. O to 9. ‘fi . Thus, I 8 (10st 00(0) ale . 2P S‘Cos (so Cesneete O “’0 O - “'5. 1“ ‘ (B.1) T s and 1 P“ 1‘ O 2gw(.) Cos 110 40 'H)\ a. i o (2.2) *23L0346 71:0 If the initial charge placed on the surface of the cylinder (gt 5.0) is a line charge,then we have ' gun!“ a s 0 ° 0 1: o hem Fig.7, we have Smu. Anus («012 4.0) .15 f ‘3-3’ 0 where f is the charge per unit length e Fig.7. In this case (B. 2) can “Iowans; s}: “1040:;‘2M ‘srfsumt mum.» q-Swmdo 3 housemate sslsll,Usv\s.‘ . ‘I APPENDIX C . p m - z a Come W Let €(9)be an arbitrary function of G that can be expanded ' as 6(9) = 29,100: no ’ (c.I') Then, 1' 11-0 ‘D 1 3,6193 (103 ‘MG do a Zfi.‘ ‘SOCosm Cosmo do v1.0 - his: cos weds 2 1 is 5(9) C05 ‘me do a). Rn ' 01 (0.2) I Now we note that16 7 So 6(0) d” Th- 0 1 . I Gos'mecle- M gofi- 361.0039 +4?) " 11%;:— Lzflf" 1 Cosmed9:1am 2 S’W +d) 2 (0.3) Thus if - ..a.‘ 6(9):)»(1-14005N0 it follows that gm, Q?" provided thnt Wu": 0 , Whenmgo, by using (C. 2) we have, 1.4. 1 e g _ TS, I named? 90‘ (CA)- But from ‘03. .3)d' . (O ) 11¢ QC—T'a'c—T-z . 172 .5 therefore "/5. erfron (C. 1) ._a (C5) 2I mean?) ZQCosmui Consequently I (C 7) Q Co: “a 'x (-I IQCOOO+?)-1 .. 1.3 .Dwieightd ”:Tables of Integrals and other hthematical Data',‘ 0 1. . 47 APPENDIX D 4W The problem is tofind the potentials inside and out- side of a sphere with a two-dimensional distribution of charge on its surface at the begining.. The charge on the surface will be denoted by taco) . It is arbitrary except that it is a function of only. It can be expanded as coca): Z?” RICCOSO) (v.1) The radius of the sphere is “L and.the dielectric constant Of the sphere is '(L , The dielectric constant of the medium outside ‘of the sphere is K0 and T‘ is the radial distance from the centre of the sphere to any point. The potentialyeither inside or outside, is the particular solution of the Laplace's equation in spherical coordinates. It can be written in the form17 ‘ V; = ViBn‘Y‘“RflCCos 6) (13.2) 7130 “2A,, R, (00$ 63 (9-3) “—'WT‘T_’ “so where \/C is the potential inside the sphere and VB is the potential outside the sphere. A“ and B,“ are the constants whose values are to be determined so that the boundary conditions are satisfied. The first boundary condition states that \/L ”\I at'T”. F{ . Therefore :3 RP“ (Cos e) .25.,133 (Case) (n.4, ‘h-o 17.8ee reference 2,p.417-418. 'OP B SR (v.5) A1n+1 The second boundary condition states that IBVh - 6(3V' W( “K fifln-o tT‘ zit . Therefore, LC? 8;! "P PCCosID)K K.(ZA(“‘ 9134““ so R *1 . =3 REE, P‘v‘CCOS 6) (9.5) 1130 , [KL BWR 'HKoA (“WW RM] En (n. 7) Using.“ (D. 5) and (D. 7). the values of A71 and BY} can be obtained: ”P. l MCKRi++ko)+ko (1)-8) B = a ; . (D > 7‘ E1