1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid, cn be written σ ellipsoid = x + y b + z = 1, (1) c, () 4πbc x + y + z where is the totl chrge. Show tht if the chrge distribution of the ellipsoid is projected onto ny of its symmetry plnes, the result is independent of the extent of the ellipse perpendiculr to the plne of projection (i.e., σ xy, the projection of σ ellipsoid on the x-y plne, is independent of prmeter c). Thus, the projection of the chrge distribution of conducting oblte or prolte spheroid onto its equtoril plne is the sme s the projected chrge distribution of conducting sphere. By considering thin conducting circulr disk of rdius s specil cse of n ellipsoid, show tht its surfce chrge density (summed over both sides) cn be written V 0 π r, (3) where the electric potentil V 0 of the disk is relted by V 0 = π/. Show lso tht if the chrge distribution on the conducting ellipsoid is projected onto ny of the coordinte xes, the result is uniform (i.e., the chrge distribution projected onto the x-xis is σ x = /). In prticulr, we expect tht the chrge distribution long conducting needle will be uniform, since the needle cn be considered s the limit of conducting ellipsoid, two of whose three xes hve shrunk to zero. Solution The chrge distribution (3) on thin, conducting disk cn be deduced in vriety of wys [1]. We record here highly geometric derivtion following Thomson []. The strting point is the elementry result tht the electric field is zero in the interior of sphericl shell of ny thickness tht hs uniform volume chrge density between the inner nd outer surfces of the shell. A well-known geometric rgument (due to Newton) for this is illustrted in Fig. 1. The electric field t point r 0 in the interior of the shell due to lmin of thickness δ nd re A 1 centered on point r 1 tht lies within nrrow cone whose vertex is point 0 is 1
Figure 1: For ny point r 0 in the interior of uniformly chrged shell of chrge, the xis of nrrow bicone intercepts the inner surfce of the shell t points r 1 nd r. The corresponding res on the inner surfce of the shell intercepted by the bicone re A 1 nd A. In the limit of smll res, A 1 /R 01 = A /R 0. given by E 1 = ρ dvol 1 ˆR R01 01, (4) where ρ is the volume chrge density, dvol 1 = A 1 δ, R 01 = r 0 r 1, nd the center of the sphere is tken to be t the origin. Likewise, the electric field from lmin of re A centered on point r defined by the intercept with the shell of the sme nrrow cone extended in the opposite direction (forming bicone) is given by E = ρ dvol R 0 ˆR 0, (5) In the limit of bicones of smll hlf ngle, the two prts of the bicone s truncted by the shell re similr, so tht A 1 = A dvol 1, = dvol, (6) R01 R0 R01 R0 nd, of course, ˆR 0 = ˆR 01. Hence E 1 + E = 0. Since this construction cn be pplied to ll points in the mteril of the sphericl shell, nd for ll pirs of surfce elements subtended by (nrrow) bicones, the totl electric field in the interior of the shell is zero. We now reconsider the bove rgument fter rbitrry scle trnsformtions hve been pplied to the rectngulr coordinte xes, x k 1 x, y k y, z k 3 z. (7) A sphericl shell of rdius s is thereby trnsformed into n ellipsoid, x s + y s + z s = 1 x s /k 1 + y s /k + z s /k 3 = 1. (8)
As prmeter s is vried, one obtins set of similr ellipsoids, centered on the origin. A smll volume element obeys the trnsformtion dvol = dxdydz k 1 k k 3 dxdydz = k 1 k k 3 dvol. (9) The three points 0, 1, nd in Fig. 1 lie long line, so tht R 01 = r 0 r 1 = CR 0 = C(r 0 r ), (10) where C is (negtive) constnt. This reltion is invrint under the scle trnsformtion (7), so tht together with eq. (9) the reltion dvol 1 R 01 = dvol R 0, (11) is lso invrint. Hence, if the ellipsoidl shell, which is the trnsform of the sphericl shell of Fig. 1, contins uniform volume chrge density, the reltion E 1 + E = 0 remins true t the vertex of ny bicone in the interior of the shell, which implies tht the totl electric field is zero there. This proof is bsed on the premise tht the ellipsoidl shell is bounded by two similr ellipsoids, nd tht the volume chrge density in the shell is uniform. If we let the outer ellipsoid of the shell pproch the inner one, lwys remining similr to the ltter, we rech configurtion tht is equivlent to thin, conducting ellipsoid, since in both cses the electric field is zero in the interior. Hence, the surfce chrge distribution on thin, conducting ellipsoid must the sme s the projection onto its surfce of uniform chrge distribution between tht surfce nd similr, but slightly lrger ellipsoidl surfce. The chrge σ per unit re on the surfce of thin, conducting ellipsoid is therefore proportionl to the thickness, which we write s δd, of the ellispodl shell formed by tht surfce nd similr, but slightly lrger ellipsoid: σ = ρ δd, (1) where constnt ρ is to be determined from knowledge of the totl chrge on the conducting ellipsoid. The thickness δd of thin ellipsoidl shell t some point on its inner surfce is the distnce between the plne tht is tngent to the inner surfce t the specified point, nd the plne tht is tngent to the outer surfce t the point similr to the specifiedl point. These plnes re prllel since the ellipsoids re prllel. In prticulr if the semimjor xes of the inner ellipsoid re clled, b, nd c, then those of the outer ellipsoid cn be written + δ, b + δb nd c + δc. Let the (perpendiculr) distnce from the plne tngent to the specified point on the inner ellipsoid to its center be clled d, nd the corresponding distnce from the outer tngent plne be d + δd, so tht δd is the desired thickness of the shell t the specified point. Then, the condition of similrity is tht δ = δb b = δc c = δd d. (13) Since the volume of n ellipsoid with semimjor xes, b, nd c is 4πbc/3, the volume of the ellispoidl shell is 4π(+δ)(b+δb)(c+δc)/3 4πbc/3 = 4πbc(δd/d), using eq. (13). 3
As the constnt ρ hs n interprettion s the uniform chrge density within the mteril of the ellipsoidl shell, we find tht the totl chrge on the conducting ellipsoid is relted by = ρ Vol shell = 4πbc ρ δd, (14) d nd hence, σ = ρ δd = d 4πbc. (15) It remins to find n expression for the distnce d to the tngent plne. If we write the eqution for the ellipsoid in the form f(x, y, z) = x + y b + z 1 = 0, (16) c then the grdient of f is perpendiculr to the tngent plne. Thus, the vector d from the center of the ellipsoid to the tngent plne is proportionl to f. Tht is, The unit vector ˆd is therefore ( x d f =, y b, z ). (17) c ˆd = ( x, y b, z c ). (18) x + y + z The mgnitude d of the vector d is relted to the vector r = (x, y, z) of the specified point on the ellipse by d = r ˆd x + y + z = b c 1 =. (19) x + y + z x + y + z be At length, we hve found the chrge density on the surfce of conducting ellipsoid to where is the totl chrge. σ ellipsoid =, (0) 4πbc x + y + z To deduce the projection of the chrge distribution of the conducting ellipse onto one of its symmetry plnes, sy the x-y plne, note tht projected re element dx dy corresponds to re da on the surfce of the ellipsoid tht is relted by dx dy = da ẑ = da d z, (1) where the unit vector ˆd tht is norml to the surfce of the ellipsoid t the point (x, y) is given by eq. (19). Thus, da = dx dy ˆd z = c dx dy x z + y 4 b + z 4 c. () 4 4
The chrge projected onto the x-y plne from both z > 0 nd z < 0 is d xy = σ ellipsoid da = c dx dy πbz = dx dy, (3) πb 1 x y b combining eqs. (1), (0) nd (). The projected chrge density (due to both hlves of the ellipsoid), σ xy = d xy /dx dy, is independent of the prmeter c tht specifies the size of the ellipsoid in z. For exmple, the chrge distributions of sphere, disk, nd both n oblte nd prolte spheroid, ll of the sme equtoril dimeter, re the sme when projected onto the equtoril plne. If the project the chrge d xy of eq. (3) onto the x xis, the result is d x = dx π b b x b b x dy = dx, (4) b b x y which is uniform in x! In prticulr, the uniform chrge distribution on conducting sphere projects to uniform chrge distribution on ny dimeter; nd the chrge distribution is uniform long conducting needle tht is the limit of conducting ellipsoid. The cse of thin, conducting ellipticl disk in the x-y plne cn be obtined from eq. (0) by letting c go to zero. For this we note tht eq. (16) for generl ellipsoid permits us to write c x 4 + y b 4 + z c 4 = ( x c + y 4 b 4 ) + 1 x y b 1 x y b. (5) The chrge density on ech side of conducting ellipticl disk is therefore σ ellipticl disk =. (6) 4πb 1 x + y b The chrge density on ech side of conducting circulr disk of rdius follows immeditely s 4π r, (7) where r = x + y. Such disk hs potentil V 0, which cn be found by clculting the potentil t the center of the disk ccording to V 0 = V (r = 0, z = 0) = 0 σ(r) πr dr = r 0 dr r = π. (8) Hence, conducting disk of rdius t potentil V 0 hs chrge density V 0 π r. (9) 3 References [1] W.R. Smythe, Sttic nd Dynmic Electricity, 3rd ed. (McGrw-Hill, New York, 1968). [] W. Thomson (Lord Kelvin), Ppers on Electricity nd Mgnetism, nd ed. (Mcmilln, London, 1884), pp. 7 nd 178-179. 5