disk as squat cylinder Kirk

Capacitance of a Thin Conducting Disk and of Conducting Spheroids Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (February 25, 2003) 1P r o b l e m Estimate the capacitance of a thin conducting disk in the form of a right circular cylinderof radius aand height 2 b,w h e r e b/lessmucha. Compare this case with that of a disk in the form of a thin oblate spheroid of diameter 2aand height 2 b. Note that an analytic expression can be given for the capacitance of an oblate or prolate spheroid of any ratio b/a. 2S o l u t i o n 2.1 Thin Conducting Disk (Right Circular Cylinder) The problem of a conducting right circular cylinder has been considered by Maxwell [1] in one of his last papers, and by Smythe [2], who obtained somewhat different results. Maxwell’s method was to calculate the electrical energy Ustored by a disk with charge Qaccording to U=1 2/integraldisplay 1/integraldisplay 2σ(r1)σ(r2) r12dArea 1dArea 2=Q2 2C, (1) where σis the surface charge density and Cis the capacitance, in Gaussian units. However, he appears to have neglected the contribution of the charge density on the narrow cylindricalsurface at r=a(while commenting on this in his final sentence). Smythe considers the charge distribution on a cylinder (where a≈b)t ob em a d eu po fa set of rings whose charges are adjusted so that the potential inside the cylinder is uniform. Here, we evaluate the potential at the center of the cylinder, V(0) = /integraldisplayσ(r) rdArea =Q C, (2) supposing (like Maxwell), that the charge density σover (most of) the flat surface the disk has same form as for an infinitely thin disk [3], namely σ(r)=Q/prime 4πa√ a2−r2. (3) If we let smeasure the distance radially inward from the circumference of the disk, then the charge density (3) varies as 1 /√ sfor small s. However, the charge density near the edge of a conductor whose surfaces intersect at an exterior angle of 3 π/2, as in the present 1 problem, is known to vary as 1 /3√ s,f o rsmeasured normal to the edge along either surface [4]. Our approximation is to suppose that the surface charge density obeys σs=K 3√ s(s≤b)( 4 ) for distance s<b on both the flat and cylindrical surfaces of the disk, and that it obeys eq. (3) for s>b,i.e.,f o rr<a−b. Continuity of the charge density at r=a−brequires that K 3√ b=Q/prime 4πa/radicalBig a2−(a−b)2≈Q/prime 4πa√ 2ab. (5) Hence, K=Q/prime 4πa√ 2ab1/6. (6) In this approximation, the total charge on the disk for s<b is Q1=4/integraldisplayb 0K 3√ s2πads =1 2πab2/3K=3/radicalBigg b 2aQ/prime=3 2/radicalBigg 2b aQ/prime. (7) The amount of charge on the cylindrical surface of the disk is not of order b/a, but of order of the much larger quantity/radicalBig b/a. Neglect of this charge result in the correction to the capacitance being of the wrong order of smallness in b/a, as seen below. The total charge on the flat surfaces of the disk for r<a−bis Q2=2/integraldisplaya−b 0Q/prime 4πa√ a2−r22πrdr =Q/prime⎛ ⎝1−/radicalBig a2−(a−b)2 a2⎞ ⎠≈Q/prime⎛ ⎝1−/radicalBigg 2b a⎞ ⎠.(8) Of course, Q1+Q2=Q,s ot h a t Q/prime≈Q⎛ ⎝1−1 2/radicalBigg 2b a⎞ ⎠=Q⎛ ⎝1−/radicalBigg b 2a⎞ ⎠. (9) Finally, we evaulate the potential Vat the center of the disk. The contribution to the potential due to the charge with s<b near the edge of the disk is, to the first approximation, V1≈Q1 a≈3/radicalBigg b 2aQ a. (10) The potential at the origin due to the charge at r<a−bis, using Dwight 380.001 and the fact that sin−1(1−/epsilon1)≈π/2−√ 2/epsilon1, V2≈2/integraldisplaya−b 0Q/prime 4πa√ a2−r22πrdr √ r2+b2=Q/prime 2a/integraldisplay(a−b)2 0dr2 /radicalBig −r4+(a2−b2)r2+a2b2 =−Q/prime 2asin−1a2−b2−2r2 a2+b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea−b 0≈Q/prime 2a/bracketleftBiggπ 2+s i n−1/parenleftBigg 1−4b a/parenrightBigg/bracketrightBigg ≈Q 2a⎛ ⎝1−/radicalBigg b 2a⎞ ⎠⎛ ⎝π−4/radicalBigg b 2a⎞ ⎠≈πQ 2a−Q 2a(4 +π)/radicalBigg b 2a. (11) 2 The total potential at the origin, and hence the potential of the conductor, is V=V1+V2≈πQ 2a−Q 2a(π−2)/radicalBigg b 2a=πQ 2a⎛ ⎝1−π−2 π/radicalBigg b 2a⎞ ⎠. (12) The capacitance is therefore,1 C≈2a π⎛ ⎝1+π−2 π/radicalBigg b 2a⎞ ⎠≈2a π⎛ ⎝1+0.26/radicalBigg b a⎞ ⎠ (b/lessmucha). (13) The result of Smythe’s numerical calculation [2] for 1 /8<b / a< 8i st h a t CSmythe≈2a π⎡ ⎣1+0.87/parenleftBigg b a/parenrightBigg0.76⎤ ⎦ (1/8<b / a< 8). (14) In contrast, Maxwell’s calculation [1] for b/lessmuchais CMaxwell≈2a π/parenleftBigg 1+1 2πb alna b/parenrightBigg (b/lessmucha). (15) If we ignore the charge density on the cylindrical surface, then Q/prime=QandV=V2where the upper limit of integration in eq. (11) is arather than a−b. Then we would find V≈2/integraldisplaya 0Q 4πa√ a2−r22πrdr √ r2+b2=−Q 2asin−1a2−b2−2r2 a2+b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea 0 ≈Q 2a/bracketleftBiggπ 2+s i n−1/parenleftBigg 1−2b2 a2/parenrightBigg/bracketrightBigg ≈πQ 2a/parenleftBigg 1−2 πb a/parenrightBigg , (16) and hence, C≈2a π/parenleftBigg 1+2 πb a/parenrightBigg (b/lessmucha). (17) Although the assumptions leading to this result are essentially the same as those of Maxwell, we do not find the logarithmic factor seen in eq. (15). And, we see from eqs. (13) and (17)that the inclusion of the charge density on the cylindrical surface changes the dependence of the correction term from b/ato/radicalBig b/a. It is amusing that Symthe’s result lies between these two cases (although his calculation does not necessarily apply for b/lessmucha). 1As a slight generalization, we might suppose that the 1 /3√ sdependence of the charge density on the flat surface of the disk extends inwards a distance kbrather than b. If so, the coefficient 0.26 in eq. (13) would be multiplied by (3 −k2/3)/2k1/6.F o r k>33/2=5.2, the sign of the correction to the capacitance would reverse. However, it is intuitive that the increase of surface area as height bincreases should be associated with an increase in capacity of the conductor. 3 2.2 Conducting Spheroids We now consider the example of a spheriod (about the zaxis), whose surface is given by r2 a2+z2 b2=1. (18) Forb<a the spheroid is oblate, while for b>a it is prolate; of course, for b=ait is a sphere. We take advantage of the result that for any spheroid of equatorial radius a, the projection onto the plane z= 0 of the charge density on the upper ( z>0) surface of the spheroid is given by eq. (3), where Q/primeequals the total charge Q[3]. Hence, the potential at the origin can be calculated similarly to eqs. (11) or (16), V=2/integraldisplaya 0Q 4πa√ a2−r22πr dr √ r2+z2=2/integraldisplaya 0Q 4πa√ a2−r22πrdr /radicalBig r2+b2−r2b2/a2 =Q 2a/integraldisplaya2 0dr2 /radicalBig −r4(1−b2/a2)+(a2−2b2)r2+a2b2. (19) The cases of b<a,b=aandb>a must be treated separately. For a solution via spheroidal coordinates, see problems 467 and 469 of [5]. 2.2.1 Oblate Spheroid ( b<a ) Again using Dwight 380.001 (or Gradshteyn and Ryzhik 2.261), the potential at the origin follows from eq. (19) as V=−Q 2a/radicalBig 1−b2/a2sin−1a2−2b2−2r2(1−b2/a2) /radicalBig (a2−2b2)2+4a2b2(1−b2/a2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea 0 =Q 2√ a2−b2/parenleftbiggπ 2+s i n−1(1−2b2/a2)/parenrightbigg =Q √ a2−b2sin−1√ a2−b2 a=Q C.(20) Forb/lessmuchathis becomes V≈πQ 2a/parenleftBigg 1−2 πb a/parenrightBigg , (21) so the capacitance of a thin oblate spheroid is C≈2a π/parenleftBigg 1+2 πb a/parenrightBigg . (22) This is the same as our result (17) for a thin conducting disk (right circular cylinder) with neglect of the charge on the cylindrical surface of that disk. Asbapproaches a,i.e., as the spheroid approaches a sphere, both the numerator and the denominator of eq. (21) approach zero. In this limit, the arcsine can be written as sin−1/parenleftBigg 2a2−b2 a2−1/parenrightBigg =−sin−1/parenleftBigg 1−2a2−b2 a2/parenrightBigg ≈−π 2+2√ a2−b2 a, (23) so the potential goes to Q/aas for a sphere. 4 2.2.2 Sphere ( a=b) The capacitance of a sphere is, of course, simply its radius a. Here, we confirm that our method of calculating the potential at the origin leads to this well-known result. When b=a, eq. (19) becomes V=Q 2a2/integraldisplaya2 0dr2 √ a2−r2=−Q a2√ a2−r2/vextendsingle/vextendsingle/vextendsingle/vextendsinglea 0=Q a, (24) as expected. 2.2.3 Prolate Spheroid ( b>a ) Equation (19) now integrates to2 V=Q 2a/radicalBig b2/a2−1ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/radicalBigg b2 a2−1/radicaltp/radicalvertex/radicalvertex/radicalbt /parenleftBiggb2 a2−1/parenrightBigg r4+(a2−2b2)r2+a2b2+2/parenleftBiggb2 a2−1/parenrightBigg r2+a2−2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea 0 =Q 2√ b2−a2lna2 2b2−a2−2b√ b2−a2=Q √ b2−a2lna b−√ b2−a2 =Q √ b2−a2lnb+√ b2−a2 a=Q 2√ b2−a2lnb+√ b2−a2 b−√ b2−a2=Q C. (26) Again, as bapproaches aboth the numerator and denominator of eq. (26) approach zero. In this limit, the logarithm can be approximated by ln/parenleftBigg 1+√ b2−a2 a/parenrightBigg ≈√ b2−a2 a, (27) so the potential goes to Q/aas expected for a sphere. Forb/greatermuchathe prolate spheroid takes the form of a conducting needle of length band radius a, and eq. (26) becomes, to a first approximation, V≈Q bln2b a,⇒ C≈b ln(2b/a). (28) Maxwell [1] has disccussed higher approximations to the capacitance of a needle. See also [7]. As noted in [3], the charge distribution as projected onto the zaxis is uniform for a conducting spheroid of any ratio b/a, so in particular, the charge distribution along the axis of our prolate needle is uniform. Jackson [8] has considered an amusing paradox concerningthe uniform charge distribution of a needle. See also [9]. 2The capacitance of a conducting prolate spheroid can also be calculated by noting that the field due to charge Qon the spheroid is the same as that due to charge Qspread uniformly along the line |x|≤c≡√ b2−a2[6]. The voltage on the conductor is conveniently evaluated at z=bto be V=/integraldisplayc −cQ 2cdz b−z=−Q 2cln(b−z)/vextendsingle/vextendsingle/vextendsingle/vextendsinglec −c=Q √ b2−a2lnb+√ b2−a2 a. (25) 5 References [1] J.C. Maxwell, On the Electrical Capacity of a long narrow Cylinder, and of a Disk of sensible Thickness ,i nScientific Papers (Dover, New York, 1965), Vol. II, p. 672, http://puhep1.princeton.edu/~mcdonald/examples/EM/maxwell_plms_9.pdf [2] W.R. Smythe, Static and Dynamic Electricity , 3rd ed. (McGraw-Hill, New York, 1968), sec. 5.39; Charged Right Circular Cylinder ,J .A p p l .P h y s . 27, 917 (1956); 33, 2966 (1962), http://puhep1.princeton.edu/~mcdonald/examples/EM/smythe_jap_27_917_56.pdf http://puhep1.princeton.edu/~mcdonald/examples/EM/smythe_jap_33_2966_62.pdf [3] K.T. McDonald, Conducting E llipsoid and Circular Disk (Oct. 19, 2002), http://puhep1.princeton.edu/~mcdonald/examples/ellipsoid.pdf [4] J.D. Jackson, Classical Electrodynamics , 3rd ed. (Wiley, New York, 1999), sec. 2.11. [5] N.N. Lebedev, I.P. Skalskaya and Y.S. Uflyand, Worked Problems in Applied Mathe- matics (Dover, New York, 1979). [6] R. Becker, Electromagnetic Fields and Interactions (Dover, New York, 1982), sec. 22. [7] R.A. Handelsman and J.B. Keller, The Electrostatic Field around a Slender Conducting Body of Revolution , SIAM J. Appl. Math. 15, 824 (1967), http://puhep1.princeton.edu/~mcdonald/examples/EM/handelsman_siamjam_15_824_67.pdf J.B. Keller, Charge density on a slender axially symmetric conducting body ,A m .J . Phys. 71, 282 (2003), http://puhep1.princeton.edu/~mcdonald/examples/EM/keller_ajp_71_282_03.pdf [8] J.D. Jackson, Charge density on thin straight wire, revisted ,A m .J .P h y s . 68, 789 (2000), http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_68_789_00.pdf Charge density on a thin straight wire: The first visit ,A m .J .P h y s . 70, 409 (2002), http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_409_02.pdf [9] J.C. Maxwell, On a Paradox in the Theory of Attractions ,i nScientific Papers (Dover, New York, 1965), Vol. II, p. 599, http://puhep1.princeton.edu/~mcdonald/examples/EM/maxwell_pcps_3_77.pdf 6