The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 17 of 39

expression (263) is, as we have seen, a solution of Laplace's equation : it is seen on inspection to be a single-valued function of position on the Biemann's surface, and to be periodic in 9 with period 4nr. Hence it is the potential-function of which we are in search. Thus

iu

f(r, 6, z, a, a, 0) =

4vr I £ _•» Vr2 - 2ar cos {6-u) + a? + z*

The details of the integration can be found in Sommerfeld's paper. The value of the integral is found to be

12, _, /^+T -^ - tan l a / ,

where t = cos \ (<fi — a), a = cos h (p — p1).

Other systems of coordinates can be treated in the same way ; details will be found in the papers to which reference has already been made.

337. The present chapter has attempted to give an account of the principal methods available for the solution of electrostatic problems. A few examples have been given of each method, but no attempt has been made to enumerate all the problems which can be solved. The reader who wishes to study particular problems more fully may be referred to the following works :

Sir "W. Thomson (Lord Kelvin). Papers on Electricity and Magnetism.

In particular a number of examples of images and inversion will be found here, with mmierical calculations.

Maxwell. Electricity and Magnetism. Vol. I. (3rd Edn.).

In Chap. ix. the theory of spherical harmonics is developed, and the problem of the distribution of electricity on a nearly spherical conductor free in space, as also that on a nearly sj^herical conductor enclosed in a nearly spherical and nearly concentric conduct- ing vessel, are solved in detail. The coefficients of capacity and induction of two spherical conductors are investigated by spherical harmonics. Chapter xi. contains examples of the method of images and inversion. Chapter xn. contains a number of examples of conjugate functions, some being of special importance in the theory of electrostatic instruments.

J. J. Thomson. Recent Researches in Electricity and Magnetism.

Chapter in. contains important examples of conjugate function transformations. In particular problems are solved which enable us to estimate the effect on the capacity of a condenser produced by the slit between a guard ring and the moveable plate of the con- denser. Transformations are given which solve the problems of (i) a condenser formed by

336, 337] Examples 287

two parallel and equal plates of finite breadth ; (ii) a condenser formed by two parallel and equal strips placed in the same plane ; (iii) a pile of plates ; (iv) a system of 2n plates arranged radially at angles ir/n with one another, alternate plates being at the same potential.

Kirchhoff. Gesammelte abhandlungen.

A formula is given for the capacity of two circular plates of an uniform thickness placed coaxially at any distance apart.

EXAMPLES.

1. An infinite conducting plane at zero potential is under the influence of a charge of electricity at a point 0. Shew that the charge on any area of the plane is proportional to the angle it subtends at 0.

2. A charged particle is placed in the space between two uninsulated planes which intersect at right angles. Sketch the sections of the equipotentials made by an imaginary plane through the charged particle, at right angles to the planes.

3. In question 2, let the particle have a charge e, and be equidistant from the planes. Shew that the total charge on a strip, of which one edge is the line of intersection of the planes, and of which the width is equal to the distance of the particle from this line of intersection, is — \e.

4. In question 3, the strip is insulated from the remainder of the planes, these being still to earth, and the particle is removed. Find the potential at the point formerly occupied by the particle, produced by raising the strip to potential V.

5. If two infinite plane uninsulated conductors meet at an angle of 60°, and there is a charge e at a point equidistant from each, and distant r from the line of intersection, find the electrification at any point of the planes. Shew that at a point in a principal plane through the charged point at a distance r^/3 from the line of intersection, the surface

density is

3 1

• ,

47rr2 \4 7 J7

6. Two small pith balls, each of mass m, are connected by a light insulating rod. The rod is supported by parallel threads, and hangs in a horizontal position in front of an infinite vertical plane at potential zero. If the balls when charged with e units of electricity are at a distance a from the plate, equal to half the length of the rod, shew that the inclination 6 of the strings to the vertical is given by

e2 tan<9 = - s -(1

A'

Amga2 \ 2*/

7. What is the least positive charge that must be given to a spherical conductor, insulated and influenced by an external point-charge e at distance r from its centre, in order that the surface density may be everywhere positive?

8. An uninsulated conducting sphere is under the influence of an external electric charge ; find the ratio in which the induced charge is divided between the part of its surface in direct view of the external charge and the remaining part.

9. A point-charge e is brought near to a spherical conductor of radius a having a charge E. Shew that the particle will be repelled by the sphere, unless its distance from

the nearest point of its surface is less than \a */ ~p% approximately.

288 Methods for the Solution of Special Problems [ch. viii

10. A hollow conductor has the form of a quarter of a sphere bounded by two perpendicular diametral planes. Find the image of a charge placed at any point inside.

11. A conducting surface consists of two infinite planes which meet at right angles, and a quarter of a sphere of radius a fitted into the right angle. If the conductor is at zero potential, and a point-charge e is symmetrically placed with regard to the planes and the spherical surface at a great distance / from the centre, shew that the charge induced on the spherical portion is approximately — beaPjirf3.

12. A point-charge is placed in front of an infinite slab of dielectric, bounded by a plane face. The angle between a line of force in the dielectric and the normal to the face of the slab is a ; the angle between the same two lines in the immediate neighbourhood of the charge is /3. Prove that a, /3 are connected by the relation

.13 / 2k . a

Sm2 = VmSm2'

13. An electrified particle is placed in front of an infinitely thick plate of dielectric. Shew that the particle is urged towards the plate by a force

k + 1 4J2' where d is the distance of the point from the plate.

14. Two dielectrics of inductive capacities kj and k2 are separated by an infinite plane face. Charges eu e2 are placed at points on a line at right angles to the plane, each at a distance a from the plane. Find the forces on the two charges, and explain why they are unequal.

15. Two conductors of capacities cx, c2 in air are on the same normal to the plane boundary between two dielectrics kj, k2, at great distances a, b from the boundary. They are connected by a thin wire and charged. Prove that the charge is distributed between them approximately in the ratio

Kj — k2 2kj J

Kltc2 2b(Kt + K2) (K1 + K2)(a + b)rK2\c

C! 2a(n1 + n2) (Kl + K2)(a + b)}

16. A thin plane conducting lamina of any shape and size is under the influence of a fixed electrical distribution on one side of it. If <ri be the density of the induced charge at a point P on the side of the lamina facing the fixed distribution, and <r2 that at the corresponding point on the other side, prove that <n — <r2 = cr0, where o-0 is the density at P of the distribution induced on an infinite plane conductor coinciding with the lamina.

17. An infinite plate with a hemispherical boss of radius a is at zero potential under

the influence of a point-charge e on the axis of the boss distant /from the plate. Find the

surface density at any point of the plate, and shew that the charge is attracted towards

the plate with a force

e2 4e2a3/3

4/2 (/4-a4)2*

18. A conductor is formed by the outer surfaces of two equal spheres, the angle between their radii at a point of intersection being 277/3. Shew that the capacity of the conductor so formed is

5^/3-4

where a is the radius of either sphere.

2N/3 ">

Examples 289

19. Within a spherical hollow in a conductor connected to earth, equal point-charges e are placed at equal distances / from the centre, on the same diameter. Shew that each is acted on by a force equal to

r_4«^3_ i -i

L(«4-/4)2 + 4/2j'

20. A hollow sphere of sulphur (of inductive capacity 3) whose inner radius is half its outer is introduced into a uniform field of electric force. Prove that the intensity of the field in the hollow will be less than that of the original field in the ratio 27 : 34.

21. A conducting spherical shell of radius a is placed, insulated and without charge, in a uniform field of electric force of intensity F. Shew that if the sphere be cut into two hemispheres by a plane perpendicular to the field, these hemispheres tend to separate and require forces equal to -^cPF2 to keep them together.

22. An uncharged insulated conductor formed of two equal spheres of radius a cutting one another at right angles, is placed in a uniform field of force of intensity F, with the line joining the centres parallel to the lines of force. Prove that the charges induced on the two spheres are ^Fa2 and — ^Fa2.

23. A conducting plane has a hemispherical boss of radius a, and at a distance /from the centre of the boss and along its axis there is a point-charge e. If the plane and the boss be kept at zero potential, prove that the charge induced on the boss is

-Ji._/l=gLl

24. A conductor is bounded by the larger portions of two equal spheres of radius a cutting at an angle ^tt, and of a third sphere of radius c cutting the two former orthogonally. Shew that the capacity of the conductor is

c + a(f-I V3)-«c{2(a2 + c2)-^-2(a2 + 3c2)-* + (a2 + 4c2)~^}.

25. A spherical conductor of internal radius 6, which is uncharged and insulated, surrounds a spherical conductor of radius a, the distance between their centres being c, which is small. The charge on the inner conductor is E. Find the potential function for points between the conductors, and shew that the surface density at a point P on the inner conductor is

E_ /I _ 3c cos 6\ 4n\a2 b*-a3)'

where 8 is the angle that the radius through P makes with the line of centres, and terms in c2 are neglected.

26. If a particle charged with a quantity e of electricity be placed at the middle point of the line joining the centres of two equal spherical conductors kept at zero potential, shew that the charge induced on each sphere is

• 2em (l-m + m2- 3m3 -f 4m4),

neglecting higher powers of m, which is the ratio of the radius to the distance between the centres of the spheres.

27. Two insulated conducting spheres of radii a, b, the distance c of whose centres is large compared with a and b, have charges Eu E2 respectively. Shew that the potential energy is approximately

J. 19

290 Methods for the Solution of Special Problems [en. vm

28. Shew that the force between two insulated spherical conductors of radius a placed in an electric field of uniform intensity F perpendicular to their line of centres is

c4 \ c3 c° c being the distance between their centres.

29. Two uncharged insulated spheres, radii a, b, are placed in a uniform field of force so that their line of centres is parallel to the lines of force, the distance c between their centres being great compared with a and b. Prove that the surface density at the point at which the line of centres cuts the first sphere (a) is approximately

F ( GZ>3 15a&3 28a263 57a363 1

■&y + -jT + -?-+-ji-+-jr-+"'r

30. A conducting sphere of radius a is embedded in a dielectric (K) whose outer

boundary is a concentric sphere of radius 2a. Shew that if the system be placed in

a uniform field of force F, equal quantities of positive and negative electricity are

separated of amount

9Fa*K

5/i +7 '

31. A sphere of glass of radius a is held in air with its centre at a distance c from a point at which there is a positive charge e. Prove that the resultant attraction is

where /9 = (A-1)/(A+1).

32. A conducting spherical shell of radius a is placed, insulated and without charge, in a uniform field of force of intensity F. Shew that if the sphere be cut into two hemispheres by a plane perpendicular to the field, a force ^ a2F2 *3 required to prevent the hemispheres from separating.

33. A spherical shell, of radii a, b and inductive capacity K, is placed in a uniform field of force F. Shew that the force inside the shell is uniform and equal to

9KF

9A-2(A-l)-2(63/a3-l)*

34. The surface of a conductor being one of revolution whose equation is

4 1 7_

r + r ~ 12 '

where r, r' are the distances of any point from two fixed points at distance 8 apart, find the electric density at either vertex when the conductor has a given charge.

35. The  curve 
    

9a f a + x a — x 11

when rotated round the axis of x generates a single closed surface, which is made the bounding surface of a conductor. Shew that its capacity will be a, and that the surface density at the end of the axis will be e/dna2, where e is the total charge.

36. Two  equal  spheres  each  of  radius  a  are  in  contact.     Shew  that  the  capacity  of  the 
    

conductor so formed is 2a loge 2.

Examples 291

37. Two  spheres  of  radii  a,  b  are  in  contact,  a  being  large  compared  with  b.     Shew 
    

that if the conductor so formed is raised to potential V, the charges on the two spheres are

Va 1 - — — -jr and la -r- — -^ . \ 6(a + 6)V \G(a + b)y

33. A conducting sphere of radius a is in contact with an infinite conducting plane. Shew that if a unit point-charge be placed beyond the sphere and on the diameter through the point of contact at distance c from that point, the charges induced on the plane and sphere are

■KCL , TT<X , TTtt . TTCL

cot — and — cot 1.

c c c c

39. Prove that if the centres of two equal uninsulated spherical conductors of radius a be at a distance 2c apart, the charge induced on each by a unit charge at a point midway between them is

where c=acosh a.

2 (-l)nsech«a, l

40. Shew that the capacity of a spherical conductor of radius a, with its centre at a distance c from an infinite conducting plane, is

QO

a sinh a 2 cosech wa, i where c = a cosh a.

41 An insulated conducting sphere of radius a is placed midway between two parallel infinite uninsulated planes at a great distance 2c apart. Neglecting ( - J , shew that the capacity of the sphere is approximately

a|l+|log2J.

42 Two spheres of radii r1} r2 touch each other, and their capacities in this position are cx, c2. Shew that

fool ool col ■)

where /= — — .

J rx + r2

43. A conducting sphere of radius a is placed in air, with its centre at a distance e from the plane face of an infinite dielectric. Shew that its capacity is

oo /X-IV1-1 a sinh a T ( t^ — ; I cosech na, 7 \A + 1/

where a=cja.

44. A point-charge e is placed between two parallel uninsulated infinite conducting planes, at distances a and b from them respectively. Shew that the potential at a point between the planes which is at a distance z from the charge and is on the line through the charge perpendicular to the planes is

\2a + 2b) \2a + 2b) \2a + 2bJ ' V 2a + 2b J \

• ;„.. .: +

{ l \2a + 2bJ \2a + 2bJ \2a + 2bJ \

19—2

292 Methods for the Solution of Special Problems [ch. viii

45. A spherical conductor of radius a is surrounded by a uniform dielectric A", which is bounded by a sphere of radius b having its centre at a small distance y from the centre of the conductor. Prove that if the potential of the conductor is V, and there are no other conductors in the field, the surface density at a point where the radius makes an angle 6 with the line of centres is

KVb f 6(iT-l)ya8cos0

}•

47ra{(ff-l)o + 6} \ ^2(K-l)a3 + (K+2)b3.

46. A shell of glass of inductive capacity A, which is bounded by concentric spherical surfaces of radii a, b (a<b\ surrounds an electrified particle with charge E which is at a point Q at a small distance c from 0, the centre of the spheres. Shew that the potential at a point P outside the shell at a distance r from Q is approximately

E 2Eo(b3-a3)(K-lf cos (9

r + 2a3 (K-lf-b3 (K+2) (2K+1) r2 '

where 6 is the angle which QP makes with OQ produced.

47. If the centres of the two shells of a spherical condenser be separated by a small distance d, prove that the capacity is approximately

ab ( abd2 "1

b^a \ + (b-a){V-a?)) '

b

48. A condenser is formed of two spherical conducting sheets, one of radius b surrounding the other of radius a. The distance between the centres is c, this being so small that (c/a)2 may be neglected. The surface densities on the inner conductor at the extremities of the axis of symmetry of the instrument are <ri, 0-2, and the mean surface density over the inner conductor is a. Prove that

o"2 ~ o"i 6ca2

cr

b3-

49. The equation of the surface of a conductor is r = a (1 + ePn), where e is very small, and the conductor is placed in a uniform field of force F parallel to the axis of harmonics. Shew that the surface density of the induced charge at any point is greater than it would be if the surface were perfectly spherical, by the amount

4<r8(2n + l){(w+1)i>^1 + (w-2)i>"-|}-

50. A conductor at potential V whose surface is of the form r=a(l + ePn) is sur- rounded by a dielectric (A") whose boundary is the surface r=b (1 +rjPn), and outside this the dielectric is air. Shew that the potential in the air at a distance r from the origin is

Kab V

1 (2n + l)eanb2n + 1 + (K-l)r1bn{nb2n + 1 + (n + l)a'in + l} Pn

(K-l)a + b_i- (l + n + »A')6s* + 1 + (A'-l)(n + l)oa» + l r

where squares and higher powers of e and 77 are neglected.

«■].

51. The  surface  of  a  conductor  is  nearly  spherical,  its  equation  being 
    

r = a (1 + oSy,

where e is small. Shew that if the conductor is uninsulated, the charge induced on it by a unit charge at a distance / from the origin aud of angular coordinates 6, <f> is approximately

Examples 293

52. A uniform circular wire of radius a charged with electricity of line density e surrounds an uninsulated concentric spherical conductor of radius c ; prove that the electrical density at any point of the surface of the conductor is

53. A dielectric sphere is surrounded by a thin circular wire of larger radius b carrying a charge E. Prove that the potential within the sphere is

^LS, 1Vl, 1+4^ 1.3.5...2n-l /r*» \

• 1 i ~ 1+2»(1 + Z) 2.4.6...2« W 2nJ*

54. If within a conductor formed by a cone of semi-vertical angle cos-1 fx0 and two spherical surfaces r=a, r = b with centres at the vertex of the cone, a charge q on the axis at distance r' from the vertex gives potential V, and if we write

• a

r=ae~\ V=Ue2, Xo=log-^>

mn u n

the summation with respect to m extending to all positive integers, and that with respect to n to all numbers integral or fractional for which Pn (^0) = 0, determine Amn. Effecting the summation with respect to m, shew that when r < /,

and that when r>r',

55. A spherical shell of radius a with a little hole in it is freely electrified to potential V. Prove that the charge on its inner surface is less than VS/8ira, where S is the area of the hole.

56. A thin spherical conducting shell from which any portions have been removed is freely electrified. Prove that the difference of densities inside and outside at any point is constant.

57. Electricity is induced on an uninsulated spherical conductor of radius a, by a uniform surface distribution, density <r, over an external concentric non-conducting spherical segment of radius c. Prove that the surface density at the point A of the conductor at the nearer end of the axis of the segment is

where B is the point of the segment on its axis, and D is any point on its edge.

58. Two conducting discs of radii a, a' are fixed at right angles to the line which joins their centres, the length of this line being r, large compared with a. If the first have potential V and the second is uninsulated, prove that the charge on the first is

2anr9- V 7r2r2 — 4aa''

59. A spherical conductor of diameter a is kept at zero potential in the presence of a fine uniform wire, in the form of a circle of radius c in a tangent plane to the sphere with

294 Methods for the Solution of Special Problems [ch. vm

its centre at the point of contact, which has a charge E of electricity; prove that the electrical density induced on the sphere at a point whose direction from the centre of the ring makes an angle -ty with the normal to the plane is

c2E sec3 its f2* k i

•   /,    Y        (a2  +  c*  sec2  ylt-  2ae  tan  4,  cos  6) "  *  dO. 
  

60. Prove that the capacity of a hemispherical shell of radius a is

61. Prove that the capacity of an elliptic plate of small eccentricity e and area A is approximately

x/®§(^*«)-

62. A circular disc of radius a is under the influence of a charge q at a point in its plane at distance b from the centre of the disc. Shew that the density of the induced distribution at a point on the disc is

q /&-<&

2n2R2 V a2-/-*' where r, R are the distances of the point from the centre of the disc and the charge.

63. An ellipsoidal conductor differs but little from a sphere. Its volume is equal to that of a sphere of radius r, its axes are 2r(l + a), 2r(l+/3), 2r(l+y). Shew that neg- lecting cubes of a, /3, y, its capacity is

64. A prolate conducting spheroid, semi-axes a, b, has a charge E of electricity. Shew that repulsion between the two halves into which it is divided by its diametral plane is

E2 , a log-

4(a2-62) &6"

Determine the value of the force in the case of a sphere.

65. One face of a condenser is a circular plate of radius a : the other is a segment of a sphere of radius R, R being so large that the plate is almost flat. Shew that the capacity is ^KR\ogtilt0 where ti, t0 are the thickness of dielectric at the middle and edge of the condenser. Determine also the distribution of the charge.

66. A thin circular disc of radius a is electrified with charge E and surrounded by a spheroidal conductor with charge E1 , placed so that the edge of the disc is the locus of the focus S of the generating ellipse. Shew that the energy of the system is

2 a 2 a

B being an extremity of the polar axis of the spheroid, and G the centre.

67. If the two surfaces of a condenser are concentric and coaxial oblate spheroids of small ellipticities e and <■' and polar axes 2c and 2c', prove that the capacity is

CC' (C -c)-*{c'-C+$ (ec' - e'c)},

neglecting squares of the ellipticities ; and find the distribution of electricity on each Burface to the same order .of approximation.

Examples 295

68. An accumulator is formed of two confocal prolate spheroids, and the specific inductive capacity of the dielectric is A7/ur, where cr is the distance of any point from the axis. Prove that the capacity of the accumulator is

where a, b and au bt are the semi-axes of the generating ellipses.

69. A thin spherical bowl is formed by the portion of the sphere #2+y2+z2 = as

!*>& qji, £j2

bounded by and lying within the cone — -2 + t» = -2, and is put in connection with the earth

(X" 0" C"

by a fine wire. 0 is the origin, and C, diametrically opposite to 0, is the vertex of the bowl ; Q is any point on the rim, and P is any point on the great circle arc CQ. Shew that the surface density induced at P by a charge E placed at 0 is

Ec CQ

where

47ra&/0P"(0Pa-0$2)4' d6

Jo (a2

(a2sin2(9 + 62cos2(9)^'

70. Three long thin wires, equally electrified, are placed parallel to each other so that they are cut by a plane perpendicular to them in the angular points of an equilateral triangle of side sJZc ; shew that the polar equation of an equipotential curve drawn on the

plane is

r6 + c6- 2r3c3 cos 3$ = constant,

the pole being at the centre of the triangle and the initial line passing through one of the

wires.

71. A flat piece of corrugated metal (y = asinmx) is charged with electricity. Find the surface density at any point, and shew that it exceeds the average density approxi- mately in the ratio my : 1.

72. A long hollow cylindrical conductor is divided into two parts by a plane through the axis, and the parts are separated by a small interval. If the two parts are kept at potentials Vx and V2, the potential at any point within the cylinder is

2 7T a2-?-2

where r is the distance from the axis, and <9 is the angle between the plane joining the point to the axis and the plane through the axis normal to the plane of separation.

73. Shew that the capacity per unit length of a telegraph wire of radius a at height h above the surface of the earth is

71 An electrified line with charge e per unit length is parallel to a circular cylinder of radius a and inductive capacity K, the distance of the wire from the centre of the cylinder being c. Shew that the force on the wire per unit length is

K-l 4aV

K+l c(c2-a2)'

75. A cylindrical conductor of infinite length, whose cross-section is the outer boundary of three equal orthogonal circles of radius a, has a charge e per unit length. Prove that the electric density at distance r from the axis is

e (3r2 + a2)(3r2-a2-/6ar)(3r2-a2+v/6a/-) 6^a r2(9r4-3a2r2 + a4)

296 Methods for the Solution of Special Problems [ch. viii

76. If  the  cylinder  xi+yi  =  ai  be  freely  charged,  shew  that  in  free  space  the  resultant 
    

force varies as

/ a8\ ~ *

( r4 + 2a4cos4<9 + -^) ,

where x=r cos 6, y = r sin 8 ; and that its direction makes with the axis of x an angle

r4 — a* \ 2

-. -. tan 2(9

r4 + a4

77. If (f) + i\ls=f(x + iy), and the curves for which <£ = constant be closed, shew that the capacity C of a condenser with boundary surfaces 0 = <£i, <£ = $o i3

KM 4tt (0i -0o)

per unit length, where [^] is the increment of ■v|/- on passing once round a 0-curve.

78. Using the transformation x + iy = c cot \ (U+iV), shew that the capacity C per unit length of a condenser formed by two right circular cylinders (radii a, b), one inside the other, with parallel axes at a distance d apart, is given by

^=2-"-'(t)

79. A plane infinite electric grating is made of equal and equidistant parallel thin metal plates, the distance between their successive central lines being tt, and the breadth

of each plate 2 sin ~ l ( -= j . Shew that when the grating is electrified to constant

potential, the potential and charge functions V, U in the surrounding space are given by the equation

sin ( U+ iV) — K sin (x+iy).

Deduce that, when the grating is to earth and is placed in a uniform field of force of unit intensity at right angles to its plane, the charge and potential functions of the portion of the field which penetrates through the grating are expressed by

U+iV-(x+ty),

and expand the potential in the latter problem in a Fourier Series.

80. A cylinder whose cross-section is one branch of a rectangular hyperbola is maintained at zero potential under the influence of a line-charge parallel to its axis and on the concave side. Prove that the image consists of three such line charges, and hence find the density of the induced distribution.

81. A cylindrical space is bounded by two coaxial and confocal parabolic cylinders, whose latera recta are 4a and 4b, and a uniformly electrified line which is parallel to the generators of the cylinder intersects the axes which pass through the foci in points distant c from them (a>c> b). Shew that the potential throughout the space is

A log

wr" cos - 77- 1 r- sin -

cosh "■ — cos

•J

7T7- cos - 77- 1 v sin - + c^-a- -b* 1 1

cosh - + cos x , |

c^-64 a?-b* J

where r, 0 are polar coordinates of a section, the focus being the pole. Determine A in terms of the electrification per unit length of the line.

Examples 297

82. An infinitely long elliptic cylinder of inductive capacity K, given by g = a where x+ iy = c cosh (£+ it]), is in a uniform field P parallel to the major axis of any section. Shew that the potential at any point inside the cylinder is

p 1-fcotha A-fcoth a*

83. Two insulated uncharged circular cylinders outside each other, given by rj = a and r/= — 8 where x + iy=ctsm ^(i + ir]), are placed in a uniform field of force of potential Fx. Shew that the potential due to the distribution on the cylinders is

ofv, xnew(,,"a)sinhn/3 + e-w(,?+^sinhna . _ x N ' sinh?i(a + /3) s

84 Two circular cylinders outside each other, given by r/=a and t)= — 8 where

a;+t>=ctanf (f+iij),

are put to earth under the influence of a line-charge E on the line x=0, y = 0. Shew that the potential of the induced charge outside the cylinders is

. r,v 1 ewasinh n(n + 8) + en sinhrc (a-n)

• 4A2, ■£, , s\ " cos n£ + constant,

n sinh n(a + 8) s '

the summation being taken for all odd positive integral values of n.

85. The cross-sections of two infinitely long metallic cylinders are the curves

(#2+#2 + c2)2-4e2.£2=a4 and (.r2+y2 + c2)2-4c2a72 = &*,

where h>a>c. If they are kept at potentials V1 and V2 respectively, the intervening space being filled with air, prove that the surface densities per unit length of the electricity on the opposed surfaces are

VlV\ J^+tf and Fg Y\ v^+? 47ra2 log - 47r62 los; -

respectively.

86. What problems are solved by the transformation

1

where a > 1 ?

87. What problem in Electrostatics is solved by the transformation

x + iy = en (4> + i^), where ^ is taken as the potential function, <£ being the function conjugate to it ?

88. One half of a hyperbolic cylinder is given by 17= ±tj1 , where | »?i | <^-, and £, rj are

given in terms of the Cartesian coordinates x, y of a principal section by the trans- formation

x+iy = c cosh (£ + irj).

The half-cylinder is uninsulated and under the influence of a charge of density E per unit length placed along the line of internal foci Prove that the suri'ace density at any point of the cylinder is

• El ij^cm cosh ^- Vcosh 2£ - COS 2ny

298 Methods for the Solution of Special Problems [ch. vm

89. Verify that, if r, s be real positive constants, z=x + iy, a = pe1^, - = - + -, the

C i s

neld of force outside the conductors x2+y2 + 2sx=0, x'z+y2-2rx=0 due to a doublet at the point z = a, outside both the circles, of strength p, and inclination a to the axis, is given by putting

^+iF=^|e*-(--2«COtCff(I-iVe-««-^COtCff(I-I)},

where z=a0 is the inverse point to 2= a with regard to either of the circles.

90. A very thin indefinitely great conducting plane is bounded by a straight edge of indefinite length, and is connected with the earth. A unit charge is placed at a point P. Prove that the potential at any point Q due to the charge at P and the electricity induced on the conducting plane is

11 _,/ 1 0-4A 11 _,/ 1 <f> + 4>'

pn-cos-M cos ' --7377; -cos~M COS

Pty n \ o- 2 J P Q TT \o- 2

where P' is the image of P in the plane, the cylindrical coordinates of Q and P are (r, <p, z), (r', <p', z1), the straight edge is the axis of z, the angles <f>, <f>' lie between 0 and 2ir, (f> = 0 on the conductor,

f(r + r')2 + (2-/)2l*

*-\—4& j '

and those values of the inverse functions are taken which lie between \ir and jr.

91. A semi-infinite conducting plane is at zero potential under the influence of an electric charge q at a point Q outside it. Shew that the potential at any point P is given by

1

' 2/7"

{cosh, - cos (*-«,)} *tan-i^cQsh|;;_co3|^

/^ ,«-!, , /cosh in + COS A (6 + d{)~\ -{cosh,-cos(^^)} Han- ^ -^—A^^t

where r, 8, z are the cylindrical coordinates of the point P, (rj, dlf 0) of the point Q, 8-0 is the equation of the conducting plane, and

2/Tj cosh t) — r3 + rx2 + *2.

Hence obtain the potential at any point due to a spherical bowl at constant potential, and shew that the capacity of the bowl is

-jl+^4,

7T [ Sill a)

where a is the radius of the aperture, and a is the angle subtended by this radius at the centre of the sphere of which the bowl is a part.

92. A thin circular conducting disc is connected to earth and is under the influence of a charge q of electricity at an external point P. The position of any point Q is denoted by the peri-polar coordinates p, 8, 0, where p is the logarithm of the ratio of the distances from Q to the two points R, S in which a plane QBS through the axis of the disc cuts its rim, 8 is the angle RQS, and <p is the angle the plane QRS makes with a fixed plane through the axis of the disc, the coordinate 8 having values between - n and + ir, and changing from + ir to — tt in passing through the disc. Prove or verify that the potential of the charge induced on the disc at any point Q (p, 8, <p) is

QP 2nsiu1^os^60^seoh^ ~QP'l

• • -sin-1 {-cos£(# + #o)sech!«} >

Examples 299

where p0, 60, cpQ are the coordinates of P, 60 being positive, the point P' is the optical image of P in the disc, a is given by the equation

cos a = cosh p cosh p0 — sinh p sinh p0 cos ((p — <p0),

and the smallest values of the inverse functions are to be taken.

Prove that the total charge on the disc is - qdojir'.

Explain how to adapt the formula for the potential to the case in which the circular disc is replaced by a spherical bowl with the same rim.

93. Shew that the potential at any point P of a circular bowl, electrified to potential G, is