XII. On Green's Function for a Circular Disc, with applications to Electro static Problems. By E. W. Hobson, Sc.D., F.R.S. [Received 7 October 1899.] The main object of the present communication is to obtain the Green's function for the circular disc, and for the spherical bowl. The function for these cases does not appear to have been given before in an explicit form, although expressions for the electric density on a conducting disc or bowl under the action of an influencing point have been obtained by Lord Kelvin by means of a series of inversions. The method employed is the powerful one devised by Sommerfeld and explained fully by him in the paper referred to below. The application of this method given in the present paper may serve as an example of the simplicity which the consideration of multiple spaces introduces into the treatment of some potential problems which have hitherto only been attacked by indirect and more ponderous methods. The System of Peri-Polar Coordinates. 1. The system of coordinates which we shall use is that known as peri-polar co ordinates, and was introduced by C. Neumann* for the problem of electric distribution in an anchor-ring. A fixed circle of radius a being taken as basis of the coordinate system ; in order to measure the position of any point P, let a plane PAB be drawn through P containing the axis of the circle and intersecting the circumference of the PA circle in A and B ; the coordinates of P are then taken to be p = log , 6 which is the angle APB, and <f> the angle made by the plane APB with a fixed plane through the axis of the circle. In order that all points in space may be represented uniquely by this system, we agree that 6 shall be restricted to have values between — it and 7r, a discontinuity in the value of 6 arising as we pass through the circle, so that at points within the circumference of the circle, 6 is equal to it, on the upper side of the circle, and to — tt on the lower side of the circle, the value of 0 being zero at all ' points in the plane of the circle which are outside its circumference. As * Theorie der Elektricitate- und WSrme-Vertheilung in einem Ringe. Halle, 1864.
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XII. On Green's Function for a Circular Disc, with applications to Electro
static Problems. By E. W. Hobson, Sc.D., F.R.S.
[Received 7 October 1899.]
The main object of the present communication is to obtain the Green's function
for the circular disc, and for the spherical bowl. The function for these cases does not
appear to have been given before in an explicit form, although expressions for the
electric density on a conducting disc or bowl under the action of an influencing point
have been obtained by Lord Kelvin by means of a series of inversions. The method
employed is the powerful one devised by Sommerfeld and explained fully by him in
the paper referred to below. The application of this method given in the present paper
may serve as an example of the simplicity which the consideration of multiple spaces
introduces into the treatment of some potential problems which have hitherto only been
attacked by indirect and more ponderous methods.
The System of Peri-Polar Coordinates.
1. The system of coordinates which we shall use is that known as peri-polar co
ordinates, and was introduced by C. Neumann* for the problem of electric distribution
in an anchor-ring. A fixed circle of radius a being taken as basis of the coordinate
system ; in order to measure the position of any point P, let a plane PAB be drawn
through P containing the axis of the circle and intersecting the circumference of the
PAcircle in A and B ; the coordinates of P are then taken to be p = log , 6 which
is the angle APB, and <f> the angle made by the plane APB with a fixed plane
through the axis of the circle. In order that all points in space may be represented
uniquely by this system, we agree that 6 shall be restricted to have values between
— it and 7r, a discontinuity in the value of 6 arising as we pass through the circle,
so that at points within the circumference of the circle, 6 is equal to it, on the upper
side of the circle, and to — tt on the lower side of the circle, the value of 0 being
zero at all ' points in the plane of the circle which are outside its circumference. As
* Theorie der Elektricitate- und WSrme-Vertheilung in einem Ringe. Halle, 1864.
278 Dr HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
P moves from an infinite distance along a line above the plane of the circle up to
any point inside the circle, and in its plane, 0 is positive and increases from 0 to ir,
whereas as P moves from an infinite distance along a line below the plane of the
Fm. l.
circle up to a point within the circumference, 6 is negative, and changes from 0 to
— tt. The coordinate (f> is restricted to have values between 0 and 2ir, and the co
ordinate p may have any value from — oo to + oo , which correspond to the points A, B
respectively. The system of orthogonal surfaces which correspond to these coordinates
consists of a system of spherical bowls with the fundamental circle as common rim, a
system of anchor-rings with the circle as limiting circle, and a system of planes through
the axis of the circle. If we denote by f the distance GN of P from the axis of
the circle, and by z the distance PN of P from the plane of the circle, the system
£ cos <p, f sin <f>, z will be a system of rectangular coordinates, which can of course be
expressed in terms of p, 0, <f>. Let the lengths PA, PB be denoted by r, r respec
tively, then r/r = log p ; we have
2rr' cos 6 = r2 + r'2 - 4a2 = 2rr' cosh p - 4a2,
2aa
hence rr =—;cosh p — cos 6
Again, z .2a = rr' sin 8,
a sin 0hence z — ; •
cosh p — cos0
also since r2 + r'2 = 2a2 + 2CP1,
we have CP2 = rr' cos 0 + a2,
Curtright
Callout
Oops! log(r/r')=rho
WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 279
, nry. oCOShp + CO8 0
whence we find Cr3 = a"—=-- A ,cosh p — cos (f
hence a'sinhV
(cosh p — cos 0)2 '
thus £, z are expressed in terms of p, 0 by means of the formulae
y a sinh p a sin 0
cosh p — cos 0 ' cosh p — cos 0 '
2. To express the reciprocal of the distance D between two points (p, d, <f>) and
(j00, do, <f>o), we substitute for f, z and f„, z0 in the expression
5 = {(* - *„)2 + fa + £>a - 2ff. cos (* - *,))-»,
their values in terms of p, 0 and p0> 60; we then find
1 _ 1 (cosh p — cos 0)>> (cosh p0 — cos 0o)l
D ~ aV2 jcosh a - cos (0 - 0^ '
where cosh a denotes the expression cosh p cosh p0 — sinh p sinh p0 cos (</> — If we
suppose the expression {cosh a — cos(0 — 0o))~* is expanded in cosines of multiples of
2 f ff cos Tiv^r
6—00, the coefficient of cosm(0 — (?„) is — I .—; ——7-r,d-Jr which is equal* tottJo (cosh a — cos y)' T 1
2 V2Qm-i (cosh o) when QOT-j denotes the zonal harmonic of the second kind, of degree
7T
m — \ ; thus i = — (cosh p — cos #)* (cosh p„ — cos 2 2Q,„_i (cosh a) cos m(0— 0O), where
2 JJ 7ra
the factor 2 is omitted in the first term, for which m = 0. The series in this expres
sion for 1/D may be summed, by substituting for Qm_j(cosha) the expression
1 f °° e-mti
VfJ. (cosh u- cosh a*du< {l0C- dt P" 519>;
we find
i = ^71 (C°Sh P " C°S 6)i (C°Sh * " C°S ^ [ (cosh n- cosh a)i ^ + 22<r'"" cos *>l *.
and thus we have the formula
* = —~= (cosh p — cos <?)4 (cosh p0 - cos j —- —_^ sinhjt ^
*' ira\2 J* vco*h « — cosh a cosh u — cos(0 — 0O)
where a is given by
cosh a = cosh p cosh p0 — sinh p sinh p0 cos (</> — </>„).
* See page 521 of my memoir "On a type of spherical harmonics of unrestricted degree, order, and argument,"
Phil. Trant. Vol. clxxxtii. (1896) A.
Curtright
Callout
OK
280 Dr HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
Green's Function for the Circular Disc.
3. In order to obtain Green's function for an indefinitely thin circular disc, which
we take to coincide with the fundamental circle of our system of coordinates, we shall
apply the idea originated and developed by Sommerfeld*, of extending the method of
images by considering two copies of three-dimensional space to be superimposed and
to be related to one another in a manner analogous to the relation between the sheets
of a Riemann's surface. In oiir case we must suppose the passage from one space to
the other to be made by a point which passes through the disc ; the first space is
that already considered, in which 0 lies between — ir and tt ; for the second space we
shall suppose that 0 lies between ir and Sir, thus as a point P starting from a point
in the first space passes from the positive side through the disc, it passes from the
first space into the second space, the value of 0 increasing continuously through the value
7r, and becoming greater than ir in the second space. In order that a point P starting
from a position P0(p0, 0O, <f>0), say on the positive side of the disc, may after passing
through the disc get back to the original position P0, it will be necessary for it to
pass twice through the disc; the first time of passage the point passes from the first
space into the second space, and at the second passage it comes back into the first
space. Corresponding to the point p0, 0O, <p0 where 0O is between — tt and ir, is the
point (p0, 0o + 2ir, <f>0) in the second space, whereas the point (p0, 0o + 4:it, <f>0) is regarded
as identical with the point (p0, 0O, </>„). The section of our double space by a plane
which cuts the rim of the disc is a double-sheeted Riemann's surface, with the line of
section as the line of passage from one sheet into the other. Let p0, 0O, <f>0, be the
coordinates of a point P in the first space, on the positive side of the disc, thus
0<^0<7r; taking the expression for the reciprocal of the distance of a point Q (p, 0, <p)
from P, given in the last article, we have, since
. , , sinh ^ u , sinh ^ usinh ?< _ 1 2 1 2
cosh it - cos (0 - 0,) ~ 2 ,1 I ,a dN + 2 ,1 \ ,a 'cosh ~u — cos s (0 — 0O) cosh ^ u + cos ^ (0 — 0O)
11 f" 1 sinh | m
pn = o /fa, (cosh P ~ cos d)h (cosh P" ~ cos e$ I 7- , r= i i du
PQ 2V2?m h VcoshM-coshacoshi«-cosh0-0o)
+ o -,-»— (cosh P ~ cos (cosh Po - cos 0$ i ~r=T==v" 1 ; du >
2V2?ra K Vcoshw-coshacoshi«-cosi(^-^0-2^)
we thus see that 1/PQ is expressed as the sum of two functions, the first of which
involves the coordinates p0, 0O, <j>0 of P, and the second is the same function of the
* See his paper "Ueber verzweigte Potentiate im Raume," Proc. Lond. Math. Soc. Vol. am
WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 281
coordinates p0, 0„ + 2ir, <£0 of the point P in the second space, which corresponds to P.
If Q moves up to and ultimately coincides with P, we have cosh a = 1 ; it will then
be seen that the first function becomes infinite at the lower limit, but that the second
one remains finite at that limit.
Consider then the function TF(p0) 0O, <j>0) given by
W(a>» 0o, <£o) = —i=— (°osh p - cos 0)1 (cosh p0 — cos 0O)*
2 v27ra
r.
j sinh i u
a Vcosh it — cosh a „, i 1 ., „ 1 ,a acosh ^ w — cos £ '
the above equation may be written
= TT(p0, 0O, <f>e) + W (Po, 0, + 2tt, <£„).
It is clear that the function W is uniform in our double space as it is unaltered
by increasing 0 by 4ir ; it will now be shewn that it is a potential function. We
may express W in the form
W = }_ (cosh p - cos 0)* (cosh p0 - cos f 1 =. jl + 22e-*mu cos^(0 - 0O)\ du,
2 v 2ira J > v cosh w — cosh a I * )
which may be written in the form
1 00 tn )= 2-n-a ^C0S^ P ~ 008 ^ ^C°sh P" ~ 003^ l^-* ^C°sh a) + 2 ^ @™ L(cosh a) cos (0 - 0o)j ,
since the formula
V2 Q, (cosh a) = (coshM_CQsha)i
holds for all values of n such that the real part of n + % is positive (loc. cit. p. 519).
Now (cosh p — cos 0)* (cosh p„ — cos cos s{0 — 0„) Q,_j (cosh a) is a potential form whatever
8 may be, and thus IT is a potential function, and is expressible in the form
W = (cosh p — cos #)* (cosh p„ — cos 0O)* (cosh a) + 2Q0 (cosh a) cos | (0 — 0O)
+ 2Qj (cosh a) cos (0 - 0„) + . . .J ,
the value of W {p„, 0O + 2ir, <£„) being
(cosh p - cos 0)1 (cosh />, — cos 0O)* |q_j (cosh o) — 2Q0 (cosh a) cos | (0 — 0O)
1+ 2Q_j (cosh a) cos (0 — 0O) - . . . j ;
the two expressions added together give the expansion of 1/D obtained in Art. 2.
Vol. XVIII. 36
282 Dr HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
4. To evaluate the definite integral in the expression for W, write cosh | u = a ,
cosh | a = a, cos i (0 — <?„) = t, then
f / 1 ^ rfu = V2 f
J. Vcosh«-coshacoshl _cosl(^_5o) ^ V^-a»(a;-T)
— —
V22 /tt . ,T\
where the inverse circular function has its numerically least value ; we thus obtain the
expression
w 1 (cosh p - cos 0)* (cosh p„ - cos 0o)i [tr , . _. ( 1 ... , 1 "llfr = =i 1-—r ,a a mi d o + 8in-Mcos s (0 - 0„) sech s ctM ,
7raV2 {cosha-co8(0-0o)|* |_2 I 2V 0/ 2 J J
which may also be written in the form
w=fq[1+1 sin_i !cos \ {d - e<,) 8ech I '
This expression W has the following properties:—it is, together with its differential
coefficients, finite and continuous for all values of p, 0, <f> in the double space, except
at the point P in the first space, and it satisfies Laplace's equation ; when Q coincides
with P, the inverse circular function approaches jr , and the function becomes infinite
as 1/PQ; when however Q approaches the point in the second space which corresponds
TTto P, the inverse circular function approaches — ^ , and the function does not become
infinite. The expression (1) is then the elementary potential function which plays the
same part in our double space as the ordinary elementary potential function 1/PQ does
in ordinary space.
5. In order to find a potential function which shall vanish over the surface of
the disc, and shall throughout the first space be everywhere finite and continuous
except at a point P (p0, 0O, <f>0) in the first space on the positive side of the disc
(0<#o<'""), we take the function W (p0, 0O, </>0) — TP (p0, 2ir — 0o, <£„) which is the
potential for the double space due to the point P and its image P'(p0, 27r-#0» </><>)>
which is situated in the second space at the optical image of P in the disc. This
function is equal to
1 (cosh p- cos 0)1 (cosh p„ - cos 0„)* Ttt , . _, f \ ,a u 1 11{cosh a-.co. (0-0^ L2+8m {C09 2^-^9ech2a|J
i (cosh g - cos 01 (cosh p +- co. gty rv {_ cos i + sech i n
tV2 |cosh o+ cos(0 + #„)}» \_2 { 2X 2 J J
WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 283
which is the same thing as
u = ¥Q [l + w sin-1 {cos \ ^ ~ e^ 8ech \ ai ~ Fq [i + \ sin-1 {~ 008 \^e+ 6^ sech I "}
• (2),
where P' is the optical image of P in the disc. On putting in this expression (2), for
U, the values 0=tt, 0 = — 7r, and remembering that over the disc PQ = PQ, we verify
at once that U vanishes on both surfaces of the disc. If Q coincides with the point
(p0, —Bo, <j>0) the function U remains finite.
The Green's function Gpq which is a function that is finite and continuous throughout
the whole of ordinary (the first) space, everywhere satisfies Laplace's equation, and is
equal to 1/PQ over both surfaces of the disc, is given by Gpq = ^.— U, hence the
required value of Gpq is
Gpq= pQ ^-^sin-'jcosi (0-0,) sech |o|J +~ ^^sin"1 ■ -cos| (0 + 0o)sech ^ a ■
= j^q . ~ cos-1 jcos | {0 - 0O) sech | aj + Jtq • ^ cos-1 ■ cos | (0 + 0„) sech | aj (3),
the numerically smallest values, as before, of the inverse circular functions being taken.
It will be observed that in interpreting these formulae (2) and (3), the second copy of
space, having served its purpose, may be supposed to be removed.
The Distribution of Electricity on a Conducting Disc under the influence
of a Charged Point.
6. If we suppose a thin conducting disc to be placed in the position of the funda
mental circle of the coordinate system, to be connected to earth, and influenced by a
charge q at the point P (p0, 0O, <f>0) on the positive side, the potential of the system at
any point Q is qU where U is given by (2), and the potential of the charge on the
disc is —q.GpQ. We shall now throw these potentials into a more geometrical form.
We have
r i i ) cos g (0-0.)
-1 jcos 2 (0 - 0.) sech £ 4 = tan"1 i . .
cosha2«-cosJ2(0-0o)
Vlcosi(0-0o)
= tan"1
(Vcosh a — cos (0 - 0O)J
now take an auxiliary point L, of which the coordinates are p0, 0 + tt, <f>a, the upper
or lower sign being taken according as 0 is positive or negative (— 7r<0<7r). Thus L
and Q are always on opposite sides of the disc ; using the formulae of Art. 1, we find
CL>-a* = -2a'C08^, a'-<7<? = " 2aW
cosh p0 + cos 0 ' cosh p — cos 0 '
PL _\ 1 +cos(0-0o) {* fcosh p - cos 0) *
PQ (cosh a - cos (0 — 0o)j (cosh p0 + cos 0
36—2
284 Dr HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
hence
sin-1 {cos I (0 - 0t) sech ^ a} = ± tan- i^qfJ \
p. ^
V J
"
Fio. 2.
in order to determine the sign on the right-hand side, we observe that the inverse
sine is positive unless 6 lies between — (ir — 0O) and — it, that is unless Q lies within the
sphere passing through P and the rim of the disc, and is on the negative side of the
disc; thus the sign on the right-hand side is to be taken positive unless Q lies within
this spherical segment.
Similarly we find
sin- {- cos 1(6 + 6a) sech i«| = ? tan- (£| *Jg^^)
where the negative sign is to be taken unless Q is on the positive side of the disc and
within the sphere which contains the rim and the point P'. We have thus as the
expression for the potential of the system at any point Q (p, 6, </>)
2PQ
2
1 + — tantr
JPL /a'-CQ\
\PQV CL*-a'J
9
2P'QIT2
IT
when the ambiguous signs are assigned in accordance with the above rules.
The auxiliary point L may be found from the following construction :
Draw a spherical bowl through the rim of the disc on the opposite side to that on which
Q lies, and equal to a similar bowl which passes through Q; draw a plane PA'R through
P and the axis, cutting the rim in A', B ; this plane intersects the bowl in a circle ; on
this circle L lies, and is found by taking it so as to satisfy the relation
LA1 : LB = PA' : PB.
WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 285
2PQ
In the case in which the influencing point is on the axis of the disc, we have p„ = 0,
hence a = p, and the auxiliary point L is on the axis of the disc at the point where this
axis is cut by the sphere through the rim and the point Q, on the opposite side of the
disc to Q; the formulae for the potential then become
v=h D + 1 sin_1 {cos \ {d ~ sech Ip}]-fq [I + 1 sin_1 f cos -2 {d + es> sech i p}_
r 2. /PL /a*-CQ*\] q V. _ 2 _. /P'i /a?-C®\\
the sign in the first bracket is positive unless Q lies in the segment ApB, and the sign in
the second bracket is negative unless Q lies in the segment Ap'B.
7. To find, in the general case, the induced charge on the disc, it is sufficient to
examine the limiting value of the potential at a point Q, as Q moves off to an infinite
distance from the disc in the direction of the axis. In the expression for — q . Gpq given
by (3), let 0 = 0, p = 0, then a=pa, and PQ, FQ become infinite in a ratio of equality;
the expression for the potential of the induced electrification on the disc has therefore
the limiting value
- ^jSq cosrI (cos \S<> sech \ P"
therefore the whole charge on the disc is
2 1
"2 -rr
which is equivalent to
cos_I^cos sech ^p^j ,
2 . (Ja*-CL>\
when I is a point in the plane of the disc which lies on the bisector of the angle APB.
This expression may be interpreted thus:—
Let PL be the bisector of the angle APB, draw the chord NLM perpendicular to
AB ; the total induced charge is
Z NPM (as-q. (»)•
286 Dr HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
QWhen the point P is on the axis of the disc, the induced charge is - q . — , where 6„
IT
is the angle subtended at P by a diameter of the disc.
When P is in the plane of the disc, the angle NPM becomes the angle between the
tangents from P to the circular boundary of the disc.
8. The surface density at any point of the disc is given by the formula
_ l_dV
9 ~ 4tt dv '
when dv is an element of normal and is given by
dv_ ± add
cosh p — cos 6 '
We thus find for the density p0 at the point (p, ir, <f>) on the positive side of the disc,
* = - £ • m i1 + 1 8111-1 (sin \e% sech i■). '
1 U.I CO3;;0„
q 1 cosh p + 1 9.
*J cosh2 - sin'
this expression can be put into a more geometrical form by introducing the auxiliary
point L (p0, 6— ir, <f>0) of Art. 6. The point L is now in the plane of the disc, and external
to the disc ; denoting this position of L by L0, its coordinates are p0, 0, <p0. We have
sin- (sin ^0 sechia) = tan- g§ sj •
which is equal to | - tan-1 sj ) ;
on reducing the second term in the expression for p, remembering that
_ a sin 0„
cosh p0 — cos #o '
we find that it becomes
2tt2 PQ> . PL, V o5^CQr '
and thus the expression for the density at any point Q on the positive side of the