av,1 Ptf. 1 I,ve. Math
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87
2.9 BIPOLAR COORDINATES (,5, z)
EXERCISES
2.7.1
Let cosh u
iii , cos v ----- (12, -.:
q3. Find the new scale factors hq, and h". ., ., q i — 11 72 ' ' It " - a , )
(qi - — 1
V= = h„., -- a
or -- 11,
(
2.8 Parabolic Cylindrical Coordinates (,:=,
q,
The transformation equations, X= = )0/2
•
(2.79)
Z=
generate two sets of orthogonal parabolic cylinders (Fig. 2.6). By solvin g, Eq. 2.79 for and ti we obtain the following: I. Parabolic cylinders, ‘; = constant.' (2.74)
2.75) ( (2.76)
2. Parabolic cylinders, iI = constant,
the li
\ -/ 1
major one.
0
tl <
3. Planes parallel to the .vi-plane, = constant,
— Co < < CO.
From Eq. 2.6 the scale factors are
= h 4 (2.77)
— Co < <
h
II,
+ /12)12, 11 2 ) 1 2 +
(2.80)
It,. -=
\
2.9 Bipolar Coordinates
(2.78)
hapter 6 ∎s hen
41)
This is an oddball coordinate system. It is not a degenerate case of the confocal ellipsoidal coordinates. Equation 2.1 is not completely separable in this system even for k 2 = 0 (cf. Exercise 2.9.2). It is included here as an example or how an unusual coordinate system may be chosen to lit a problem. The parabolic e N lindcr c.); constant is invariant to the negati ve to cover negative values of .v.
sign of
We must let ,; (or id
go
88
2 COORDINATE SYSTEMS
The transf(
Dividing Eq.
Using Eq. 2.
Usin g Eq. 2.
From Eqs I. Circula
2. CircuL
3. Planes
When 17 y = 0. Simil to a point, t Eq. 2.84 pa y = 0 are s( The scale
FIG. 2.6 Parabolic cylindrical coordinates. (Top) Cross section
To see h (a, 0), (—a
BIPOLAR COORDINATES (5, 77 , z)
2.9
89
The transformation equations are x= =
a
cosh
sinh — cos 'C.
(2.81a)
a sin C' cosh q — cos
(2.81b)
z = z.
(2.81c)
Dividing Eq. 2.81a by 2.81h, we obtain x
sinh sin;
y
(2.82)
Usin g Eq. 2.82 to eliminate C from Eq. 2.81a, we have (x — a coth ) + 2
y2
= 2 C/
CSC112
(2.83)
• Eq. 2.82 to eliminate q from Eq. 2.81h, we have X
2 + (3' -
a
cot ,;) 2 = a 2
csc2
(2.84)
From Eqs. 2.83 and 2.84 we may identify the coordinate surfaces as follows:
•
I. Circular cylinders, center at i =
a
= constant, 2. Circular cylinders, center at .v =
a
cot *C, 0 5 C coth -
= constant,
27t.
CC < 1/ < CC .
3. Planes parallel to .vy-plane, =
constant,
- SO < < Cf-
when coth q —* I and csch q 0. Equation 2.83 has a solution x = a, q — cr.>, a solution is .v = —a, r = 0, the circle degenerating 0. Similarly, when y= to a point, the cylinder to a line. The family of circles (in the .vy-plane) described by Eq. 2.84 passes through both of these points. This follows from noting that .v = +a, y = 0 are solutions of Eq. 2.84 for any value of C. The scale factors for the bipolar system are h ,
•
a cosh q — cos c
h: =
a cosh q — cos c
11, =
=
11 3 =
-= 1.
(2.85)
To see how the bipolar system may be useful let us start with the three points and i )„,_ at angles of 0, 0), (—a, 0), and (.v, y) and the two distance vectors
(a,
90
2 COORDINATE SYSTEMS
FIG. 2.7 Bipolar coordinates
and 0 2 from the positive .v-axis. From Fig.2.8
Pi = (x — a) 2 + y2, = (x + a) 2 + y2
( 2.86 )
and tan 0, = x— (2.87) tan 0 2
FIG. 2.8
= x
+a
We define' 012 =
P2
in
(2.88a)
Pi (2.88b)
02.
= Ol
By taking tan 5 I2 and Eq. 2.87 tan C, 2
=
tan 0, — tan 02 + tan 0 1 tan 02 1 , /( x — I
1 The notation In is used to indicate loge.
a) — y/(V
+ y 2 /(X
(2.89) a)
(1 )( x + ( 1)
2.9 BIPOLAR COORDINATES
(6,
7) ,
91
z)
From Eq. 2.89, Eq. 2.84 follows directly. This identifies ,; as 5 12 = 0, — 0 2 . Solving Eq. 2.88a for p 2 /p, and combining this with Eq. 2.86, we get + +y2 o z, v(2.90) + y2 pi (x Multiplication by e - " 2 and use of the definitions of hyperbolic sine and cosine produces Eq. 2.83, which identifies q as th, = In (0,1p,). The following example exploits this identification. EXAMPLE 2.9.1
An infinitely long strai g ht wire carries a current I in the negative :-direction. A second wire, parallel to the first, carries a current / in the positive z-direction. Using dA =
.8
dk
47r
(2.91)
r
find A, the magnetic vector potential, and B, the magnetic inductance. From Eq. 2.91 A has only a z-component. Integrating over each wire from 0 to P and oo, we obtain taking the limit as P
FIG. rents
clz
Rol
Poi ( lirn 2 In —
p—
/
P +
+ P 2
2 In
p2
P +
Cur-
(2.92)
\/ + Z2
+
1 — In(z + + \/1) :1, + z 2 )1 0'
— lim n-1 47t 1"--• 47
2 Jo
Antiparallel electric
CI= \
P
11 ° 1 lim( j. A_ =-47r p—. (),//); + z2
A"
2.9
p,
.
(2.93)
Pi.
This reduces to .-1_
=—
/101 1 1 2 P0 I , i 11 — = — — il.
_7 1
(2.94)
p i 27
So far there has been no need for bipolar coordinates. Now. however. let us calculate the magnetic inductance B from B = V x A. From Eqs. 2.22 and 2.85
(cosh ii — cos (;) 2 B= u 22
1l440
Ito °
c,
il -- _.
(IL,'"
(. 11
0
0
k 7.— ('.7
—I1„1
(cosh — cos = %,o
a
(2.95) )TE
92
2 COORDINATE SYSTEMS
The magnetic field has only a 4 0 -component..The reader is ur g ed to try to compute B in some other coordinate system. We shall return to bipolar coordinates in Sections 2.13 and 2.14 to derive the toroidal and bispherical coordinate systems.
EXERCISES
2.9.1 Verify that the surfaces 6
h i and c; are orthogonal by the following methods: (a) Show that the slope of one surface (the intersection with a = constant)-plane) is the ne gative reciprocal of the slope of the other surface. (b) Calculate q„.
2.9.2 (a) Show that Laplace's equation, V 0(6, 7),
— 0 is not completely separable in bipolar coordinates. (b) Show that a complete separation is possible if we require that tb — 0(6, 7)), that is, if we restrict ourselves to a two-dimensional system.
2.9.3 Find the capacitance per unit length of two conducting cylinders of radii b and c and of infinite length, w ith axes parallel and a distance
C- 2.9.4
apart.
277E0
-
7/2
As a limiting case of Exercise 2.9.3, find the capacitance per unit length between a conducting cylinder and a conductin g infinite plane parallel to the axis of the cylinder. 27reo
n 2.10 Prolate Spheroidal Coordinates (a, 1 ,, (p) Let us start with the elliptic coordinates of Section 2.7 as a two-dimensional system. We can generate a three-dimensional system by rotating about the major or minor elliptic axes and introducing (I) as an azimuth angle (Fig. 2.10). Rotating first about the major axis gives us prolate spheroidal coordinates with the following nate surfaces : coordinate 1. Prolate spheroids, =
constant,
0 u
< v4.
2. Hyperboloids of two sheets, c= constant,
0
3. Half planes through the .7-axis, = constant,
0
2n.
