A method employing the use of toroidal functions is introduced for calculating the scalar potential and the electric field from a charged conducting ring. This method is an alternative to the well-known elliptic integral formulation and is usually easier to formulate than the elliptic integral solution.

1.
F. R.
Zypman
, “
Off-axis electric field of a charged ring
,”
Am. J. Phys.
74
,
4
,
295
300
(
2006
).
2.
J. D.
Jackson
,
Classical Electrodynamics
(
Wiley
, New York,
1999
), 3rd ed., pp.
117
119
, 181–183.
3.
J. A.
Stratton
,
Electromagnetic Theory
(
McGraw-Hill
, New York,
1941
), Probs. 3, 4, 7.
4.
W. R.
Smythe
,
Static and Dynamic Electricity
(
McGraw-Hill
, New York,
1968
), 3rd ed., pp.
204
, 290–291.
5.
P. M.
Morse
and
H.
Feshbach
,
Methods of Theoretical Physics
(
McGraw-Hill
, New York,
1953
), Part I, p. 744, Part II, pp.
1301
1304
.
6.
Z. X.
Wang
and
D. R.
Guo
,
Special Functions
(
World Scientific
, Teaneck, NJ,
1989
), Problem 64.
7.
N. N.
Lebedev
,
Special Functions and Their Applications
(
Prentice Hall
, Englewood Cliffs, NJ,
1965
), pp.
186
188
.
8.
P.
Moon
and
D. E.
Spencer
,
Field Theory for Engineers
(
Van Nostrand
, Princeton, NJ,
1961
), pp.
368
376
.
9.
R. W.
Brown
,
H.
Fujita
,
S. M.
Shvartsman
,
M. R.
Thompson
,
M. A.
Morich
,
L. S.
Petropoulos
, and
V. C.
Srivastava
, “
New applications of inverse methods in the design of MRI coils
,”
Int. J. Appl. Electromagn. Mech.
9
,
277
290
(
1998
).
10.
C.
Snow
, “
Magnetic fields of cylindrical coils and annular coils
,”
Natl. Bur. Stand., Appl. Math. Series
38
,
1
29
(
1953
).
11.
O.
Kwon
,
C.
Surussavadee
,
M. V.
K Chari
,
S.
Salon
, and
K.
Sivasubramaniam
, “
Analysis of the far field permanent-magnet motors and effects of geometric asymmetries and unbalance in magnet Design
,”
IEEE Trans. Magn.
40
,
2
,
435
442
(
2004
).
12.
J.
Selvaggi
,
S.
Salon
,
O.
Kwon
, and
M. V. K.
Chari
, “
Calculating the external magnetic field from permanent magnets in permanent-magnet motors–An alternative method
,”
IEEE Trans. Magn.
40
(
5
),
3278
3285
(
2004
).
13.
J.
Selvaggi
,
S.
Salon
,
O.
Kwon
, and
M. V. K.
Chari
, “
Application of a toroidal harmonic expansion for computing the magnetic field from a balanced 6-pole permanent-magnet motor
,” ICEM 2006, XVII International Conference on Electrical Machines, Paper Code PSA1-20 (
2006
).
14.
J.
Selvaggi
,
S.
Salon
,
O.
Kwon
, and
M. V. K.
Chari
, “
A general method for calculating the external magnetic field from a cylindrical magnetic source using toroidal functions
,” Electric Machines Technology Symposium, Philadelphia, PA, May 22–24,
2006
.
15.
H. S.
Cohl
,
J. E.
Tohline
, and
A. R. P.
Rau
, “
Developments in determining the gravitational potential using toroidal functions
,”
Astron. Nachr.
321
,
363
372
(
2000
).
16.
H. S.
Cohl
and
J. E.
Tohline
, “
A compact cylindrical Green’s function expansion for the solution of potential problems
,”
Astron. J.
527
,
86
101
(
1999
).
17.
H. S.
Cohl
,
A. R. P.
Rau
,
J. E.
Tohline
,
D. A.
Browne
,
J. E.
Cazes
, and
E. I.
Barnes
, “
Useful alternative to the multipole expansion of 1r potentials
,”
Phys. Rev. A
64
,
052509
1
(
2001
).
18.
C. J.
Bouwkamp
and
N. G.
de Bruijn
, “
The electrical field of a point charge inside a cylinder, in connection with wave guide theory
,”
J. Appl. Phys.
18
,
562
577
(
1947
).
19.
E. W.
Hobson
, “
On Green’s function for a circular disk, with applications to electrostatic problems
,”
Trans. Cambridge Philos. Soc.
18
,
277
291
(
1900
).
20.
I. S.
Gradshteyn
and
I. M.
Ryzhik
,
Table of Integrals, Series, and Products
(
Academic Press
, New York,
1980
), 4th ed., pp.
732
, 709.
21.
C.
Snow
,
Formulas for Computing Capacitance and Inductance
(National Bureau of Standards Circular, 544,
1954
), pp.
13–17
.
22.
C.
Snow
,
Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory
(National Bureau of Standards Applied Mathematics Series 19,
1952
), pp.
228
252
.
23.
Tecplot ⟨www.tecplot.com/⟩.
24.
Maple, ⟨www.maplesoft.com/⟩.
25.
H. E.
Fettis
, “
A new method for computing toroidal harmonics
,”
Math. Comput.
24
,
111
,
667
670
(
1970
).
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