Simple Analytic Expressions for the Magnetic Field of a Circular Current Loop

3 Pages·2001·0.16 MB·English

by  Simpson, James C.

Simple Analytic Expressions for the Magnetic
Field of a Circular Current Loop
James Simpson, John Lane, Christopher Immer, and Robert Youngquist
as shown (Fig. I.). It is assumed that the cross section of the
Abstract - Analytic expressions for the magnetic induction and
its spatial derivatives foracircular loop carrying astatic current conductor is negligible.
z.
are presented in Cartesian, spherical and cylindrical coordinates.
The solutions are exact throughout all space outside the
,L/0
conductor.
I
Index Terms - Circular Current Loop, Magnetic Field, Spatial _v
Derivatives.
I. INTRODUCTION
nflaulxytidcenesxitpyr,essiBo)nsof faorsimthpelemapglanneatirc ciirncduulactrioncur(rmenatgnelotiocp Fig. 1. Circular current loop geometry.
have been published in Cartesian and cylindrical coordinates The vector potential, A, of the loop is given by [3]:
[1,2], and are also known implicitly in spherical coordinates
[3]. In this paper, we present explicit analytic expressions for
12oia [2_ cos_' d_o'
B and its spatial derivatives in Cartesian, cylindrical and
A_r,O) =--_-x .lo _[a2+r2_2arsin Ocos_o"
spherical coordinates for a filamentary current loop. These (l)
results were obtained with extensive use of Mathematica TM _ ta,, 41a [(2-k2)K(k2)-2E(k2)]
and are exact throughout all space outside of the conductor. 4x 4a2+r2+ 2arsinO [ k2 ]'
The field expressions reduce to the well-known limiting cases
and satisfy V •B = 0 and V×B = 0outside the conductor.
where r, 0, and _ are the usual spherical coordinates, and the
These results are general and applicable to any model using
filamentary circular current loops. Solenoids of arbitrary size argument of the elliptic integrals is
may be easily modeled by approximating the total magnetic
induction as the sum of those for the individual loops [4]. The k2 _ 4arsinO (2)
inclusion of the spatial derivatives expands their utility to a2 +r 2 +2arsinO
magnetohydrodynamics where the derivatives are required. Note that we use k2for the argument of the elliptic integrals.
The equations can be coded into any high-level This choice is consistent with the convention of Abramowitz
programming language. It is necessary to numerically evaluate and Stegun [5] where m= k 2.
complete elliptic integrals of the first and second kind, but this For a static field with constant current, the B components in
capability is now available with most programming packages. spherical coordinates are [3]:
II. SPHERICALCOORDINATES l 3
Br - rsin0 _)O(sinOA_) (3)
We start with spherical coordinates because this is the
system usually used in the standard texts. The Cartesian and
cylindrical results in Sections III and IV were derived from the Bo = _r__r(r A_ ) (4)
spherical coordinate results.
The current loop has radius a, is located in the x-y plane,
Be = 0. (5)
centered at the origin, and carries a current / which is positive
Analytic expression for the field components and their
Manuscript received February I, 2001. This work was supported in part
derivatives in spherical coordinates are given below. For
by NASA under Contract No. NAS 10-9800 I.
simplicity we use the following substitutions:
J. C. Simpson is with Dynacs, Inc., KSC, FL 32899 USA (telephone: 321-
867-6937, e-mail: [email protected]). a,2-=a2+r 2-2arsinO,fl 2-a 2+r 2+2arsinO ,k 2=-I-aZ/fl 2, and
J. E. Lane iswith Dynacs, Inc., KSC, FL 32899 USA (telephone: 321-867-
C=-izoI/x.
6939, e-mail: John.Lane-I @ksc.nasa.gov).
C. D. Immer is with Dynacs, Inc., KSC, FL 32899 USA (telephone: 321-
867-6752, e-mail: Christopher.lmmer-1 @ksc.nasa.gov).
R. C. Youngquist is with NASA., KSC, FL 32899 USA (telephone: 321-
867-1829, e-mail: Robert.Youngquist- [email protected]).
Field Components: Field Components:
(12)
Ca2 cosO E(k 2) (6) Cxz _a 2+r 2)E(k 2)_a 2K(k?)]
B._ = 2a-'tip 2
Br = a2fl
_ C [,2 (7) 8,- c,_,: i 2+/_e(k,)_ -'x(k._)]=zs_ (13)
Bo 2azfl-sinO [r +aZ c°s20)E(k2 )-aZK(k 2)] " 2a2flp 2 x
Spatial Derivatives of the Field Components: 8_=7C_._b-_--r 2 )_2.o:_-',1 (14)
OBr _ Ca2 cos8 {[a 4_7r4_6a2r "_cos 20]E(k 2)+
(8) Spatial Derivatives of the Field
Or 2ra 4fl 3
[o2(r__o-')kc(k2)} Components:
qoB_. _ CZ {[a4(_7(322 +a-')+p.(8x__, _ v 2 ))-
08r_ --Co" {[o4_7,_-'rr4 c)x 2a4 fl3 p 4
00 4sin_?fl 3 - + + a2(p4(5x2 +y2)_2p2z-'(2x 2+ y-' )+ 3z4?)_
(9) (15)
cos20(-3a 4+2a2r -'-3r4)+ rZ(2x4 +_,(y2 + z2))]E(k2)+
o_rc-'o.sOil(k2)+
[2_2(+or2-')c2o0s1_(_}2)
r2(2x 4+,(v2+ze))la'K(k-')}
3Bo _ -c {[a__3 4r-'+a2r4 + OBx_ Cxyz {[3a4(3p2_2zZ)_r4(2r-'+p2)_
Or 4a 4fl3rsin 0 _' 2a4fl3p 4
2r 6 +a2(3r2-a2)(a-' +r 2)cos 20+ 2a6-2a-(2p, 4-p 2z-' +3z 4)]E(k2)+ (16)
3a4r 2cos 40]E(k 2)+ (1o) [r2(2r2 + p2 )_a2(5p2_4z 2)+ 2a4]a2K(k2) }
[,,-,(_a4+, 2r2_2r4+
o2_2a_3_2)c2o2s0)]__(_)} _B.r
-- Cx {_D2 a2)2(p2+a2) +
OZ 2a4,83p 2
2z'(a 4 - 602p 2+p4)+z 4(a" + p2 )]E(k 2)_ (17)
OBo b2_,,;_-+_w-+o-')1o-_}(_-')
- -Ccos0 {[5a6 +3a4r2_3a2r4
30 4a4,83sin2 0
+2r 6 +(-3a 6 +2a4r 2 +9a2r 4)cos 20+
OB__ 3B., (18)
a4r 2 cos 40]E(k 2)_ Ox 3y
(11)
3a 6 +2a4r2+a 2r 4 +2r6+
a2(5r -' -a 2)(a2 +r 2)cos 20+ OB3 _ Cz {[a4(y(3z2+a2)+p2(8y2 x2)) -
Oy 2a4f13p 4
( -7aSr+7a3r 3 -4or _ )sin 0+
a2(p4( 5)'-' +x2)- 2p2z 2(2y-' +x-' )- 3z4y) -
o3r(a 2-5r 2)sin 30]K(k 2)} (19)
r4(2y4- 7.2 +Z2))]E(k-')+
[a-'(-7(a-' + 2z2 )-p2(3y2- 2x-' ))+
III. CARTESIAN COORDINATES
rZ(2y4 - 7(x2 + Z2))]a2 K(k -')}
The field components and their derivatives in Cartesian
coordinates are given below. These are easier to use when t)By _ y OB.r (20)
Oz x 3z
rotations or translations are needed and obviate the need to
transform the basis vectors. The following substitutions are
3B: _ OBx (21)
used for simplicity: p2 =_x2 +y2 r 2 _x2+y2 +Z2 ' Ox az
a-' =-a2 +r2-2ap,fl -'- a2 +r2 + 2ap ,k 2=-l-a-'/fl 2, 7---x 2_ y2
3B. 3B,
and C =-/_, 1/_. Note that p and r are non-negative. (22)
_y Oz
38: Cz ff , _ , +r4_.(k2)+
Oz = _6a'(p" - z- )- 7a4 (23)
,,2[,,2_r2]Ka-')
IV. CYLINDRICAL COORDINATES Far from the loop (r>>a):
=__ cos0 (33)
The following substitutions are used for simplicity: Br bt°(Ilra2 ) 3
2_r r
a2=_aZ+pZ+z-__ap,fl-=_a'-+p2+ze+2ap,k2 l-a2/fl -', C=_,uul/,,r
Bo = llO (llfij 2 ) sin0 (34)
Field Components: 4_r r 3
_ C z _a2+p2 +z2)E(k2)_a2K(k2)] (24) Vl. CONCLUSION
B,, 2a2flp
We have presented simple, closed-form algebraic formulas
B: = a- -z-)E(k2)+a2K(k2)] (25)
2_[ *_p2 for the magnetic induction and its spatial derivatives of a
filamentary current loop that are exact everywhere in space
Spatial Derivatives of the Field Components: outside the conductor. Although these formulas are exact, they
do require the numerical evaluation of elliptic integrals.
_Bp Solenoids with circular cross sections of arbitrary size and
_ -Cz {[ar+(p2+z2)2(2p2+z2)+
_p 2p2a4fl 3 configuration can be modeled by simply summing the
(26) contributions of each individual loop.
a4(3z 2-8p 2) +a2(5p a-4p2z 2+
There are, of course, other ways to obtain B for the basic
3z4)_(k2)-a2[a4-3a2p2 +2p4 + circular current loop. For example, series expansions are
available [3] and numerical integration via a finite element
approach can be performed [6]. However, these suffer from
limitations such as truncating the series expansions after some
_Bp - C _a 2+p_),(Z 4 +(a 2_p2)2 )+ tolerance is reached or accepting some graininess when using a
3Z 2pa4fl 3
discrete grid. Our approach has neither of these limitations and
2z (a -6a p +pa) (k2 (27)
. 4 _ - )_ yields results are that exact up to the limitations of the
a2_a2-p2 )2+(a2 + p2 )z2_K(k2 )} numerical arithmetic and the elliptic integral routines.
The inclusion of the spatial derivatives allows convective
OB. CZ Jr[ _- 2 _ derivatives to be found exactly and may prove useful for
3z" - 2-77-_ _[_6a"(p - z- )- 7a + (28) magnetohydrodynamics applications.
(p2 + Z2)2]E(k 2)+a2[a2_p2 ,2]K(k2)}
REFERENCES
OBz _ 3Bp (29) [11 Y. Chu, "'Numerical Calculation for the Magnetic Field in Current-
Op 3z Carrying Circular Arc Filament," IEEE Trans. Magn., vol. 34, pp. 502-
504, 1998.
V. LIMITING CASES [2] M.W.Garrett, "Calculation of Fields, Forces, and Mutual Inductances
of Current systems by Elliptic Integrals," J. Appl. Phys.. vol.34 no. 9,
pp.2567-2573, 1963.
Several special limiting cases are given for completeness. [3] "Classical Electrodynamics", J.D. Jackson, John Wiley & Sons, 3'd
We have confirmed that our results given above do indeed Edition, 1998, pp. 181-183.
converge to these formulas. We also give additional
[41 J. Lane, R. Youngquist, C. Immer and J. Simpson. "Magnetic Field,
expressions for Bx and B vnear the axis that may prove Force, and Inductance Computations for an Axially Symmetric
Solenoid", unpublished, 2001.
useful.
[5] "Classical Electrodynamics", J.D. Jackson, John Wiley & Sons, 3rd
Edition, 1998, pp. 181-183.
Along the axis of the loop:
[6] M. Abramowitz and I. A. Stegun, "Hanndbook of Mathematical
Functions," Dover, 1972, pp. 590.
B. - ,uola2 (30)
2(a 2+ z"
Near the axis of the loop (x,y<<a):
B._ - 3a2p°Ixz (31)
4(a 2+ Z2)'_'_2
3a -kq)lyz (32)
B_
4(a 2+ z2)/2