Simple Analytic Expressions for the Magnetic Field of a ...

[Pages:3]Simple Analytic Expressions for the Magnetic Field of a Circular Current Loop

James Simpson, John Lane, Christopher Immer, and Robert Youngquist

Abstract - Analytic expressions

for the magnetic induction and

its spatial derivatives for a circular loop carrying a static current

are presented in Cartesian, spherical and cylindrical coordinates.

The solutions are exact throughout all space outside the

conductor.

Index Terms - Circular

Derivatives.

Current

Loop, Magnetic

Field, Spatial

as shown (Fig. I.). It is assumed that the cross section of the conductor is negligible.

z.

,L/0

I

_v

I. INTRODUCTION

nflaulxytidcenesxitpyr,essiBo)nsof faorsimthpele mapglanneatirc ciirncduulactrioncur(rmenatgnelotiocp have been published in Cartesian and cylindrical coordinates [1,2], and are also known implicitly in spherical coordinates [3]. In this paper, we present explicit analytic expressions for B and its spatial derivatives in Cartesian, cylindrical and spherical coordinates for a filamentary current loop. These results were obtained with extensive use of Mathematica TM

and are exact throughout all space outside of the conductor. The field expressions reduce to the well-known limiting cases and satisfy V ?B = 0 and V ? B = 0 outside the conductor.

These results are general and applicable to any model using filamentary circular current loops. Solenoids of arbitrary size may be easily modeled by approximating the total magnetic induction as the sum of those for the individual loops [4]. The inclusion of the spatial derivatives expands their utility to magnetohydrodynamics where the derivatives are required.

The equations can be coded into any high-level programming language. It is necessary to numerically evaluate complete elliptic integrals of the first and second kind, but this capability is now available with most programming packages.

II. SPHERICAL COORDINATES

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 spherical coordinate results.

The current loop has radius a, is located in the x-y plane, centered at the origin, and carries a current / which is positive

Fig. 1. Circular current loop geometry.

The vector potential, A, of the loop is given by [3]:

12oi a [2_

cos_' d_o'

A_r,O) = --_-x .lo _[a2 + r 2 _ 2arsin Ocos_o"

_ ta,,

41a

[(2-k2)K(k2)-2E(k2)]

4x 4a 2 + r 2 + 2arsinO [

k2

(l) ]'

where r, 0, and _ are the usual spherical coordinates, and the argument of the elliptic integrals is

k2 _

4arsinO

(2)

a 2 +r 2 +2arsinO

Note that we use k2for the argument of the elliptic integrals. This choice is consistent with the convention of Abramowitz

and Stegun [5] where m = k 2. For a static field with constant current, the B components in

spherical coordinates are [3]:

l3

Br - rsin0 _)O(sinOA_)

(3)

Bo = _r__r(r

A_ )

(4)

Be = 0.

(5)

Manuscript received February I, 2001. This work was supported in part

by NASA under Contract No. NAS 10-9800 I. J. C. Simpson is with Dynacs, Inc., KSC, FL 32899 USA (telephone: 321-

867-6937, e-mail: James.Simpson-3@ksc.).

J. E. Lane is with Dynacs, Inc., KSC, FL 32899 USA (telephone: 321-867-

6939, e-mail: John.Lane-I @ksc.).

C. D. Immer is with Dynacs, Inc., KSC, FL 32899 USA (telephone: 321-

867-6752, e-mail: Christopher.lmmer-1

@ksc.).

R. C. Youngquist is with NASA., KSC, FL 32899 USA (telephone: 321-

867-1829, e-mail: Robert.Youngquist-

1 @ksc.).

Analytic expression for the field components and their derivatives in spherical coordinates are given below. For

simplicity

we use the following

substitutions:

a,2 -=a 2 +r 2 -2arsinO,fl 2 -a 2 +r 2 +2arsinO ,k 2 =-I-aZ/fl 2, and

C =-izo I/x.

Field Components:

Field Components:

Ca2 cosO E(k 2 ) B r = a2fl

_

C

[,2

Bo 2azfl-sinO [r +aZ c?s20)E(k2 )-aZK(k 2 )]

(6)

Cxz B._ = 2a-'tip

_a 2+r 2)E(k 2)_a 2

2K(k?)]

(12)

(7) 8,- c,_,: i 2+/_e(k,)_ -'x(k._)]=zs_

(13)

" 2a2flp 2

x

Spatial Derivatives of the Field Components:

8_=7C_._b-_ --r 2 )_2.o: _-' ,1

(14)

OB r _ Ca2 cos8 {[a 4_7r4_6a2r

Or

2ra 4fl 3

"_cos 20]E(k 2)+

(8)

[o2(r__o-')kc(k2)}

08r_ --Co" {[o4_7,_-'rr4

00

4sin_?fl

3

-+ +

(9)

cos20(-3a

4+2a2r -'-3r4)+

o_rc-'o.sOil(k2)+

[2_2(+or2-')c2o0s1_(_}2)

Spatial Derivatives Components:

of the Field

qoB_. _

CZ

{[a4(_7(322

c)x

2a4 fl3 p 4

a2(p4(5x2

+y2)_2p2z-'(2x

rZ(2x4

+_,(y2

+ z2))]E(k2)+

+a-')+p.(8x__,

_ v 2 ))-

2 + y-' )+ 3z4?)_ (15)

r2(2x 4+,(v2+ze))la'K(k-')}

3Bo _

-c

{[a__3 4r-'+a2r4 +

Or

4a 4fl3rsin 0

2r 6 +a2(3r2-a2)(a-'

+r 2)cos 20+

3a4r 2 cos 40]E(k 2 )+

(1o)

[,,-,(_a4+, 2r2_ 2r4 +

o2_2a_3_2)c2o2s0)]__(_)}

OBo

-

-Ccos0

30

4a4,83sin2

{[5a6

0

+3a4r2_3a2r4

+2r 6 +(-3a 6 +2a4r 2 +9a2r 4)cos 20+

a4r 2 cos 40]E(k 2 )_

3 a 6 +2a4r2+a

2r 4 +2r6+

(11)

a2(5r -' -a 2)(a2 +r 2)cos 20+

( -7aSr+7a3r

3 -4or _ )sin 0+

o3r(a 2-5r 2)sin 30]K(k 2)}

III.

CARTESIAN

COORDINATES

The field components

and their derivatives

in Cartesian

coordinates are given below. These are easier to use when rotations or translations are needed and obviate the need to

transform the basis vectors. The following substitutions

are

used

for

simplicity:

p2 =_x2 +y2

r 2 _x2+y2 +Z2 '

a-' =-a2 +r2-2ap,fl

-' - a2 +r2 + 2ap ,k 2 =-l-a-'/fl 2 , 7---x 2 _ y2

and C =-/_, 1/_. Note that p and r are non-negative.

OBx_ _'

Cxyz 2a4fl3p

{[3a4(3p2_2zZ)_r4(2r-'+p2)_ 4

2a6-2a-(2p , 4 -p 2z-' +3z 4)]E(k2)+

[r2(2r2 + p2 )_a2(5p2_4z

2 )+ 2a4]a2K(k2) }

_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)_

b2_,,;_-+_w-+o-')1o-_}(_-')

OB__ 3B., Ox 3y

OB3 _ Cz

Oy

2a4f13p

{[a4(y(3z2+a2)+p2(8y2 4

x2)) -

a 2(p4( 5)'-' + x 2 )- 2p2z 2( 2y-' + x-' )- 3z4y) -

r4(2y4- 7.2 +Z2))]E(k-')+

[a-'(-7(a-' + 2z2 )-p2(3y2-

2x-' ))+

r Z(2y4 - 7(x2 + Z2 ))]a2 K(k -' ) }

t)By _ y OB.r

Oz x 3z

3B: _ OBx

Ox az

3B. 3B,

_y

Oz

(16)

(17) (18)

(19) (20) (21) (22)

38:

Cz

ff , _

,

+r4_.(k2)+

Oz = _6a'(p"

- z- )- 7a 4

(23)

,,2[,,2_r2]Ka-')

IV. CYLINDRICAL COORDINATES

The following substitutions

a2=_aZ+pZ+z-__ap,fl-=_a'-+p2+ze+2ap,k2

Field Components:

are used for simplicity:

l-a2/fl -', C=_,uul/,,r

Far from the loop (r>>a):

= __

cos 0

Br bt? (Ilra2 ) 3

2_r

r

Bo = llO (llfij 2 ) sin 0

4_r

r3

(33) (34)

_ C z _a 2 +p2 + z 2)E(k2)_a2K(k2)]

(24)

B ,, 2a2 flp

Vl. CONCLUSION

B: =

a-

-z-)E(k2)+a2K(k2)]

(25)

2_[

*_p2

Spatial Derivatives of the Field Components:

_Bp _ -Cz

{[ar+(p2+z2)2(2p2+z2)+

_p 2p2a4fl 3

a4(3z 2-8p 2) +a2(5p a-4p2z 2+

(26)

3z4)_(k2)-a2[a4-3a2p2

+2p4 +

_Bp - C _a 2 +p_),(Z 4 +(a 2 _p2)2 )+

3Z 2pa4fl 3

2z . (a 4 -6a _ p - + pa ) (k 2 )_

a2_a2-p2

) 2 +(a2 + p2 )z2_K(k2 )}

OB .

CZ Jr[ _ - 2 _

3z" - 2-77-_ _[_6a"(p - z- )- 7a +

(p2 + Z 2)2]E(k 2)+a2[a2_p2

,2]K(k2)}

OB z _ 3Bp

Op

3z

(27) (28) (29)

V. LIMITING CASES

Several special limiting cases are given for completeness. We have confirmed that our results given above do indeed converge to these formulas. We also give additional expressions for B x and B v near the axis that may prove useful.

Along the axis of the loop:

B. -

,uol a 2

(30)

2(a 2 + z"

We have presented simple, closed-form algebraic formulas for the magnetic induction and its spatial derivatives of a filamentary current loop that are exact everywhere in space outside the conductor. Although these formulas are exact, they do require the numerical evaluation of elliptic integrals.

Solenoids with circular cross sections of arbitrary size and configuration can be modeled by simply summing the contributions of each individual loop.

There are, of course, other ways to obtain B for the basic 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 tolerance is reached or accepting some graininess when using a discrete grid. Our approach has neither of these limitations and yields results are that exact up to the limitations of the numerical arithmetic and the elliptic integral routines.

The inclusion of the spatial derivatives allows convective derivatives to be found exactly and may prove useful for magnetohydrodynamics applications.

REFERENCES

[11 Y. Chu, "'Numerical Calculation for the Magnetic Field in CurrentCarrying Circular Arc Filament," IEEE Trans. Magn., vol. 34, pp. 502504, 1998.

[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.

[3] "Classical Electrodynamics",

J.D. Jackson, John Wiley & Sons, 3 'd

Edition, 1998, pp. 181-183.

[41 J. Lane, R. Youngquist, C. Immer and J. Simpson. "Magnetic Field,

Force, and Inductance Computations

for an Axially Symmetric

Solenoid", unpublished, 2001.

[5] "Classical Electrodynamics",

J.D. Jackson, John Wiley & Sons, 3rd

Edition, 1998, pp. 181-183.

[6] M. Abramowitz and I. A. Stegun, "Hanndbook of Mathematical Functions," Dover, 1972, pp. 590.

Near the axis of the loop (x,y ................
................

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