On the magnetic field near the center of Helmholtz coilsM. S. Crosser, Steven Scott, Adam Clark, and P. M. Wilt Citation: Review of Scientific Instruments 81, 084701 (2010); doi: 10.1063/1.3474227 View online: http://dx.doi.org/10.1063/1.3474227 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/81/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analyzing the uniformity of the generated magnetic field by a practical one-dimensional Helmholtz coils system Rev. Sci. Instrum. 84, 075109 (2013); 10.1063/1.4813275 A current-carrying coil design with improved liquid cooling arrangement Rev. Sci. Instrum. 84, 065115 (2013); 10.1063/1.4811666 The magnetic field lines of a helical coil are not simple loops Am. J. Phys. 78, 1117 (2010); 10.1119/1.3471233 Influence of balancing parameters in achieving magnetic field uniformity in a large cylindrical volume J. Appl. Phys. 104, 014908 (2008); 10.1063/1.2949259 An improved Helmholtz coil and analysis of its magnetic field homogeneity Rev. Sci. Instrum. 73, 2175 (2002); 10.1063/1.1471352

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On the magnetic field near the center of Helmholtz coilsM. S. Crosser,a� Steven Scott,b� Adam Clark,c� and P. M. WiltPhysics Program, Centre College, Danville, Kentucky 40422, USA

�Received 13 May 2010; accepted 12 July 2010; published online 24 August 2010�

We develop a series expansion for the calculation of the magnetic field near the center of Helmholtzcoils and apply the result to a magnet of our design. Our analysis considers geometric details of thecoils, the magnetic properties of the form and windings, conductor insulation effects, and severalwinding imperfections. We also consider the relaxation of coil symmetry which happens when themean radius of each coil and the coil midplane separation distance are unequal. We compute thefield uniformity near the coil’s center for three cases, including one where axial symmetry remainsbut geometric imperfections of the order of 10−3 of the coil “radius” exist. © 2010 AmericanInstitute of Physics. �doi:10.1063/1.3474227�

I. INTRODUCTION

The term Helmholtz coils refers to the geometrical ar-rangement of two identical, parallel, circular, coaxial,current-carrying coils whose midplane separation is equal totheir mean radius R. Although there is some uncertainty1

regarding the origin of this geometric configuration, thesecoils are named2,3 in honor of Hermann von Helmholtz�1821–1894�, and are a common arrangement used to pro-duce magnetic fields needed for many laboratory applica-tions. The geometric symmetry4 of the coils produces a fieldof high uniformity near their center which is nominallycalculable5 from the coil radius and current. Higbie6 has ex-amined the uniformity of the field in the midplane of thecoils, and Purcell7 has considered this field with reference tomagnetic multipole moments and an emphasis on the fieldfar from the coils’ center. Wang et al.8 have examined thefield uniformity for Helmholtz coils, Maxwell’s tricoil, andan “improved Helmholtz coil” which they suggest. Cacakand Craig9 studied field uniformity as a function of coil pairseparation for circular and square coils. Garrett10 more gen-erally has discussed magnetic fields produced by axiallysymmetric current distributions.

We have designed and built Helmholtz coils for a par-ticular experiment in which it is desirable to compute thecoils’ field to better than one part in 105 and to know quan-titatively the uniformity of the field near their geometric cen-ter. We follow the approach of others4,11–13 who have consid-ered this problem and have given descriptions of the fieldstrength in the form of a power series involving ratios ofvarious coil geometric parameters to the coil radius.

Below we present design details of our magnet and theanalysis of its field. Additionally, we report extensions to thepower series calculation that include corrections for severaleffects present in our magnet’s construction. In Sec. II the

power series calculation is extended to include the reductionin coil symmetry if the mean radii of the two sets of wind-ings and their midplane separation are different. This reduc-tion in symmetry reduces the field uniformity. Results fromthis calculation will be utilized throughout the rest of thispaper. For instance, by using these calculations, it can beshown that a second set of correction Helmholtz coils maybe used to further improve field uniformity. Section III pre-sents these arguments. Section IV refines the power seriescalculation by considering gaps between adjacent layers ofwindings caused by the insulation of the wire. Since themagnetostatic properties of materials within the magnet alsoaffect the field, we present these corrections in Sec. V. Thesecalculations include materials used in constructing the mag-net and the windings of the magnet itself. Sections VI andVII present the effects of winding imperfections and thermalexpansion, respectively, on the uniformity of the field. Fi-nally, Sec. VIII describes one of our experimental tests of ouranalysis of the field. Throughout, we provide the magnitudeof each effect and in one instance compute the correspondingreduction in field uniformity.

II. CALCULATION OF FIELD NEAR THE CENTEROF HELMHOLTZ COILS

Our interest is in the calculation of the magnetic field, B,at field point P off the axis near the center of Helmholtzcoils, as shown in Fig. 1�a�. The position is defined by �x ,y�or �r ,�� in a coordinate system whose origin is at the coils’center. The initial approach is to calculate the magnetostaticscalar potential � at a point �x ,0� located on the axis �seeFig. 1�b�� due to a single turn of the coil passing through the

point �x� ,y�� within the right coil. A suitable average �̄ overthe rectangular area �a ,b� is then found. Similar argumentsare made for the left coil. Next, the axial substitutionrnPn�cos �� /Rn, where Pn�cos �� is the nth Legendre polyno-

mial, is made for xn /Rn to obtain �̄ in the off-axis positionfor the case r�R. B then follows as

a�Present address: Department of Physics, Linfield College, McMinnville,Oregon.

b�Present address: Humana Corporation, Louisville, Kentucky.c�Present address: Department of Mechanical Engineering, University of

Kentucky, Lexington, Kentucky.

REVIEW OF SCIENTIFIC INSTRUMENTS 81, 084701 �2010�

0034-6748/2010/81�8�/084701/7/$30.00 © 2010 American Institute of Physics81, 084701-1

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B = − �0 � �̄ . �1�

The components Br and B� are computed and transformed tothe axial and radial components Bx and By, respectively.Maxwell3 has used this approach; likewise, except for aver-aging over the cross-sectional area of the coils, hasJefimenko.4

We define the separation between the midplanes of thecoils as R and the mean radii of the left and right coil as R−� and R−�, respectively. This addition to the model allowsfor the possibility that the two coils, perhaps due to imperfectmachining and/or nonuniform thermal expansion, have dif-ferent radii. This geometry preserves axial symmetry. If �=�=0, we describe true Helmholtz coils. The resulting fieldcomponents Bx and By are found in a series expansion interms of the form

�Series term� = �constant�a�bxy����

R�+++�+�+ +1 . �2�

For the symmetry of true Helmholtz coils, �+++� iszero or an even positive integer; we refer to the value of �+++�+�+ as the order of a term and find all terms inthe expansion of B through fourth order. Higher-order termsprove unimportant for our purpose.

To begin the calculation, let N be the number of circularturns per coil. Consider a turn of zero cross-sectional areacarrying current NI and passing normally through the point�x� ,y�� of the right coil in Fig. 1�b�. The resulting scalarpotential �2 �x� ,y� ,x� at the point on the axis a distance xfrom the center is

�2�x�,y�,x,y = 0� = NI�/4� = �NI/2��1 − cos �2� , �3a�

where � is the solid angle subtended by the turn and

cos �2 =�R/2 − x + x��

��R/2 − x + x��2 + �R − � + y��2. �3b�

The average scalar potential over the rectangular cross-section of the coil, �2�x ,0�, is found by expanding �2 tosuitable order in a two-dimensional Maclaurin series in x�and y�, and then integrating over the rectangle and dividingby its area, ab. The resulting �2�x ,0� is combined with a

similar function �1�x ,0� for the left coil. Axial substitution

then yields �̄�r ,�� off-axis and B follows from Eq. �1�.The magnetic field at the off-axis point �x ,y� is, through

fourth order

Bx�x,y� =8�0NI

5�5R�1 −

b2

60R2 + Fcx +x

125RF1x

+�2x2 − y2�

125R2 F2x +�3xy2 − 2x3�

125R3 F3x

−18

125R4 �8x4 − 24x2y2 + 3y4� + . . .� �4a�

and

By�x,y� =8�0NI

5�5R� y

125RF1y +

xy

125R2F2y

+y�4x2 − y2�

125R3 F3y +xy

125R4 �288x2 − 216y2�

+ . . .� �4b�

where

Fcx = −�18a4 + 13b4�

1250R4 +31a2b2

750R4 +� + �

R�1

5

+2�a2 − b2�

250R2 � −��2 + �2�250R2 25 +

52b2 − 62a2

R2 −

8

25

��3 + �3�R3 −

52

125

�4 + �4

R4 , �4c�

F1x =�� − ��

R�150 +

�24b2 − 44a2�R2 � +

165��2 − �2�R2

+96��3 − �3�

R3 , �4d�

F2x =�31b2 − 36a2�

R2 +60�� + ��

R+

186��2 + �2�R2 , �4e�

F3x =88�� − ��

R, �4f�

F1y =�� − ��

R75 +

12b2 − 22a2

R2 +165��2 − �2�

2R2

+48��3 − �3�

R3 , �4g�

F2y =2�36a2 − 31b2�

R2 −120�� + ��

R−

372��2 + �2�R2 , �4h�

and

F3y =66�� − ��

R. �4i�

For true Helmholtz coils, �=�=0 and these expressions sim-plify substantially. In that case Ruark and Peters12 have givensimilar results. Note that if ���, the midplane x=0 of the

FIG. 1. Cross-sectional views of Helmholtz coils to define geometry andcoordinate directions used in the calculations. Figure 1�a� identifies many ofthe symbols used in the text and Fig. 1�b� identifies the angles �1 and �2.

084701-2 Crosser et al. Rev. Sci. Instrum. 81, 084701 �2010�

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coils is not a plane of symmetry. The magnitude and direc-tion of B�x ,y� are easily found from these equations.

III. MAGNET CONSTRUCTION AND FIELDUNIFORMITY

Our experimental application involves an optical pump-ing experiment wherein a Pyrex cell containing an atomicvapor is placed in an external magnetic field whose strengthcan be varied in the range 0–30 G. Our cell is cylindricalwith 21 mm radius and 22 mm height, is centered at x=0,and is coaxial with the magnet’s axis. The magnet’s form is acylinder approximately 15 cm in radius and 20 cm in heightonto each end of which is attached a coaxial cylindrical diskcontaining a winding channel of mean radius 0.24 m. Theform has an axial hole with a radius of about 5 cm, and atransverse, square cross-section �10 cm side� hole whichpasses through the form’s center. This second hole allowspositioning of the cell.

We chose the epoxy-laminate G-10 as form material pri-marily because of the small magnetic susceptibility and ma-chining properties of G-10. Additionally, an “A”-frame, usedto align the magnet’s axis with the local field, and the x-y-zadjustable holder for the Pyrex cell were all constructed fromthis material. The magnet has a mass of about 60 kg. Acurrent of 5 A produces a field of about 30 G at the magnet’scenter.

We used square cross-section copper wire14 of 20 AWGand wound each coil with 14 layers of 13 turns each. Note inEqs. �4e� and �4h� the presence of the term �31b2−36a2�. Ifb /a=�36 /31�1.0776, this term vanishes. Therefore, usingthe layer to turn ratio of 14 /13=1.0769 makes the aboveterm negligible at the level of one part in 105.

Independently, we wrote three separate computer pro-grams in double precision to compute B within the cell. Fig-ure 2�a� shows the results for our magnet for the case �=�=0 and actual values aL /RL and bL /RL equal to 4.58�10−2

and 4.91�10−2, respectively; the surface plots �B /Bo�−1, ameasure of the difference in the magnitude of B relative toBo at the magnet’s center, versus position in the cell. Thex-axis of the figure points along the magnet’s axis while thefigure’s y-axis is radially outward. We show results only forone half of the cell’s volume because the x-axis and the planex=0 are symmetry elements. The largest variation of about25 parts per million �ppm� occurs at the circumference of thecell’s windows.

Another improvement can readily be made by includinga second, smaller set of correction Helmholtz coils of meanradius RS that are cocentered and coaxial with the larger setof radius RL. Consider the case in which identical current Icirculates in the smaller coils in the opposite direction andthe radius RS of these smaller correction coils satisfies thecondition

NS/RS5 = NL/RL

5 , �5�

where NS�L� is the number of turns for the small �large� coil.Note that under these conditions, the effect of the remainingxy dependent terms independent of � and � in Eqs. �4� wouldbe canceled.13

As part of our design, we considered improvementswhich may result from the use of correction Helmholtz coilsin accordance with Eq. �5�. Given our value of RL, we pickedNS=16 and thus RS=0.1476 m. Since computations show itis unnecessary to satisfy the ratio b /a=1.0776 for the cor-rection coils, their cross-section ratio was chosen to be 1.0,or the square of four layers of four turns each. The computeduniformity of the field using both sets of coils is shown inFig. 2�b�, for the case NS=16, aS /RS=bS /RS=2.4�10−2, and�=�=0. The largest deviation in the magnitude of B, aboutsix parts in 108, occurs in the cell’s midplane at the walls.Inside the cell, the largest variation in the direction of Brelative to the axis is about 10−7 rad. Although the corre-sponding reduction in the magnitude of B is about 14%, ourcomputations show that the use of coaxial Helmholtz correc-

(a)

0 .02 .04 0 .04 .08x/R y/R

B/B

O-1

0

2E-5

-2E-5

0y/R

.04.020x/R

0

B/B

O-1

2.5E-8

-5.0E-8

(b)

0 04-.04x/R

0 .04 .08y/R

0

4E-5

-2E-5

B/B

O-1

(c)

.08.04

.

FIG. 2. �Color online� Calculated plots of the uniformity of the magneticfield near the center of our Helmholtz coils, as designed. Figure 2�a� iswithout the use of correction coils, Fig. 2�b� shows the designed improve-ment resulting from correction coils, while Fig. 2�c� shows the reduceduniformity produced by unequal coil radii. Bo is the field magnitude at thecenter of the coils. See text for additional information.

084701-3 Crosser et al. Rev. Sci. Instrum. 81, 084701 �2010�

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tion coils can produce at least a 400-fold improvement infield uniformity in the volume enclosed by our cell.

IV. INSULATION EFFECTS

The model described by Eq. �4� assumes that currentflows uniformly within the entire cross-sectional area ab ofthe magnet form. In reality, there are voids within this areadue to the insulation of the wire and occasional gaps betweenedge turns and form walls. Our magnet is wound with insu-lated wire of square cross-section to minimize these voids.The wire cross-section is nominally square; the corners arerounded with a small radius. We call the insulation thicknesst and the wire edge h and assume the ratio r� t /h�1. Ourmagnet has an even number 2n �n=1,2 , . . .� of horizontallayers, each of N� turns. The cross-sectional area of thewindings thus contains horizontal rows and vertical columnsof thickness 2t occupied by insulation. At the intersection ofa row and column, the rounded corners of the turns create asmall diamondlike gap. Figure 3�a� shows the geometry, andoveremphasizes the wire’s rounded corners. Here we esti-mate the effect of wire insulation on the field computed atthe cell’s center from Eq. �4� to second order. We neglect thesusceptibility of the insulation, and here take �=�=0.

We focus at the point x=y=0 on terms through secondorder which involve a and b. The interest is in the quantity�1−b2 /60R2� from Eq. �4a�. We neglect the columns of in-sulation of thickness 2t because they do not contribute tosecond order. We also ignore the small corner radius of theconductor. Thus we model the system shown in Fig. 3�b�.

We begin by grouping the even number 2n of layers�horizontal rows in Figs. 3� of windings into n pairs, suchthat the central two rows form the first pair. Each successivepair is composed of the two rows of windings each equidis-tant from the central insulation row. We next compute thefield produced at the cell’s center only by the central pair ofwindings. Later, through induction, we extend this result tothe full n pairs.

We define the number of turns in a row as N� and set thenumber of turns in a row per unit distance measured alongthe distance b to be N� /h. We consider a hypothetical magnetof turns �N� /h� �2h+2t�=2N��1+r�, distributed without in-sulation over a rectangular cross-section �a , �b=2h+2t��. Wecompute B for this distribution.

Next we consider the field produced by a second hypo-thetical magnet with one horizontal row of N�= �N� /h��2t�turns contained in rectangular cross-section �a ,b=2t�. Sub-tracting the second result from the first gives, to first order inr

B =8�0I

5�5R�2N���1 −

�2h + 2t�2

60R2 �1 + r�� . �6�

This result is B at the cell’s center for the central two rows ofwindings in Fig. 3�b�; these rows are separated by the insu-lation row of thickness 2t. We note that �2h+2t� is the di-mension b for this part of the magnet. Therefore, the effect ofthe central insulation row is to change the second order term�−b2 /60R2� to �−b2�1+r� /60R2� and hence to reduce thefield.

Using this process and induction, one can obtain the ef-fective b value for any rectangular cross-section Helmholtzcoil having 2n layers of windings. The result is

beff2 = b2 1 + ��2n − 1�/n2�r� , �7�

which shows that the effect of the insulation layers is toincrease beff and hence decrease B. This effect increases withr and decreases with n. In practical application, the effect isusually small if r�1.

The same approach may be used for coils with an oddnumber of layers. Any other insulation effects would be oforder higher than second.

Following this development, in our magnet we have n=7, t=1.7�10−5 m, h=8.13�10−4 m, and r=0.021. Withthese parameters, we compute that the presence of insulationon our windings decreases the field computed from Eqs. �4�by 0.2 ppm.

V. MAGNETOSTATIC EFFECTS

In order to predict the magnetic field B near the magnet’scenter, we use Eq. �4� to calculate B for known magnet ge-ometry and current I. The actual field may differ from thiscalculation because of the magnetization effects of the mag-

b

h

}

}

2t

centraltworows

2t

≈ ≈

≈

≈≈

≈

a(a)

≈≈

b }

centraltworows

2t

h

≈ ≈

≈

≈a

(b)

FIG. 3. Geometric parameters used to model the effects of wire insulationon the calculated magnetic field. Figure 3�a� shows actual magnet geometry.The radii of the corners of the square cross-section wire are exaggerated forclarity. Figure 3�b� shows the system actually modeled. Dimensions are notto scale.

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net form and windings and other materials within the mag-net. Here we summarize how to compute the size of theseeffects, assumed to result from linear materials.

In International System of units �SI�, the magnetic sus-ceptibility �m for linear materials is the proportionality factorbetween magnetization M and the field H

M = �mH , �8�

where �m�0 for paramagnetic and �m�0 for diamagneticmaterials.

The magnetization M of the magnetized material givesrise to a bound volume current density Jb calculable from

Jb = � � M �9a�

and a bound surface current density Kb whose value is

Kb = M � n̂ , �9b�

where n̂ is an outward unit vector normal to the surface ofany material of interest. One may compute the effects ofmagnetized linear materials, given their susceptibilities, if His known. Since in our case only weakly magnetizable ma-terials are involved, their effects on B near the magnet’scenter are small. Where r�R, H may be approximated fromEq. �4�; alternatively, at any field point, one may approxi-mate H by numerically integrating over infinitesimal circularcurrent loops15 confined to the rectangular cross-sectional ar-eas of our Helmholtz coils. We used this second approachonly for parts of the magnet form where r�0.7R. Once H isfound, one uses the infinitesimal current elements originatingfrom Jb and Kb in the volume or on the surface of the ma-terial to compute the corresponding fields due to magnetizedmaterial. Also, since ��H=Jf where Jf is the conductioncurrent density, Jb=�mJf and the magnetization effects of thecurrent-carrying winding material can be computed.

Following this approach, we compute the change in thefield at the magnet’s center due to the magnetization of sev-eral materials within the magnet: the form material, the cop-per windings, a Pyrex absorption cell, and a cell holder withair baffles used to inhibit the escape of heated air. To esti-mate the uniformity of some of these effects, we repeated thecalculations on the magnet’s axis at a cell window. The re-sults in units of 8�oNI�m /5�5R are shown in the first row ofTable I. We note that these results depend on the details of

the material’s geometry, are directly proportional to �m of thematerial, and are usually the sum of several partially cancel-ing results.

Some additional comments are in order: First, near themagnet’s center, the radial magnetization component My, be-ing a second- or higher-order position-dependent quantity,contributes effects much smaller than the axial magnetizationcomponent Mx whose effects dominate the results. Howeverin the form immediately adjacent to the windings, both Mx

and My are relatively large. Second, since Jf =0 in the form,cell, and cell holder, the bound current Jb contributes nothingin these cases and only surface magnetization currents arepresent. Third, those surfaces perpendicular to the symmetryaxis have Mx� n̂=0 and generally contribute comparativelyweakly. Finally, the negative sign of an entry means that if�m�0, the magnetization of that material reduces the fieldwithin the cell. This nonintuitive result is the net effect ofsurface magnetization currents of varying strengths flowingin opposite directions and being positioned at different dis-tances from the cell. We used symbolic computational soft-ware to perform the rather complex geometric integrals aris-ing in these computations.

We repeated the computation for the material of thePyrex cylindrical cell. The windows contribute little and thecylindrical inner and outer walls of differing heights andnearly equal radii produce the nearly offsetting effects shownin the second column of the table. The magnetic effects ofcell-holders and baffles are shown in the third column of thetable.

The magnetization of the square copper wire used in thecoil windings produces bound magnetization currents whichadd or subtract to the B field an amount proportional to �m

for the wire. As Table I shows, the magnetic susceptibility ofcopper ��m�−10−5� appears to reduce the calculated valueof Bm by about 10 ppm because of the bound current density.However, outside a long straight wire of circular cross-section, the effect of a uniform Jb is exactly offset by effectsof the surface current Kb. Detailed numerical calculations forthe geometry of our conduction current show that, at ourmagnet’s center, the effect of Kb exceeds that of Jb by onepart in 103.

The magnetic susceptibilities shown in Table I weremeasured using a Johnson Matthey susceptibility balance.16

For our Cu wire we measured �m and obtained the acceptedvalue17 within the uncertainty of the measurement. We usedthe accepted value with confidence that our wire sample hasthe susceptibility of pure copper.

Thus the computed magnetization effect for the G-10magnet form is to increase ��−0.050��−3.4�10−6�=+1.7�10−7� the field at the cell’s center by about 0.2 ppm. ThePyrex cell produces an increase of about 0.1 ppm. The cellholder and baffles produce an increase of about 0.3 ppm. Themagnetization effect of the copper conductor is estimated tobe of the order of +10−8 of the field due to the conductioncurrent.

TABLE I. Magnetization effects for materials near our magnet’s center, inunits of 8�0NI�m /5�5R. The values of �m are measured �Ref. 16� or, asindicated �Ref. 17� are accepted values. The uniformity of these effects isestimated, where possible, by comparing effects on the cell axis at the centerand top of the cell.

Magnetform

Pyrexcell

Cellholder

and bafflesCopper

wire

Change in B �0.050 +0.03 �0.10 �−10−3

Material G-10 Pyrex G-10 Cu17

�m��10−6� �3.4 +3.0 �3.4 �9.7Estimateduniformity �6% �60% �25%

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VI. WINDING IMPERFECTIONS

Above we modeled the windings of our magnet as cir-cular turns which close on themselves. In fact, in a givenlayer no turn quite closes but has, over a small fraction of aturn, a slight lateral offset which provides the transition tothe adjacent turn. We call this winding imperfection a “lat-eral offset.” For the last turn touching a side of the channel inthe magnet form, a lateral offset is impossible and that wind-ing will “pop up” into the next layer prematurely. Theseimperfections produce a small void in the first and last turnsof a layer, and in our magnet resulted in a fraction of the lastturn of the top layer having an increased radius. Thus, for areason other than insulation effects, the model of currentuniformly distributed throughout a rectangular cross-sectionof dimensions �a ,b� is not strictly correct. Here we makecorrections for these other effects.

Because of the pop-up effect caused by the lateral offsetof each turn, the effective coil radii R−� and R−� in Eq. �4�are slightly larger than the average of the carefully measuredinner and outer radii of the windings. Instead of an integralnumber of turns in a layer, there are slightly fewer, and thesemissing pieces of layers produce an outer layer with fewerthan the full number of turns. We thus compute the radii inEq. �4� as a weighted average of the radius of each layer,with the weight being the nonintegral number of turns in thatlayer. The average midplane spacing R of Eq. �4� is com-puted in an analogous manner. Also, the effective value of bin Eqs. �4� and �7� should be slightly larger because of thepresence of a �usually small� fraction of an outer layer.

Each current-carrying lateral offset of a turn could beconsidered as a combination of two parts. The first would bea small circular arc which would complete each circular turn;the second part would be a current element of length equal tothe wire edge and direction parallel to the magnet’s symme-try axis. Since the lateral offsets of turns in a layer are adja-cent to each other, the combined effect of the current ele-ments in a layer parallel to the magnet’s axis is of ahypothetical short wire segment of length slightly less than ain Fig. 3. This hypothetical current element would produce afield at the magnet’s center which is perpendicular to theaxis. However, it is possible to have offsets of comparablelayers of the two coils run in opposite directions and producecanceling results at the magnet’s center.

In our magnet, instead of 13 turns per winding layer, wemeasure only 12.981 because of the pop-up effect. There are0.264 turns in a fifteenth layer. Following the weighting pro-cedure described above, we find that the effective value of Rin Eq. �4� is about 40 ppm larger and correspondingly thezeroth-order term �1 /R� is 38.4 ppm smaller.

The effective b value in Eq. �7� should be slightly largerbecause of the small fraction of a 15th layer. Using a similarweighting technique, we find that the term 1−b2 / �60R2� de-creases only 0.1 ppm. The effect of the larger effective bvalue is nearly canceled by the effective increase in R. Wealso wound our magnet so that the lateral offsets in compa-rable layers of the two large coils run in opposite directionsto produce canceling effects at the magnet’s center.

VII. THERMAL EXPANSION

During use of our magnet, its core is heated to 60 °C. Inthe course of this work, we discovered for ourselves whatothers have known: G-10 has strongly nonisotropic thermalexpansion caused by different glass fiber counts in differentdirections in a layer and a distinctive structure in the direc-tion normal to layers. Because of a small machining errorand the nonisotropic thermal expansion of G-10, carefulmeasurements showed that, during use, � /RL=6.26�10−4,� /RL=1.02�10−3. In absolute units, the corresponding val-ues of � and � are only about 200 �m. As accurately as wecould measure �about 20 �m�, our magnet retained axialsymmetry, but the mean coil radii and the coil midplaneseparation differed slightly. The plane x=0 was no longer asymmetry plane.

Figure 2�c� shows the computed field uniformity in thecase NS=0 �no use of correction coils� and these nonzerovalues of � and �. The lack of symmetry about the plane x=0 is evident and the largest nonuniformity, about 40 ppm,occurs on one cell window at its perimeter. Use of the cor-rection coils in this case �not shown� improves uniformityonly by about a factor of 2, in contrast to the case shown inFig. 2�b�. Thus, one conclusion is that the desired improve-ment in field uniformity obtained through correction coils isgreatly reduced if the larger Helmholtz coils have small geo-metric defects. One must also be aware that the net field nearthe coils’ center may have even larger inhomogeneities if theenvironmental field due to the magnet’s surroundings is suf-ficiently nonuniform.

VIII. EXPERIMENTAL TEST

Here we briefly report an experimental test of Eq. �4�;this test used our magnet to measure the MF=2↔3, �2–3�,Zeeman transition in the F=3 hyperfine component of theelectronic ground state of Rb85. Benumof18 has described thegeneral measurement technique and also stated the Breit–Rabi equation which relates the Zeeman frequency to themagnetic field strength which splits the magnetic substates.We used measured frequencies to determine the field pro-duced by our magnet at nominal currents of 2.0 and 3.0 A,and compared these fields to the corresponding ones calcu-lated from Eq. �4� for the appropriate magnet current andgeometry. To eliminate the effects of magnetizable materialsin a surrounding building, our measurements were made out-side, 2 m above ground in a carefully screened area removedfrom structures. We first measured the environmental field Be

at out magnet’s center by measuring Zeeman frequencies atzero magnet current. Then we measured the �2–3� frequencyat a nominal current of 2.0 A with the magnet’s field Bm bothparallel and opposite to Be. From this information we ob-tained two measured values of our magnet’s field at its cen-ter; we repeated the measurement at 3.0 A with the two fieldsbeing parallel. For the corresponding currents and the othereffects discussed above, we calculated Bm at the center fromEq. �4�. Table II shows the results. To extract transition fre-quencies from the observed asymmetrical transition profiles,we fitted them to an exponentially modified Gaussian �EMG�shape. From the line-width parameter of the EMG profile, we

084701-6 Crosser et al. Rev. Sci. Instrum. 81, 084701 �2010�

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obtained full-widths at half height �FWHH� of the observedlines. These were about 315 Hz at 2.0 A and 460 Hz at 3.0 A.We expect our observed widths to result predominately fromaxial field inhomogeneities; contributions from the Dopplereffect �a few hertz�, light intensity, rf fields, and othersources such as relaxation due to collisions, are muchsmaller. In Rb87, under somewhat different conditions, aFWHH of 2.6 Hz has been observed.19 Since the transitionfrequencies are approximately proportional to Bm, we includethe ratio FWHH/transition frequency in Table II as an ap-proximate measure of nonuniformity.

Examination of Table II shows that in this test Eq. �4�predicts our magnet’s field with an accuracy of a few parts in105. Comparison of the FWHH/transition frequency ratioswith Fig. 2�c� shows general agreement between observedand predicted field nonuniformity. We did not measure thenonuniformity of Be which could contribute to the observedvalue. These tests are sensitive mostly to zeroth-, first-, andsecond-order constant and position dependent terms in Eq.�4�.

In two subsequent papers we will explain in detail ourexperimental methods and present the results of several hun-dred Zeeman measurements taken with Bm values rangingfrom 6 to 26 G; these results show that Eq. �4� accuratelypredicts the field at the center of our magnet, using currentsin the range of 1.0–3.5 A, with an average accuracy of theorder of one part in 105. In Table II the 3.0 A test is repre-sentative of the outer range of agreement we have experi-enced in our use of Eq. �4�.

IX. SUMMARY AND CONCLUSIONS

We have derived through fourth-order a theoreticalpower series for the computation of the magnetic field nearthe geometric center of Helmholtz coils. This expression as-sumes that the coils’ windings uniformly fill an area of rect-angular cross-section of dimension �a ,b�. Our result is gen-eralized for a reduced symmetry which retains a symmetryaxis, but allows for the mean radius of each coil and theirmidplane separation to all be unequal.

For geometrically perfect Helmholtz coils with R=0.24 m, b /a�1.0769 and positions near the center definedby x /R�5�10−2, y /R�9�10−2, the field is uniform inmagnitude to 25 ppm. Use of a second set of smaller correc-tion Helmholtz coils can theoretically increase field unifor-mity by at least a factor of 400.

The winding defects we have described as lateral offsetsand pop-ups can produce changes in the field of the order of

100 ppm. The presence of insulation on the coil windings isestimated to produce a much smaller change of the order of0.1 ppm in the field.

The magnetostatic effects of the Helmholtz coil formmaterial are computed to alter the field at the center by theorder of 0.1 times the susceptibility of the form material; fora G-10 form this change is about +0.3 ppm. For the magne-tization of circular turns of current-carrying winding mate-rial, the volume and surface effects almost completely can-cel; for copper the estimated magnetization effect is of theorder of one part in 108. A cylindrical Pyrex cell placed inthe central region is computed to alter the field by +0.1 ppm.

For “Helmholtz coils” with axial symmetry but withgeometric imperfections described by � /R and � /R of theorder of 10−3, field nonuniformity is nearly doubled and theimprovement in uniformity through the use of correctioncoils is greatly reduced. Geometric imperfections of this or-der can easily be introduced by moderate heating of G-10form material which has nonisotropic thermal expansion. Inthis case, the field’s computed magnitude at the center can bealtered by the order of 100 ppm. Thus, the nonuniform ther-mal expansion properties of G-10 material may produce rela-tively large changes in field uniformity and magnitude evenin a perfectly machined magnet if the form is used in analtered thermal environment. This behavior offsets the ad-vantage of the relatively small susceptibility of G-10. Alumi-num may be a superior form material.

ACKNOWLEDGMENTS

We thank Professor Keith MacAdam and Professor Jo-seph Straley of the University of Kentucky for helpful guid-ance. The assistance of Mr. Gary Crase of Centre Collegewas essential for the fabrication of much of our apparatus.Finally, we are grateful to Centre College for financial sup-port.

1 L. W. McKeehan, Nature �London� 133, 832 �1934�.2 L. Koenigsberger, Hermann von Helmholtz �Clarendon, Oxford, 1906�.3 J. C. Maxwell, A Treatise on Electricity and Magnetism �Dover, NewYork, 1954�, Vol. II, p. 356.

4 O. D. Jefimenko, Electricity and Magnetism: An Introduction to theTheory of Electric and Magnetic Fields �Merideth, New York, 1966�, pp.379–380.

5 P. Tipler and G. Costa, Physics for Scientists and Engineers, 5th ed. �Free-man, New York, 2004�.

6 J. Higbie, Am. J. Phys. 46, 1075 �1978�.7 E. M. Purcell, Am. J. Phys. 57, 18 �1989�.8 J. Wang, S. She, and S. Zhang, Rev. Sci. Instrum. 73, 2175 �2002�.9 R. K. Cacak and J. R. Craig, Rev. Sci. Instrum. 40, 1468 �1969�.

10 M. W. Garrett, J. Appl. Phys. 22, 1091 �1951�.11 J. C. Maxwell, A Treatise on Electricity and Magnetism �Dover, New

York, 1954�, Vol. II, pp. 337–338.12 A. E. Ruark and M. F. Peters, J. Opt. Soc. Am. 13, 205 �1926�.13 G. G. Scott, Rev. Sci. Instrum. 28, 270 �1957�.14 MWS Wire Industries, 31200 Cedar Valley Dr., Westlake Village, CA

91362.15 W. R. Smythe, Static and Dynamic Electricity �McGraw Hill, New York,

1950�, pp. 270–271.16 Johnson Matthey, 436 Devon Park Drive, Wayne, PA 19087.17 Handbook of Chemistry and Physics, 45th ed., edited by R. C. Weast

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TABLE II. Data for experimental test of Eq. �4�.

Nominalcurrent

�A�Field

alignment

Magnet’s field �G� FWHH

Observed Calculated Freq.

2.0 Parallel 13.6166 13.6163 5.3�10−5

2.0 Opposite 13.6162 13.6163 4.8�10−5

3.0 Parallel 20.4152 20.4171 4.8�10−5

084701-7 Crosser et al. Rev. Sci. Instrum. 81, 084701 �2010�

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2016 00:45:03