3794:
31:
377:
3571:
twice the frequency requires twice the voltage across the coil. Instead of directly driving the coil with a high voltage, a series resonant circuit may be used to provide the high voltage. A series capacitor is added in series with the coils. The capacitance is chosen to resonate the coil at the desired frequency. Only the coils parasitic resistance remains. This method only works at frequencies close to the resonant frequency; to generate the field at other frequencies requires different capacitors. The
Helmholtz coil resonant frequency,
3345:
82:
365:
357:
3067:
field and living tissue). The required magnetic fields are usually either pulse or continuous sinewave. The magnetic field frequency range can be anywhere from near DC (0 Hz) to many kilohertz or even megahertz (MHz). An AC Helmholtz coil driver is needed to generate the required time-varying magnetic field. The waveform amplifier driver must be able to output high AC current to produce the magnetic field.
39:
3570:
Generating a static magnetic field is relatively easy; the strength of the field is proportional to the current. Generating a high-frequency magnetic field is more challenging. The coils are inductors, and their impedance increases proportionally with frequency. To provide the same field intensity at
3066:
Most
Helmholtz coils use DC (direct) current to produce a static magnetic field. Many applications and experiments require a time-varying magnetic field. These applications include magnetic field susceptibility tests, scientific experiments, and biomedical studies (the interaction between magnetic
3055:
2709:
2038:
2354:
340:
When a
Helmholtz pair of coils carry an equal electric current in the opposite direction, they create a region of nearly uniform magnetic field gradient. This is known as anti-Helmholtz coil, and is used for creating magnetic traps for atomic physics experiments.
3291:
1120:
992:
62:
on the same axis, carrying an equal electric current in the same direction. Besides creating magnetic fields, Helmholtz coils are also used in scientific apparatus to cancel external magnetic fields, such as the Earth's magnetic field.
3163:
3889:
has the same structure as
Helmholtz coils, but with the magnets separated further apart so that the field expands in the middle, trapping charged particles with the diverging field lines. If one coil is reversed, it produces a
3783:
2818:
726:
1643:
3684:
3491:
2386:
1715:
2151:
360:
Magnetic field lines in a plane bisecting the current loops. Note the field is approximately uniform in between the coil pair. (In this picture the coils are placed one beside the other: the axis is horizontal.)
813:
1402:
2391:
1720:
380:
Contours showing the magnitude of the magnetic field near a coil pair, with one coil at top and the other at bottom. Inside the central "octopus", the field is within 1% of its central value
219:
94:
A Helmholtz pair consists of two identical circular magnetic coils that are placed symmetrically along a common axis, one on each side of the experimental area, and separated by a distance
335:
267:
3206:
1035:
2810:
2143:
289:
reduces the difference in field between the center and the planes of the coils, at the expense of worsening the field's uniformity in the region near the center, as measured by
1521:
1283:
835:
475:, the distance from the central point of the coil-pair along the axis. By symmetry, the odd-order terms in the expansion are zero. By arranging the coils so that the origin
3860:
1026:
1683:
622:
4118:
3201:
756:
2738:
2067:
1431:
3596:
557:
530:
3336:
3313:
1194:
1171:
1144:
585:
499:
165:
3945:
3833:
2378:
1707:
473:
287:
132:
112:
3077:
3050:{\displaystyle (x/R)(96{\sqrt {5}})/125-(x/R)^{3}(512{\sqrt {5}})/625+{\mathcal {O}}((x/R)^{5})\approx 1.72(x/R)-1.83(x/R)^{3}+{\mathcal {O}}((x/R)^{5})}
3690:
2704:{\displaystyle {\begin{aligned}B(x)&={\frac {\mu _{0}nI}{2R}}\left\\&={\frac {\mu _{0}nI}{2R}}\left(^{-3/2}-^{-3/2}\right)\\\end{aligned}}}
2033:{\displaystyle {\begin{aligned}B(x)&={\frac {\mu _{0}nI}{2R}}\left\\&={\frac {\mu _{0}nI}{2R}}\left(^{-3/2}+^{-3/2}\right)\\\end{aligned}}}
645:
3815:
showed in 1873 that a third larger-diameter coil located midway between the two
Helmholtz coils with the coil distance increased from coil radius
1533:
4199:
3604:
3358:
2349:{\displaystyle (16{\sqrt {5}})/25-(x/R)^{4}(2304{\sqrt {5}})/3125+{\mathcal {O}}((x/R)^{6})\approx 1.43-1.65(x/R)^{4}+{\mathcal {O}}((x/R)^{6})}
269:
as explained below), but leaves about 7% variation in field strength between the center and the planes of the coils. A slightly larger value of
4134:
4139:
3927:
3961:
167:, which is what defines a Helmholtz pair, minimizes the nonuniformity of the field at the center of the coils, in the sense of setting
4150:
3989:
766:
4023:
66:
When the pair of two electromagnets of a
Helmholtz coil carry an equal electric current in the opposite direction, it is known as
4186:
3801:
on three perpendicular axes used to cancel the Earth's magnetic field inside the vacuum tank in a 1957 electron beam experiment
1323:
4047:
3793:
4233:
4223:
3862:
can reduce the variance of the field on the axis to zero up to the sixth derivative of position. This is sometimes called a
3348:
Using a function generator and a high-current waveform amplifier driver to generate high-frequency
Helmholtz magnetic field
4086:
3901:
electromagnetic composite testing laboratory in 1993, for testing of composite materials to low-frequency magnetic fields.
3811:
To improve the uniformity of the field in the space inside the coils, additional coils can be added around the outside.
30:
447:
The calculation of the exact magnetic field at any point in space is mathematically complex and involves the study of
3942:
170:
759:
292:
224:
3286:{\displaystyle 4\pi \times 10^{-7}{\text{ T}}\cdot {\text{m/A}}=1.257\times 10^{-6}{\text{ T}}\cdot {\text{m/A}},}
1115:{\displaystyle 4\pi \times 10^{-7}{\text{ T}}\cdot {\text{m/A}}=1.257\times 10^{-6}{\text{ T}}\cdot {\text{m/A}},}
4228:
627:
The calculation detailed below gives the exact value of the magnetic field at the center point. If the radius is
345:
376:
3898:
3167:
Use the above equation in the mathematics section to calculate the coil current for a desired magnetic field,
2743:
2076:
4098:
1029:
825:
Start with the formula for the on-axis field due to a single wire loop which is itself derived from the
75:
55:
1436:
1201:
987:{\displaystyle B_{1}(x)={\frac {\mu _{0}IR^{2}}{2(R^{2}+x^{2})^{3/2}}}=\xi (x){\frac {\mu _{0}I}{2R}}.}
4196:
4005:
826:
3812:
3838:
3344:
3985:
4071:
1003:
81:
1651:
590:
4161:
3965:
3923:
3179:
734:
451:. Things are simpler along the axis of the coil-pair, and it is convenient to think about the
2717:
2046:
1410:
502:
364:
135:
3574:
535:
508:
356:
4203:
4111:
3949:
3886:
448:
4145:
3320:
3297:
1178:
1155:
1128:
562:
478:
144:
17:
4177:
3818:
3158:{\displaystyle I=\left({\frac {5}{4}}\right)^{3/2}\left({\frac {BR}{\mu _{0}n}}\right)}
2363:
1692:
458:
272:
117:
97:
51:
559:. The inflection point for a simple coil is located along the coil axis at a distance
4217:
3993:
3880:
2070:
1524:
505:
for the field strength due to each coil separately, one can guarantee that the order
452:
59:
3891:
3863:
3806:
3778:{\displaystyle C={\frac {1}{\left(2\pi f_{0}\right)^{2}\left(L_{1}+L_{2}\right)}}}
4090:
639:, then the magnetic field B at the midpoint between the coils will be given by
4208:
4165:
85:
A beam of cathode rays in a vacuum tube bent into a circle by a
Helmholtz coil
38:
721:{\displaystyle B={\left({\frac {4}{5}}\right)}^{3/2}{\frac {\mu _{0}nI}{R}},}
1638:{\displaystyle \xi (x)=1-{\frac {3}{2}}(x/R)^{2}+{\mathcal {O}}((x/R)^{4}).}
4157:
3679:{\displaystyle f_{0}={\frac {1}{2\pi {\sqrt {\left(L_{1}+L_{2}\right)C}}}}}
3486:{\displaystyle V=I{\sqrt {{\bigl }^{2}+{\bigl (}R_{1}+R_{2}{\bigr )}^{2}}}}
348:, producing a region with a magnetic field intensity much closer to zero.
3875:
71:
4169:
1686:
532:
term is also zero, and hence the leading non-constant term is of order
372: = 0 is the point in the middle of the distance between coils
368:
Magnetic field induction along the axis crossing the center of coils;
3352:
Then calculate the required
Helmholtz coil driver amplifier voltage:
1147:
27:
Two circular coil device which creates a homogeneous magnetic field
3792:
80:
29:
344:
In some applications, a
Helmholtz coil is used to cancel out the
2714:
The points near the center (halfway between the two coils) have
2043:
The points near the center (halfway between the two coils) have
808:{\displaystyle 4\pi \times 10^{-7}{\text{ T}}\cdot {\text{m/A}}}
1293:
turns of wire, so the equivalent current in a one-turn coil is
3012:
2919:
2311:
2235:
1597:
4087:"The Calculation and Measurement of Helmholtz Coil Fields"
587:
from its centre. Thus the locations for the two coils are
387:. The eight contours are for field magnitudes of 0.5
70:, which creates a region of nearly uniform magnetic field
4024:"High frequency Helmholtz coils generate magnetic fields"
3922:(2nd ed.). Amsterdam: Elsevier/Newnes. p. 195.
1397:{\displaystyle B_{1}(x)=\xi (x){\frac {\mu _{0}nI}{2R}}.}
4179:
The Calculation and Measurement of Helmholtz Coil Fields
2360:
In an anti-Helmholtz pair, the B-field strength at any
4176:
DeTroye, David J.; Chase, Ronald J. (November 1994),
3841:
3821:
3693:
3607:
3577:
3361:
3323:
3300:
3209:
3182:
3080:
2821:
2746:
2720:
2389:
2366:
2154:
2079:
2049:
1718:
1695:
1654:
1536:
1439:
1413:
1326:
1285:
is the distance dependent, dimensionless coefficient.
1204:
1181:
1158:
1131:
1038:
1006:
838:
769:
737:
648:
593:
565:
538:
511:
481:
461:
295:
275:
227:
173:
147:
120:
100:
50:
is a device for producing a region of nearly uniform
4158:
Calculation of Magnetic Field inside Plasma Chamber
3541:
are the inductances of the two Helmholtz coils, and
3920:Hall-effect sensors : theory and applications
3854:
3827:
3777:
3678:
3590:
3485:
3330:
3307:
3285:
3195:
3157:
3049:
2804:
2732:
2703:
2372:
2348:
2137:
2061:
2032:
1701:
1677:
1648:In a Helmholtz pair, the two coils are located at
1637:
1515:
1425:
1396:
1277:
1188:
1165:
1138:
1114:
1020:
986:
807:
750:
720:
616:
579:
551:
524:
493:
467:
329:
281:
261:
213:
159:
126:
106:
455:expansion of the field strength as a function of
4085:J, DeTroye, David; J, Chase, Ronald (Nov 1994).
3897:Helmholtz coils were designed and built for the
214:{\displaystyle \partial ^{2}B/\partial x^{2}=0}
4048:"High-Frequency Electromagnetic Coil Resonant"
1196:= coil distance, on axis, to point, in meters,
221:(meaning that the first nonzero derivative is
3470:
3439:
3423:
3415:
3385:
3375:
330:{\displaystyle \partial ^{2}B/\partial x^{2}}
262:{\displaystyle \partial ^{4}B/\partial x^{4}}
8:
1313:in the above formula gives the field for an
4209:http://physicsx.pr.erau.edu/HelmholtzCoils/
3598:, and capacitor value, C, are given below.
3986:"Earth Field Magnetometer: Helmholtz coil"
3327:
3304:
1512:
1274:
1185:
1162:
1135:
1017:
4140:Axial field of a real Helmholtz coil pair
4117:CS1 maint: multiple names: authors list (
3842:
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3208:
3187:
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3139:
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3110:
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3079:
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3026:
3011:
3010:
3001:
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2918:
2917:
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2855:
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2682:
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2153:
2124:
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2078:
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2011:
2004:
1994:
1982:
1968:
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1933:
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1911:
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1821:
1792:
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1331:
1325:
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1157:
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1005:
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926:
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881:
868:
861:
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837:
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792:
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768:
742:
736:
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690:
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661:
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592:
569:
564:
543:
537:
516:
510:
480:
460:
321:
309:
300:
294:
274:
253:
241:
232:
226:
199:
187:
178:
172:
146:
119:
99:
4135:On-Axis Field of an Ideal Helmholtz Coil
3343:
375:
363:
355:
134:of the coil. Each coil carries an equal
37:
4185:, Army Research Laboratory, ARL-TN-35,
3910:
2805:{\displaystyle \xi (x-R/2)-\xi (x+R/2)}
2138:{\displaystyle \xi (x-R/2)+\xi (x+R/2)}
4107:
4096:
631:, the number of turns in each coil is
3894:, which also traps charged particles.
3561:are the resistances of the two coils.
3203:is the permeability of free space or
635:and the current through the coils is
7:
4017:
4015:
4192:from the original on April 18, 2013
3341:n = number of turns in each coil.
54:, named after the German physicist
4151:The Wolfram Demonstrations Project
4006:"Magnetic Field of a Current Loop"
314:
297:
246:
229:
192:
175:
25:
4168:to compute off-axis fields, from
1516:{\displaystyle \xi (x)=^{-3/2}\;}
1278:{\displaystyle \xi (x)=^{-3/2}\;}
1289:The Helmholtz coils consists of
78:for atomic physics experiments.
42:Helmholtz coil schematic drawing
4072:"ログイン - ASACUSA MUSASHI group"
3566:High-frequency series resonant
3044:
3035:
3020:
3017:
2998:
2983:
2974:
2960:
2951:
2942:
2927:
2924:
2903:
2890:
2881:
2866:
2852:
2839:
2836:
2822:
2799:
2779:
2770:
2750:
2672:
2662:
2633:
2624:
2601:
2591:
2562:
2553:
2500:
2480:
2471:
2451:
2403:
2397:
2343:
2334:
2319:
2316:
2297:
2282:
2267:
2258:
2243:
2240:
2219:
2206:
2197:
2182:
2168:
2155:
2132:
2112:
2103:
2083:
2001:
1991:
1962:
1953:
1930:
1920:
1891:
1882:
1829:
1809:
1800:
1780:
1732:
1726:
1629:
1620:
1605:
1602:
1583:
1568:
1546:
1540:
1492:
1488:
1479:
1464:
1455:
1449:
1443:
1358:
1352:
1343:
1337:
1254:
1244:
1229:
1220:
1214:
1208:
951:
945:
919:
892:
855:
849:
1:
3855:{\displaystyle {\sqrt {3}}R}
3511:is the angular frequency or
3315:= coil current, in amperes,
3943:Helmholtz Coil in CGS units
3062:Time-varying magnetic field
2740:, and the Taylor series of
1433:, the distance coefficient
74:, and is used for creating
4250:
3899:Army Research Laboratory's
3804:
3338:= coil radius, in meters,
3071:Driver voltage and current
1021:{\displaystyle \mu _{0}\;}
760:permeability of free space
1678:{\displaystyle x=\pm R/2}
1305:-turn coil. Substituting
1173:= coil radius, in meters,
617:{\displaystyle x=\pm R/2}
18:Quadrupole magnetic field
4197:Magnetic Fields of Coils
4052:www.accelinstruments.com
3918:Ramsden, Edward (2006).
3196:{\displaystyle \mu _{0}}
751:{\displaystyle \mu _{0}}
3948:March 24, 2012, at the
138:in the same direction.
4106:Cite journal requires
3988:by Richard Wotiz 2004
3856:
3829:
3802:
3779:
3680:
3592:
3487:
3349:
3332:
3309:
3287:
3197:
3159:
3059:
3051:
2806:
2734:
2733:{\displaystyle x\ll R}
2705:
2374:
2358:
2350:
2139:
2063:
2062:{\displaystyle x\ll R}
2034:
1703:
1679:
1646:
1639:
1517:
1427:
1426:{\displaystyle x\ll R}
1398:
1279:
1190:
1167:
1140:
1116:
1022:
988:
809:
752:
722:
618:
581:
553:
526:
495:
469:
444:
373:
361:
346:Earth's magnetic field
331:
283:
263:
215:
161:
128:
108:
86:
43:
35:
4234:Hermann von Helmholtz
4224:Electromagnetic coils
4146:Helmholtz-Coil Fields
3857:
3830:
3796:
3780:
3681:
3593:
3591:{\displaystyle f_{0}}
3488:
3347:
3333:
3310:
3288:
3198:
3160:
3052:
2814:
2807:
2735:
2706:
2375:
2351:
2147:
2140:
2064:
2035:
1704:
1680:
1640:
1529:
1518:
1428:
1399:
1280:
1191:
1168:
1141:
1117:
1030:permeability constant
1023:
989:
810:
753:
723:
619:
582:
554:
552:{\displaystyle x^{4}}
527:
525:{\displaystyle x^{2}}
496:
470:
379:
367:
359:
332:
284:
264:
216:
162:
129:
109:
84:
58:. It consists of two
56:Hermann von Helmholtz
41:
33:
3839:
3819:
3691:
3605:
3575:
3505:is the peak current,
3359:
3321:
3298:
3207:
3180:
3078:
2819:
2744:
2718:
2387:
2364:
2152:
2077:
2047:
1716:
1693:
1652:
1534:
1437:
1411:
1324:
1202:
1179:
1156:
1129:
1036:
1004:
836:
767:
735:
646:
591:
563:
536:
509:
479:
459:
293:
273:
225:
171:
145:
118:
114:equal to the radius
98:
3813:James Clerk Maxwell
3331:{\displaystyle R\;}
3308:{\displaystyle I\;}
1523:can be expanded in
1189:{\displaystyle x\;}
1166:{\displaystyle R\;}
1146:= coil current, in
1139:{\displaystyle I\;}
580:{\displaystyle R/2}
494:{\displaystyle x=0}
160:{\displaystyle h=R}
68:anti-Helmholtz coil
4202:2015-04-30 at the
4162:elliptic integrals
4156:Kevin Kuns (2007)
3992:June 28, 2007, at
3962:"Electromagnetism"
3852:
3825:
3803:
3775:
3676:
3588:
3483:
3350:
3328:
3305:
3283:
3193:
3155:
3047:
2802:
2730:
2701:
2699:
2370:
2346:
2135:
2059:
2030:
2028:
1699:
1675:
1635:
1513:
1423:
1394:
1297:times the current
1275:
1186:
1163:
1136:
1112:
1018:
984:
805:
748:
718:
614:
577:
549:
522:
491:
465:
445:
374:
362:
327:
279:
259:
211:
157:
124:
104:
87:
44:
36:
3847:
3828:{\displaystyle R}
3773:
3674:
3671:
3481:
3278:
3270:
3243:
3235:
3149:
3100:
2901:
2850:
2546:
2441:
2373:{\displaystyle x}
2217:
2166:
1875:
1770:
1702:{\displaystyle x}
1566:
1389:
1107:
1099:
1072:
1064:
979:
937:
803:
795:
713:
669:
468:{\displaystyle x}
282:{\displaystyle h}
127:{\displaystyle R}
107:{\displaystyle h}
16:(Redirected from
4241:
4229:Magnetic devices
4193:
4191:
4184:
4149:by Franz Kraft,
4123:
4122:
4115:
4109:
4104:
4102:
4094:
4093:on June 2, 2018.
4089:. Archived from
4082:
4076:
4075:
4068:
4062:
4061:
4059:
4058:
4044:
4038:
4037:
4035:
4034:
4019:
4010:
4009:
4002:
3996:
3983:
3977:
3976:
3974:
3973:
3964:. Archived from
3958:
3952:
3940:
3934:
3933:
3929:978-0-75067934-3
3915:
3861:
3859:
3858:
3853:
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3797:Helmholtz coils
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1972:
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503:inflection point
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449:Bessel functions
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191:
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166:
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136:electric current
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125:
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105:
34:A Helmholtz coil
21:
4249:
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4204:Wayback Machine
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3950:Wayback Machine
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3887:magnetic bottle
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827:Biot–Savart law
823:
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4129:External links
4127:
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4108:|journal=
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3805:Main article:
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76:magnetic traps
60:electromagnets
52:magnetic field
48:Helmholtz coil
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3994:archive.today
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3968:on 2011-06-03
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3881:Halbach array
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3789:Maxwell coils
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3618:
3613:
3609:
3601:
3600:
3599:
3583:
3579:
3565:
3556:
3547:
3543:
3536:
3527:
3523:
3519:
3515:
3507:
3503:
3499:
3498:
3497:
3476:
3462:
3458:
3454:
3449:
3445:
3434:
3429:
3408:
3404:
3400:
3395:
3391:
3380:
3368:
3365:
3362:
3355:
3354:
3353:
3346:
3342:
3339:
3324:
3316:
3301:
3293:
3280:
3272:
3262:
3259:
3255:
3251:
3248:
3245:
3237:
3227:
3224:
3220:
3216:
3213:
3210:
3188:
3184:
3174:
3171:
3165:
3151:
3145:
3140:
3136:
3130:
3127:
3121:
3115:
3111:
3107:
3102:
3097:
3094:
3089:
3084:
3081:
3070:
3068:
3061:
3058:
3039:
3031:
3027:
3023:
3007:
3002:
2994:
2990:
2986:
2980:
2977:
2971:
2967:
2963:
2957:
2954:
2946:
2938:
2934:
2930:
2914:
2911:
2907:
2898:
2893:
2885:
2877:
2873:
2869:
2863:
2860:
2856:
2847:
2842:
2833:
2829:
2825:
2813:
2796:
2792:
2788:
2785:
2782:
2776:
2773:
2767:
2763:
2759:
2756:
2753:
2747:
2727:
2724:
2721:
2693:
2687:
2683:
2679:
2676:
2666:
2658:
2654:
2650:
2647:
2644:
2640:
2636:
2630:
2627:
2621:
2616:
2612:
2608:
2605:
2595:
2587:
2583:
2579:
2576:
2573:
2569:
2565:
2559:
2556:
2549:
2542:
2539:
2534:
2531:
2526:
2522:
2515:
2513:
2504:
2497:
2493:
2489:
2486:
2483:
2477:
2474:
2468:
2464:
2460:
2457:
2454:
2448:
2444:
2437:
2434:
2429:
2426:
2421:
2417:
2410:
2408:
2400:
2394:
2383:
2382:
2381:
2367:
2357:
2338:
2330:
2326:
2322:
2306:
2301:
2293:
2289:
2285:
2279:
2276:
2273:
2270:
2262:
2254:
2250:
2246:
2230:
2227:
2223:
2214:
2209:
2201:
2193:
2189:
2185:
2179:
2176:
2172:
2163:
2158:
2146:
2129:
2125:
2121:
2118:
2115:
2109:
2106:
2100:
2096:
2092:
2089:
2086:
2080:
2072:
2071:Taylor series
2056:
2053:
2050:
2022:
2016:
2012:
2008:
2005:
1995:
1987:
1983:
1979:
1976:
1973:
1969:
1965:
1959:
1956:
1950:
1945:
1941:
1937:
1934:
1924:
1916:
1912:
1908:
1905:
1902:
1898:
1894:
1888:
1885:
1878:
1871:
1868:
1863:
1860:
1855:
1851:
1844:
1842:
1833:
1826:
1822:
1818:
1815:
1812:
1806:
1803:
1797:
1793:
1789:
1786:
1783:
1777:
1773:
1766:
1763:
1758:
1755:
1750:
1746:
1739:
1737:
1729:
1723:
1712:
1711:
1710:
1696:
1688:
1672:
1668:
1664:
1661:
1658:
1655:
1645:
1632:
1624:
1616:
1612:
1608:
1592:
1587:
1579:
1575:
1571:
1563:
1560:
1555:
1552:
1549:
1543:
1537:
1528:
1526:
1525:Taylor series
1507:
1503:
1499:
1496:
1483:
1475:
1471:
1467:
1461:
1458:
1452:
1446:
1440:
1420:
1417:
1414:
1391:
1385:
1382:
1377:
1374:
1369:
1365:
1355:
1349:
1346:
1340:
1332:
1328:
1320:
1319:
1318:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1269:
1265:
1261:
1258:
1248:
1240:
1236:
1232:
1226:
1223:
1217:
1211:
1205:
1198:
1182:
1175:
1159:
1152:
1149:
1132:
1125:
1124:
1109:
1101:
1091:
1088:
1084:
1080:
1077:
1074:
1066:
1056:
1053:
1049:
1045:
1042:
1039:
1031:
1012:
1008:
1000:
999:
998:
981:
975:
972:
967:
962:
958:
948:
942:
939:
931:
927:
923:
913:
909:
905:
900:
896:
889:
882:
878:
874:
869:
865:
858:
852:
844:
840:
832:
831:
830:
828:
820:
818:
797:
787:
784:
780:
776:
773:
770:
761:
743:
739:
715:
710:
706:
703:
698:
694:
685:
681:
677:
671:
666:
663:
658:
652:
649:
642:
641:
640:
638:
634:
630:
625:
611:
607:
603:
600:
597:
594:
574:
570:
566:
544:
540:
517:
513:
504:
488:
485:
482:
462:
454:
453:Taylor series
450:
439:
432:
425:
418:
411:
404:
397:
390:
383:
378:
371:
366:
358:
351:
349:
347:
342:
338:
322:
318:
310:
306:
301:
276:
254:
250:
242:
238:
233:
208:
205:
200:
196:
188:
184:
179:
154:
151:
148:
139:
137:
121:
101:
89:
83:
79:
77:
73:
69:
64:
61:
57:
53:
49:
40:
32:
19:
4178:
4144:
4099:cite journal
4091:the original
4080:
4066:
4055:. Retrieved
4051:
4042:
4031:. Retrieved
4027:
4000:
3981:
3970:. Retrieved
3966:the original
3956:
3938:
3919:
3913:
3864:Maxwell coil
3810:
3807:Maxwell coil
3798:
3569:
3554:
3545:
3534:
3525:
3517:
3513:
3501:
3495:
3351:
3340:
3317:
3294:
3175:
3169:
3166:
3074:
3065:
2815:
2713:
2359:
2148:
2042:
1647:
1530:
1406:
1317:-turn coil:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1288:
996:
824:
730:
636:
632:
628:
626:
446:
437:
430:
429:, 1.05
423:
422:, 1.01
416:
415:, 0.99
409:
408:, 0.95
402:
395:
388:
381:
369:
343:
339:
140:
93:
67:
65:
47:
45:
4166:derivatives
401:, 0.9
394:, 0.8
352:Mathematics
90:Description
4218:Categories
4164:and their
4057:2016-02-25
4033:2016-01-27
4022:Yang, KC.
3972:2007-11-20
3906:References
2380:would be:
2069:, and the
1709:would be:
821:Derivation
3892:cusp trap
3716:π
3631:π
3381:ω
3273:⋅
3260:−
3252:×
3238:⋅
3225:−
3217:×
3214:π
3185:μ
3137:μ
2978:−
2955:≈
2864:−
2777:ξ
2774:−
2757:−
2748:ξ
2725:≪
2677:−
2622:−
2606:−
2577:−
2523:μ
2478:ξ
2475:−
2458:−
2449:ξ
2418:μ
2277:−
2271:≈
2180:−
2110:ξ
2090:−
2081:ξ
2054:≪
2006:−
1935:−
1906:−
1852:μ
1807:ξ
1787:−
1778:ξ
1747:μ
1685:, so the
1662:±
1556:−
1538:ξ
1497:−
1441:ξ
1418:≪
1366:μ
1350:ξ
1259:−
1206:ξ
1102:⋅
1089:−
1081:×
1067:⋅
1054:−
1046:×
1043:π
1009:μ
959:μ
943:ξ
866:μ
798:⋅
785:−
777:×
774:π
740:μ
695:μ
601:±
315:∂
298:∂
247:∂
230:∂
193:∂
176:∂
4200:Archived
4187:archived
3990:Archived
3946:Archived
3876:Solenoid
3870:See also
141:Setting
72:gradient
4170:PBworks
4160:, uses
3799:(hoops)
3269: T
3234: T
1687:B-field
1301:in the
1148:amperes
1098: T
1063: T
794: T
758:is the
3926:
3496:where
3176:where
1028:= the
731:where
501:is an
4190:(PDF)
4183:(PDF)
3249:1.257
1078:1.257
997:Here
4119:link
4112:help
3924:ISBN
3552:and
3532:and
2981:1.83
2958:1.72
2280:1.65
2274:1.43
2228:3125
2210:2304
1407:For
1309:for
4028:EDN
3835:to
3516:= 2
3277:m/A
3242:m/A
2912:625
2894:512
2861:125
2812:is:
2145:is:
2073:of
1527:as:
1106:m/A
1071:m/A
802:m/A
4220::
4103::
4101:}}
4097:{{
4050:.
4026:.
4014:^
3885:A
3866:.
3518:πf
3256:10
3221:10
3173:.
2843:96
2177:25
2159:16
1307:nI
1085:10
1050:10
1032:=
829::
817:.
781:10
624:.
337:.
46:A
4172:.
4153:.
4121:)
4114:)
4110:(
4074:.
4060:.
4036:.
4008:.
3975:.
3932:.
3850:R
3845:3
3823:R
3769:)
3763:2
3759:L
3755:+
3750:1
3746:L
3741:(
3735:2
3730:)
3724:0
3720:f
3713:2
3709:(
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3698:=
3695:C
3669:C
3665:)
3659:2
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3646:1
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3614:0
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3584:0
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3459:R
3455:+
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3409:2
3405:L
3401:+
3396:1
3392:L
3386:(
3376:[
3369:I
3366:=
3363:V
3325:R
3302:I
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3246:=
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3211:4
3189:0
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3146:n
3141:0
3131:R
3128:B
3122:(
3116:2
3112:/
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3095:5
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2641:/
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2634:(
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2570:/
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2563:(
2560:+
2557:1
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2550:(
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2527:0
2516:=
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2498:2
2494:/
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2484:x
2481:(
2472:)
2469:2
2465:/
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2455:x
2452:(
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2401:x
2398:(
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2327:/
2323:x
2320:(
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2268:)
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2244:(
2241:(
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2215:5
2207:(
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2183:(
2173:/
2169:)
2164:5
2156:(
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2130:2
2126:/
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2051:x
2023:)
2017:2
2013:/
2009:3
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1996:2
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1988:2
1984:/
1980:1
1977:+
1974:R
1970:/
1966:x
1963:(
1960:+
1957:1
1954:[
1951:+
1946:2
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1938:3
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1810:(
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1759:I
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1730:x
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1621:)
1617:R
1613:/
1609:x
1606:(
1603:(
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1375:n
1370:0
1359:)
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1341:x
1338:(
1333:1
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1209:(
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963:0
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946:(
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883:2
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763:(
744:0
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707:I
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678:3
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667:5
664:4
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653:=
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608:/
604:R
598:=
595:x
575:2
571:/
567:R
545:4
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486:=
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443:.
441:0
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434:0
431:B
427:0
424:B
420:0
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406:0
403:B
399:0
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311:/
307:B
302:2
277:h
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243:/
239:B
234:4
209:0
206:=
201:2
197:x
189:/
185:B
180:2
155:R
152:=
149:h
122:R
102:h
20:)
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