NASA TN D-.4.65 _D < < Z TECHNICAL NOTE D-665 THE By MAGNETIC Edmund FIELD E. OF Callaghan Lewis and Research Cleveland, NATIONAL AERONAUTICS WASHINGTON A FINITE SOLENOID Stephen H. Maslen Center Ohio AND SPACE ADMINISTRATION Oetober 1960 6 NATIONAL AERONAUTICS AND TECHNICAL THE By O O MAGNETIC Edmund E. FIELD Callaghan SPACE NOTE OF A ADMINISTRATION D-465 FINITE and SOLENOID Stephen H. Maslen ! SUMMARY The axial and radial fields at any point inside or outside a finite solenoid with infinitely thin walls are derived. Solution of the equations has been obtained in terms of tabulated complete elliptic integrals. For the axial field an accurate approximation is given in terms of elementary functions. Fields internal and external to the solenoid are presented in graphical form for a wide variety of solenoid lengths. INTRODUCTION ! O The recent great interest in plasmas either as a source of energy or as a propulsion device has resulted in a greatly renewed interest in the magnetic fields produced by various configurations of electromagnets. Of the possible methods of plasma confinement by far the most promising appears to be the use of magnetic fields (ref. I). The calculation of the fields generated by various electromagnetic configurations such as loops_ finite helical solenoids, and infinite solenoids has been treated by the early classical physicists, but only the simplest cases such as the single loop have been calculated for the entire field both inside and outside the loop (e.g., ref. 2). In other cases such as the helical solenoid or the finite solenoid the calculations have been limited to the axis (ref. 3). axis positions have been done by Foelsch (ref. are obtainable by means of a large number which are valid over restricted ranges of Derivations 4), and the of the offsolutions of approximate expressions size or position. The princi- pal difficulty in the calculation of the fields of nearly all configurations has resulted from the fact that the integral solution cannot be achieved without the use of various elliptic integrals. Even though many of these are tabulated_ the calculations involved are laborious. Such calculations can, however, be made using modern high-speed computers since machine programs have or can be written for many of the elliptic functions. The purpose of this report is twofold: (i) to derive the equations of the axial and radial field at any point within or outside a finite solenoid in terms of standard tabulated functions and (2) to plot these fields for a number of solenoids. SYMBOLS A@ magnetic a coil Br,Bz radial E complete i current K complete vector potential component in @-direction magnetic induction component radius and axial elliptic in each integral, second kind filament elliptic integral, first kind k L coil n number r_ e_ z cylindrical ho Heuman g permeability z_ length of turns per unit coll length coordinates lambda function L 2 I qD DERIVATION Consider a solenoid as shown OF E:QUATION3 in the L/2 _b z,Z follewing sketch: 3 The magnetic potential A field by due to this coil is given in terms of the vector (I) =VXA where; for the geometry assumed_ Then equation (i) yields simply Br only the A8 component can be nonzero. = _ (2) I. ;_Ae Bz = l _(rAe) r For a single circular filament_ Ae I ;gr one has = 4_i_ifa c°sR 8 d8 O ! where R is the distance from the local point on the filament to the field point. For a solenoid made up of a series of n filaments per unit length_ we have then <9 L/2 cos Ae=_/ a_o V(z Z)2 +'r 2 8 d8 + a 2 _ 2at cos 8 or A0= f /o 2_ where origin cos + r 2 + a 2 _ = z - Z, _ = z ± L/2; and Z to the filament. On integrating i o = _ cos e in 2_ 8 d8 (3) d_ + _2 - 2ar is the axial with respect cos 8 distance from the to _ this becomes + r 2 + a 2 _ 2ar cos __ ae (4) 0 A more Ae convenient form = a_nilsin2_ 0 inI_ can be found by integrating by parts: + e=0 ar +r 2 +a 2-2ar sin?8 cos de e)V_2 ]_+ +r 2 +a?_ 2at cos 8J{_ 4 The first _2 Ae r2 + V_2 use term vanishes. a 2 + + r 2 + a2 of = the Onmultiplying _ 2dr cos 8 - _ _ 2dr cos @ - limits a2_ nit eliminates rearranging one term, terms, there and I follows = _Ao - _-_- = _ sin28 + d8r2 8)V_2 cos can now equation Bz, first as in the - 2dr cos The 0 d0 _ alani from equation equation (5), _ cos r 2 + a2 - 2ar cos O(r -a 0)_'2 :os radial (G) 2_ evaluate _A@/_r case of obtazning (5) 8] __ cos be easily obtained. (Z) and yields _ - + a2 _2 + r 2 + a 2 _ 2dr _AO_ that {+ I( a 2 + r 2 - Zar The two magnetic-field components field is found by differentiating To get ceeding observing _ 2_ Br the integrand by cos The (4). is __ result_ pro- O)dO t- r 2 + a2 - 2dr cos __ (7) If equations Bz _ i r _(rA8) _r (S) and = 2,_ (7) are f[ put into (2), th_ result is _(a-0) ?cose)de (r2 +a2 2atcos +r2+a2 2= cos0]_+ (8) Equations (6) and (8) describe the magnetic field due to a finite sole- noid. Numerical results can readily be found by integrating these equations on a computer. However, the results eal also be expressed in terms of standard elliptic integrals_ which as already tabulated. This we proceed to do. 412.01 Consider (noting B r. the This can be special case evaluated by Ise of formulas of _2 = k 2) )f reference 5. 291.05 and One has successively, Br = m __r " + m_. (r + a) 2 IK(k) m_._._2)sn2u di:+ 1(.2- - k2sn2u _ I co O O 5 + (r + a)z i - K(k) Br O o ob ! = _ Tr E(k)][+ 2k - (9) k J__ where k2 = 4ar (a + r) 2 _Z + In a similar integrals. First then use formulas successively manner, change 233.19 B z (eq. the and (8)) can (10) be reduced variable of integration 413.06 of reference 5. to standard elliptic to t = cos @, There follows and 9 Bz __ Bz r22+ - t a2 (1 - t _) _2 +2arr2 1 _ni - 2_(a + r) _k Sz = F i (a _ [._K(k) + r 4ar g_ du (a + r)2 snZu [k - t sn2u , a 0 n__. + a 2 - _- r)[ + ]"(a r)_l (11) ;%(m,kj__ where (12) As before, ho(_,k k is given ) is tabulated equation (I0). in references by 5 and For many purposes_ it is convenient of the fields near the axis. As r _ 0, to the following well-known expressions: Br : ,_ _ni [(_2 The Heuman lambda function 6. simply to equations a2,3/_, - +a2r / . ._]_+ know the variation (9) and (ii) reduce (13) (1A) _2+a2 A convenient to i percent approximation for in this range, is Bz = m(l + Bz, 2k') valid + __ whenever 4 r _< a and accurate * C C I CO 0 0 where actly m = (i- k')/(l + k'), k' =_to equation (14) at the axis. k 2. Equation (15) ex- reduces CALCULATIONS with Equations (9) and (ii) are readily written in dimensionless the distances given in units of the solenoil radius. Then tions (9) to (14) _+ throughout. Plots of -4Br/_ni , are still hold the dimensionless given in figures made for the ratio of toll Note that in figures 1 and the coil radius (r/a) and, coil half-length (2z/L). but with ij r/a, _h/a replacing form equaa, axial and radial fields_ +4Bz/_ni _ 1 and 2, respectively. Calculations length to radius in the range from i to 2 the radial distance is given in terms the axial distance is given in terms of r, were 25. of the Discussion The figures clearly show that increasing solenoid length decreases the radial variation of the axial field. This r_sult is expected since an infinitely long solenoid has a uniform field ;hroughout. For short solenoid lengths (fig. l(a)), the axial field increases rapidly from the center to the wall for positions near the ee:Iter of the solenoid. In fact, at the center the curve approaches very closely that for a simple loop. It should be noted that the radial field is always infinite 2z/L = 1. O and r/a = 1. This point corresponds to the edge of current sheet and would be expected to produce s lch a result. In general, uniform fields with total variations be achieved over as much as 60 percent of the internal solenoid if the length is 25 radii or greater. at the of i percent can volume of the I The calculations presented herein are limited to solenoids with infinitely thin walls but the results can readily be used to find approximate solutions for various shaped solenoids of finite thickness with almost any current distribution. Since superposition principles apply, it is only necessary to approximate any odd-shaped solenoid by a number of thin-walled solenoids and add the fields resulting from each. The accuracy of the answer is, of course, dependent on the number of separate solenoidal rings used to approximate the actual shape. oo It is interesting to note that the results obtained herein for magnetic fields are closely related to the velocity fields produced by a lifting helicopter rotor (e.g., ref. 7). The physical model is the same but the detailed methods of solution are widely different. Lewis Research Center National Aeronautics and SpaceAdministration Cleveland, Ohio, May 23, 1960 REFERENCES !. Bishop, AmasaS.: Project Sherwood- The U.S. Program in Controlled Fusion. Addison-Wesley Pub., 1958. 2. Scott, William T.: The Physics of Electricity Wiley & Sons, Inc., 1959. and Magnetism. John 3. Mapother, Dillon E., and Snyder, JamesN.: The Axial Variation of the Magnetic Field in Solenoids of Finite Thickness. Tech. Rep. 5, Univ. Iii., Nov. 16, 1954. (Contract DA-II-O22-ORd-992.) 4. Foelsch, Kuno: Magnetfeld und Induktivitaet einer zylindrischen Spule. Archly f. Elektrotech., Bd. XXX, Heft 3, Mar. i0, 1936, pp. 139-157. 5. Byrd, Paul F., and Friedman, Morris D.: Handbookof Elliptic Integrals for Engineers and Physicists. Springer-Verlag (Berlin), 1954. 6. Heuman,Carl: Tables of Complete Elliptic Phys., vol. 20, Apr. 1941, pp. 127-207. Integrals. ,our. Math. 7. Castles, Walter, Jr., and DeLeeuw,Jacob Henri: The Normal Component of the Induced Velocity in the Vicinity of a Lifting Rotor and Some Examples of Its Application. NACARep. 1184, 1954. 8 GI K_ iiii i__+_:__$i_-t _J I CO o 0 /,., , U!Y ., :x -_ _B:Li2tj£i _p d c' ® ii!! i_!l-;._:i ;: .... I!i_: !: i. I:' , ]!il]iii_t_: _T£ _co " 2:4i ilii i U :'4_2. T ,. ........_ I:,H _H _" , '__ ._..,:z!_u: .... @ 4_ -__:ia!2"i,.'_.'''÷_'-:-- o JR7 .... • ÷ @ rq @ C c6 ;:. p v ::W.:: _..::{i !L' _:;_;: !:_I":I'G.:: i;;: t;:T:{!!.d£!:T_!_F!_ :_t!i:_,_ 'h,J-,_"_111I_: r, _ " @ 0 __'; _ .... ! ..... ...................t!l : imq:, ' :::ti :: Y: ;: IT":i __ __ ::::t1 44_ ;i'::!: ,_' :_ : ': .... I_i!//,tlH _! I ,+.... [ , ,l _ _2 ......+" IWir/iiti::}I:dl!:lvtN_ i_,td: ::i_t i ' _:: : : : ' : ' ' '_ I:::';_ ' ' .":--.i,}" ::l:-'J2_ : ]:: -Ti4:,_':' : ' _ : " ":',|:.*_t,, '_:" .... ! ] I_ _ ,li_ _- t ...., 2+' -,m 'i !t';_:Ti,_!, :,_: ,:: iT'- '' _ _ , ' i : : , :Hh,I " ,Y_!!!:!::qt ,_ _ :t : i :: ::_':._ E I ": ::: 4 £ b_ F:iH:::ili!.hhH:h!i:-fl,:_iu_FtU:-H,:. , :" : _., ¢1 ,. '1 ; ....... _ :;t t -t:':;t ,} i:. : : ;!_: _ ''2} %.. , ,,, !. Yti: , _ 1! :!i ............. ' : "_:f : I_ $1 t:!!: ' ::_'_,,,I, [ : !:_f: O 'i[ : ' co , : i r" T ......... r ] i :1: [ ................ i! .......... I ; :t ti : rI_1 ..... _ "" ' ' - "_f :Tf q ; ' 0 CO _ "_ C_! 0 cO LiD _ Oa 0 bQ C,.I Cq O_ O,1 O1 H H r--I H _1 : ........ ! :i _Lt_ " _.......... t r_ [ :2'I!_E , , ;TH_IFH!I!:I.:'X _ [:i= '_!. t ' ; a3 i'!2 f "_....... ..... t ..... _:i i :i: ! H 4'_ CO ' t ' ...... _ _ :i: lt'-',l ' 04 ' Or-< I o © oh ! #q ,g [a _H c_ 0 I o" oJ V II c_ Q) A 6) ,.-i rj I H 0 I0 I <D 0 0 0 • u/a "snTp_ ° s¢_IUOTSU_G 11 0 0 Cb I 'K1 gl _d 0 ,q ! 4 _ 0 © o) £7 O r,m I © .H Cr_ 12 ! LO 0 0 0 • _/_ C_n_P_ sgaIUO_UemIC ° 15 0 0 C_ I I 0 0 0 0 • • I IS 0 0 0 G) I M 0 16 :4; ffl _F F_ o a,+f J_t !;4 .i;i 2.8 .4 Dimensionless radial (a) Figure 2. - Dimensionless L/a radial field, = 4J r/_ni i. field of a finite solenoid. ! r.o o o 17 .4 .8 1.2 Dimensionless 1.6 radial (b) Figure 2. - Continued. Dimensionless "_/a 2.0 field, = 2.4 2.8 4Br/#ni 2. radial field of a finite solenoid. 3.2 18 I <0 0 0 .4 .8 1.2 Dimensionless 1.6 radial 2.8 2.4 2.(, field, 4Br, _ni (o) L/_ = _. Figure 2. - Continued. Dimensionless radial field of a finite solenoid. 3.2 19 0 0 O_ I 2.8 2.6 1o .4 .8 1.2 1.6 Dimensionless radial 2.0 field, 2.4 2.8 4Br/_ni (d) L/_= 4. Figure 2. - Continued. Dimensionless radial field of a finite solenoid. 3.2 2O 2.8 ! QO (D CD 2.6 2. 2. 2. C 1.8 l.E 1.4 i. i.( .4 .8 1.2 Dimensionless I.G radial 2.0 field_ 2.4 2.8 iBr/_ni (e)L/a = _. Figure _, - Continued. Dimensionless radial field of a finite solenoid. 3.2 21 3.0 0 0 2.8 4 2.6 2.4 2.2 2.0 1.8 % 1.6 °H n "_ n 1.2 1.0 .8 .6 .& .2 (f) L/a = lO. Fi_e 2. - Continued. Dimensionless radial field of a finite solenoid. 22 II :!!! i!tt itH IUt I,w i;i! o_ M_ lift A _r i! If' ._r .4 .8 1.2 1.6 Dimensionless (g) Figure 2. - Continued. 2.0 radial Dimensionless L/a field, = 2.8 2.4 Z Br/_ni 15, radial field of a finite solenoid. 5.2 ! t..O 0 0 23 .4 .8 1.2 Dimensionless Figure hASA - L.cFloy }ielcl, Va. _-,_<. 2. - Concluded. 1.6 radial Dimensionless 2.0 field, radial 2.4 2.8 ABr/_ni field of a finite solenoid. 3.2

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