Some Solutions for Electromagnetic Problems. Involving Spheroidal, Spherical,
and Cylindrical Bodies. James R. Wait. (September 29, 1959). Solutions are ...
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JOURNAL OF RESEARCH of the National Bureau of Standards-B_ Mathematics and Mathematical Physics Vol. 64B, No. I , January-March 1960
Some Solutions for Electromagnetic Problems Involving Spheroidal, Spherical, and Cylindrical Bodies James R. Wait (Septe mber 29, 1959) Solu tions ar e prese nted for t he low-frequency elect romagn etic response t o a n oscillaL ing m agnetic dipol e by condu cting bodies of s imple shape. The qu as i-stationa ry a pprox imation is employed t hroughout, which is valid when the relevant dimension s of t he proble m are all sm a ll compared to t he fr ee-space wavelength. Thi s a moun ts to matching solu t io ns of t he wave eq uation wi thin t he bod ies to so lu tio ns of Lap lace's eq uat ion outs ide. Th e results have a pplication to geo ph ysical prosp ectin g.
1. Introduction ElecLromag neLic method s of geophys ical exploration utiliz e th e fact that the co nductivity of massive ore bodies is mu ch greater th an the surrounding barren rock. The gen eral sch eme is to set up a primary or exciting fLCld by a curl'ent-calTyiug loop and then to detect th e seconclary fielcl or r esponse of th e body by m eans of a receiving loop. The operating frequen cy should be sufficiently low that the attenuation by the surrounding b arren rock is n egl igibl e_ This usually r equires frequencies in the audio ran ge. The literature on th e subject is exten sive and her e only ce rtain representative p ap ers are referenced [1- 4]1 TheorcLical approaches to th e subject are usually restricted to highly simplified situations. For example the exciting field is often assumed to b e uniform or th e body is taken t o be p erfectly conductin g. Wl1ile such solution do indeed provide much useful informatioll, there is a need to consider situations of a more general natme. For example, ill the practical m ethods of electrom ag netic prosp ecting Lh e transmitting and receiving coils or loops m ay be lo cated at dista nces from an ore body which are comparable to i ts m aximum dimension and thu s the uniform-exciting-field assumpLion is not valid . Fur th ermore, th e frequency which is of th e order of 500 cps is sufficiently low th at th e ore bodies seldom b eh ave as if they were of pOl'fecL conductivity. For the above r easo ns, it seems worthwhile to set up solu tions for certain idealized cases which do no t s uffer from s uch over-simplifying assumptions which are usually present. The geometrical forms considered are th e prolate spheroid, the spher e, and t he circular cylinder. Because of complexity the spheroid is taken to b e p erfectly conducting. The sph ere and cylinder arc assigned a co nductivity a nd magnetic permeabili ty which a re finite. In each case the solution is presented for the case of an arbitrarily lo cated mag netic dipole. The r esults are in a form which is sui table for computation. In each case treated, special attention is paid to t he equatorial plane which contains th e source and observer. In the tlU'ee instances, th e source is at 0 and the observer at P as indicated in figure 1. For a (y-dil'ecLed) magn etic dipol e of stre ngth K at 0 , the primary field at P is
where !Cdl is the magnetic moment. Note that Kdl = (amp-tuI'lls) X (coil area) . In what follows the secondary fields are expressed in terms of x (distance from coil axes to center of body) , y and y' (coordinates of source and observer), 8 (distance between 0 and P), rr conductivity of body (mhos/m) , and }.I / }.Io (magnetic p ermeability ratio). 1
Figures in brackets ind icate the literature references at the end of th is paper.
15
y
P(x,y,o)
s
C(x , y',o) x
f.L FIG URE l.-
Cl'OSS
(0-
°
°
~
0)
section of spheroid, cylinder, or sphere in equatorial plane (z = O), showing tmnsmitting and l'eceiver coils,
2. Low-Frequency Electromagnetic Response of a Highly-Conducting Spheroid Prolate spheroidal coordinates are introduced (p,
T} ,
8).
They ar e defined by
(eq of spheroids), (1)
(eq of hyperboloids), where p and z are the us ual radial and axial coordinates in a cylindrical coordinate system. It thus follows th at (2) and where c is the semifocal distance, These prolate spheroidal coordinates are taken to be confocal with th e spheroidal body whose surface is defi ned by T} = T} o (see fig, 2),
15=+1
major axis = c"o conducting spheroid "=,, o [ 2 minor axes = a=c
~ c
~ -+-+--+-+-I-----=--_=_ P 15=0 ____-y-_ 1--- - " = can s t.
15
=canst.
15=-1 FIGURE
2,- Th e spheroid and spheroidal coordinate system,
16
J
In the r egion exterior to the body (i. e., '1/ > '1/0) the fi elds arc a solu tion of Laplace's equation if the frequen cy is sufficiently low (i.e., signifi cant di stances should b e much less t han the wavelength). Furthermore, in this case the field s ca n be derived from a magnetic poten tial It. Thu s, ...;
H =- grad It wher e for N ow, solutions of Laplace's equation in spheroidal coordinates are well known and are of t he form (for integral valu es of TIl, and n) (3)
where A, B , A', and B' arc constants and, followin g Smythe [5], P ';: (}J. ) and Q%' (J.L ) ar e associated Legendre functions of argument J.L . Th ey are defin ed by
(4)
or
(5)
in terms of t he (ordinary) Legendre fun ctions P n(J.L ) and Q,,(J.L ). It should be noted Hobson introduces a factor (- 1)'" on tbe right-b and side of t he latter two equations and his definitions are thus slightly different. At the point C it is assumed that t her e is a magn etic point ch arge of strength , J( (see fig· 3). The magnetic potential at P can now be written
(6)
FIGURE:
3. -Conducling spheroid and source C and observer P .
17
where R is the linear distance b etween Cat (1']' , 0', 1/) and P at (1'] , 0, cf» and ns is th e secondary influence of the body . To express n in prolate-spheroidal coordinates, we make use of th e inverse-distan ce formul a
~=.!. £ R
-t, (2-om)( -
l) m [ (n - m)!J
(n+ m)!
C n= Om = O
2
(2n+ 1) P~' (0)Qr;: (1']' )P';: (1'])P~' (o' )
cos m(cf> - cf>' ),
(7 )
which is valid for 1'] < 1']'. [00 = 1, om= O (m~ O)]. The po ten tial n is a solution of Laplace's equation and since 1/R is also, it follows that \72nS= 0 for 1'] > 1']0' Thus ns m ust contain terms of the type Q~' (1'])P';:(o)
only since P~( 1'] ) is infinite at 1'] -'>00 and to the spheroid is thus written, for 1'] < 1']',
cos m(cf>- cf>' )
Q~ (o )
is infinite at 0= ± 1.
The po ten tial exterior
(8)
where (9)
Th e unknown coefficien t is now found from th e boundary condition , th at on the surface of t he spheroid t he normal component of the total magnetic field must vanish . Thus, (9)
at This is satisfied if
(10)
where t he prime indicates the deri vati ve with respect to the argument of t he Legendre fun ction . The solution is thus given by
[1
n ~ ~ M P ,;:'( 1'] o) Qm( )pm() (-I. -I.') J , ".=4j{7r f"L...J L...J 1 mn Qm'( ) n 1'] n 0 cos m '1' - '1' I, n= O m = O n 1'] 0
valid for 1»1']0' To simplify things a bi t we will consider our source to be a y-directed m agnetic dipole T lo cated in the equatorial plane of the spb eroid (z= O) (sec fig . 4). The potential at H (x, y , z) is thu s given by
o
4>= - dl " u y" =+Kdl 47r
X P r;: (o)
n
{ _~~+£ -t, P~:(1'] o) oy R
n= O m=O
Qn (1'] 0)
Qr;: (1))
[O~~n cos m(cf>- :cf>' ) +o~, cos m(cf> -
cf>').MmnJ} .
(12)
The magnetic fi eld at R is obtained from -4
H =-grad 4>. In th e case of the y-component, for example, we have 04> o2n H y= - oy = dl oy'oy·
18
(13)
p
( x , y, z)
y
c
FIGuRE
(x',
y',
0)
4.-The conducting spheroid with locations oj source C and observer P.
But
on = oQ . Ory + oQ . oa,
(23)
oQ
_
J' sin eo¢'
wh er e ifio is the stream function p ertin ent to the exterior r egion and is related to the magnetic potential by Q_ _ _
- -
1_ Ofo,
i /.low or
22
(24)
The str eam functions satisfy [ l'
2
0 1 0 ( . 0) 1 0 -01'2 +--S IJ1 0 +---sin 0 00 00 sin 0 02 2
1/;= 'Y21'21/; for l' < a 1/;0= 0 for l' > a.
2 ]
2
(25)
Thus they arc of the form
~n('Y1')
P ;" (cos 0)
for
1'< a
(26)
1'- n 1'+ (n+l) p~n(eos 0)
for
1'> a .
(27)
K n('Y]')
and
In the above J nand
fc, are
_~ [
,
a, (3 0)
where the primary so urce is denoLed QP and the influence of Lhe sp hcre by Q S. Assuming a magnetic charge (strength K ) located at point C with coordin ates (1", 0' , '), t he po tential Q P at point P with coordin ates (1', 0, ' , z') we have
o n·
X' =1" cos cf>'
y = 1' sin cf>
y' = 1" sin cf>' ,
and thus,
. e/> -0+ cos e/>- 0 - o = Sll1 pOe/> oy or
(43 )
and
()
.
,0+
{)y, = SIl1e/> or'
, 0 cos e/> p'Ocp,'
(44)
The field is thus obtained conveniently from
+.,
. , . {)2 02 H v= dl [ sme/> Sll1e/>o1"or Sll1e/> cos e/>por'oe/>
+cos e/> ,.Sll1e/> p'0e/>'o1' 0 +' 0 J cose/> cose/> p'POcpOcp' n. 2
2
(45)
24
It should be noted that
for
p' = 1"
1:)' = 7r/2
(46 )
for
p = 1'
Writing it is seen that
-=-Jll
02
1
H~= K47r oyoy' R
R = ~(X_X ') 2+ (y _ y') 2
where
for
z=z ' = 0,
(47)
and thus,
Kdl = - 47rs3
s=ly- y' l·
where
(48)
The secondary fi eld is derived from s
K
QO= -4
00
(n - m)!
n
aZ"+'n
~O ",~ ~m (n + m. )' (1,,),,+ 1,,+ 1 S" 7r n= =0 l'
P';: (cos I:))P';: (cos 1:)' ) n+ 1
,
cos m(1)- 1>).
(49)
Finally (for 1';;; CL) (50)
wher e 1
U nm =..::, [(n+ 1)2 si n 1>' sin 1> cos m(1) - 1>') 1'1'
+ m(n+ 1) sin 1>' cos 1> sin m (1)-1>' ) - m(n+ 1) cos 1>' sin 1> sin m(1) - 1>')
+m
2
cos 1>' cos 1> cos m (1)- 1>')]'
(5 1)
wh ere
Y' tan 1>' =_. x The quantity Sn plays rather an essential role in the final result. When the conductivity is great or at high frequen cy, the argument 'Ya of the Bessel fun ctions may be large. In fact if l'Yal> > 1, Sn approaches unity for all n. The behavior of Sn, as a fun ction of 'Ya and !J./!J.o, is a convenient way to illustrate the frequency dependence of the secondary fields. Numerical values of the real and imaginary parts of Sn are given in some detail for n = 1 in previous papers [2, 8] and less adequately [9] for n = 2, 3, and 4. Fortunately, if the source C or the observer P (for any respective orientations) are a distan ce from the center of the sphere which is large compared to the radius, it is seen that only the term corresponding to S[ is important. The response of the sphere is thus proportional to the quantity
S 1
'Yal;('Ya) - 2Kl,ha) 'Ya1;('Ya) + Kl l ('Ya)
wb er e
The real and imaginary part.s of S, are plotted in figure 8 as a function of the "r elative radius" X defined by or
25
For small valu es of X correspondin g to the lower frequencies, 8) is negative and, in fact, it approaches the value - 2(J{- 1) / (J{ + 2). This could have been derived directly from magnetostatics. At the higher frequencies 8) approach es unity which corresponds to no penetration of magnetic flux into the body.
4 . Low-Frequency Response of a Conducting Cylinder With Arbitrary Permeability ,iVe now wish to calculate the electrOinagnetic response of an infinitely long conducting cylinder with any permeability. (See fig . 9.) Exterior to the cylinder p>a, the fields are a solution of Laplace's equation and are derivable from a potential. Inside the cylinder the --> fields are a solution of the wave equation and are derivable from a magnetic H ertz vector 11* and an electric Hertz vector IT with only z-components. Thus (for r < a) -->
-->
2
-->
I-I= (-'i+grad div)l1*+ -,2- curl 11,
(52)
'/, fJ.W
where -->
(53)
11*= (0,0,11;')
and -->
(54)
11= (0,0,11,),
Solutions of the wave cquation (55)
are of the form (56)
where a = Ci+ h2 )! and 1m and J{m are cylindrical Bessel functions of tbe modified kind defin ed by G. N. Watson [7].
P(r,;,cp)
C (r'
FIGURE
7T
'2'
cp' )
7.-SpeciaZ configuration of (dipole) source C and observer P when they are located in equatorial plane of sphe1·e.
26
0.7 0.6 0.5
cii .; -0.5
en
a::
E -1 .0
0.4 0.3 0.1
-1 .5 0.1 -1.0 10
I
10
50
100
100
10
500 RFI ATIVE RAD I US X
FIGURE
8.- The variation of the response oj a conducting 8pheTe as a function oj X (=( >1this simplifies further
A",(h) ~
to (82)
The square bracket term in eq uation (81) thus characterizes the frequency dependence of the response and is a function only of K and 'Ya. The secondary magnetic field Fls for the ease of finite 'Ya can then be written in terms of -7
value of Fls for infinite 'Ya in the following way (for a given mode numbrr m) (m = 1,2,3, .. . ),
(83)
where
T = _ [ mKI", (/t X) _·/{ XI,~ (/t X) ] ", mKI", (--/i X) + ,Ii XI,;, (~i X)
(84)
and wh ere X = (IT/1-w) ~a. The total field is, of course, a sum over all modes, but because of the factor I m(ha )/K m(ha) , the relative response decreases rapidly \vith in cl'casi!lg vflJUP. of m. This folloviTS from the approximation 1m(ha) (ha)2m (85) K m (ha) ,...., (m!) (m - 1) ! 22n - 1 ' which is valid for (ha)2« l. To shed further light on the above approximate form , it is constructive to examine a related problem . If the so urce is an infinite line source of electric curren t 10 located at (Po, CPo ) and parallel to t he z-axis, it is known tha t the secondary fi eld s due to the presence of the cylin de']" are [10]
1 '"
a 2m
I-I~= -2° ~ Tm m ",+1 sin m (cp - CPo) 7r m~ 1
Po P
(86)
and (87)
30
The structure of thi s exact two-dimensional solu tion is very similar to tbe approximated form of tltO three-dim el1sional solution ment ioned above. Apparently if a is reasonably small, the induced currents flow mainly in the axial direction even for a lo calized excitation and thus Lite fr equen cy depenrlence of the induced eddy currents are adeq uatcly desc rib ed by t he form of the solution of tlte corresponding two-dimensional problem . TltO 11 umerical values of the real and imaginary parts of the function Tm a re illustrated in figure 11 , where they are plotted as a funetion of X, t ltO " relative radiu s" of the cylinder , for vfl,rious val ues of the permeability ratio K. It is noted t hat the factor T I , corresponding to the dominant mode, is very similar to the factor 8 1 for the sphere shown in fi g ure 8. 0.5 08 0.6
OA
OA 0.1
0.3
..:
;-.-
'"
E
a: -0.1
0.1
- 0.4 -0.6
0.1
zoO
-0.8
1000 -I
0.5 0.8 0.6 -
OA
0.4 0.1
0.3
~
~
.§
'" a: -0.1
0.1
-OA -0.6
0.1
-0 8 -I
0.5 0.8 OA
0.6 OA -
0.3
0.1
'"
~ a: '"
;-.
E 0.1
-0.1 -OA -0.6
0.1
-0 8 1000 -I
0.5
10
200 0 0.5
30
RELATIVE RADIUS
FIGURE
x
ll.-The variation oJ the res ponse oJ a conducting cylinder as a Junction permeability ratios K ( = .u/.uo).
rrhc real and imagin ary p arts of "/\, 7'2, and T a arc sho\\"n .
31
oJ XC = Ca.uw)t
a) for variolls
The work in this paper was carried out while the author was employed by N ewmont Exploration Limited, Danbury, Conn. The author thanks Drs. A. A. Brant and N. F . Ness of this organization for their h elpful suggest.ions and advice.
6 . References [1] C. A. H eila nd, Geophysical exploration (Prentice-Hall, Inc., New York , N.Y., 1940). [2] J . R. Wait, A co ndu cting sphere in a time varying magnetic field , Geoph ysics 16, 666 (195 1). [31 J . R . 'W ait, Mutual electromagnetic cou pling of loops over a homogeneo us ground , Geophysics 20, 630 (1955); contains many references to related work. [4] S. H . 'Yard and H . A. Harvey, E lectromagnetic s urveying of di a mo nd drill holes, Can. Mining Manual, 1 (1954). [5] \'V. R. Smythe, Static an d dynamic electri city, 2d ed. (McGraw-Hill Boo k Co ., New York, N.Y., 1950); such functions are tabulated in Tables of assoc iated Legendre pol y nomials (Columbia Univ . Press, New York, N.Y.). [6] S. A. Schelkunoff, E lectro magnetic waves (D . Van Nostrand Co ., In c., New York, N.Y., 1943). [7] G. N. vVatson , Theo ry of Bessel fun ctions (Cambridge Un iv. Press, Engla nd, 1945). [8] S. H . vVard, A method for meas urin g the electri cal co ndu ctivi ty of diamond drill core spec imens, Geoph ysics 18, 434 (1953) . [9] J. ll. vVait, A co ndu ctin g permea ble sphere in t he prese nce of a coil cal"l'ying an oscillating current, Can. J. Phys. 31, 670 (1953) . [lO J J . ll. vVait, The cy lindrical ore body in the prese nce of a cable car rying an oscillating current, Geoph ys ics 17, 378 (1952) .
(Paper 64Bl- 16)
B OULDER, COLO.
32
