Transversely Magnetized Oblate Spheroids Troy C. Richards Defence R&D Canada - Atlantic [email protected]September 12, 2005 Abstract The use of magnetized spheroids has attracted considerable interest as a building block to model and predict the static magnetic fields around steel structures. De- fence R&D Canada (DRDC) has made effective use of magnetized prolate spheroids for the near field magnetic modelling of naval ships as required for mine simulation studies. For some geometries however, oblate spheroids may provide a more real- istic representation of the structure. In particular, presented here is the derivation of the scalar magnetic potential and magnetic fields arising from a transversely magnetized oblate spheroid. Additionally, the solution is verified to produce; the results for a sphere, a point dipole in the farfield and lastly a transversely magne- tized prolate spheroid. 1 Introduction Modelling of the static magnetic fields of naval ships is required to assess the vessel’s vulnerability to sea mines, which exploit this signature for mine detonation. Magnetized prolate spheroids are routinely used by DRDC to model a ship’s static magnetic signa- ture and to conduct ship-mine simulation studies. The technique uses a distribution of magnetized prolate spheroids about a volume representing the vessel and magnetization weights are obtained by fitting the model to collected magnetic field data. For some structures an oblate spheroid (or a combination of prolate and oblate spheroids) may provide a more realistic representation of the structure being modelled. In particular, presented here is the derivation of the scalar magnetic potential and magnetic fields aris- ing from a transversely magnetized oblate spheroid. In addition, the solution is verified asymptotically to produce the results for a sphere and a point dipole through the use of the appropriate limits. Lastly the transversely magnetized prolate spheroid solutions are derived using a transformation of the coordinate parameters. Maple worksheets with the solution details are provided as an appendix. 1
The use of magnetized spheroids has attracted considerable interest as a buildingblock to model and predict the static magnetic fields around steel structures. De-fence R&D Canada (DRDC) has made effective use of magnetized prolate spheroidsfor the near field magnetic modelling of naval ships as required for mine simulationstudies. For some geometries however, oblate spheroids may provide a more real-istic representation of the structure. In particular, presented here is the derivationof the scalar magnetic potential and magnetic fields arising from a transverselymagnetized oblate spheroid. Additionally, the solution is verified to produce; theresults for a sphere, a point dipole in the farfield and lastly a transversely magne-tized prolate spheroid.
1 Introduction
Modelling of the static magnetic fields of naval ships is required to assess the vessel’svulnerability to sea mines, which exploit this signature for mine detonation. Magnetizedprolate spheroids are routinely used by DRDC to model a ship’s static magnetic signa-ture and to conduct ship-mine simulation studies. The technique uses a distribution ofmagnetized prolate spheroids about a volume representing the vessel and magnetizationweights are obtained by fitting the model to collected magnetic field data. For somestructures an oblate spheroid (or a combination of prolate and oblate spheroids) mayprovide a more realistic representation of the structure being modelled. In particular,presented here is the derivation of the scalar magnetic potential and magnetic fields aris-ing from a transversely magnetized oblate spheroid. In addition, the solution is verifiedasymptotically to produce the results for a sphere and a point dipole through the use ofthe appropriate limits. Lastly the transversely magnetized prolate spheroid solutions arederived using a transformation of the coordinate parameters. Maple worksheets with thesolution details are provided as an appendix.
1
1.1 Magnetized spheroids
Both prolate and oblate spheroids are volumes formed by revolving an ellipse around oneof its principal axes. Prolate spheroids are egg-like while oblate spheroids are similarto discs (or flying saucers). Their use in magnetic modelling of ships is constrainedto the case where the spheroid has a uniform permanent magnetization or where thespheroid’s magnetic permeability is constant and the spheroid is placed in a uniformapplied magnetic field, the latter is referred to as the induced case. The solutions for themagnetized or induced cases are shown in [1] to differ by only a geometric constant, inother words the induced magnetic field of a spheroid with constant magnetic permeabilitywhen placed in a uniform field, is also uniform. In contrast to this, the report shows thatthe solutions for a magnetization transverse to the axis of revolution must be treatedseparately from that when the magnetization is longitudinal to the axis of revolution.The solution for an oblate spheroid with a transverse magnetization is derived here, thelongitudinally magnetized oblate spheroid although not presented can be derived in asimilar manner.
1.2 Magnetostatic review
Under the condition of a steady magnetic state and assuming that no electrical cur-rents are present Maxwell’s equations simplify to the magnetostatic case (see [2] for amore complete description of magnetostatics) described by the following two vector fieldequations
∇×H = 0 (1)
and∇ ·B = 0. (2)
where H is the magnetic field intensity in (A/m), B is the magnetic field in (T), andthe curl (∇×) and divergence (∇ · ) operators are defined in their usual manner. Anyvector field with a curl of zero can be expressed as the gradient of a scalar potential andtherefore the magnetic intensity can be written as
H = −∇ψ (3)
where ∇ is the gradient operator and ψ is the scalar magnetic potential.Magnetized spheroids can be treated as permanent magnets with a magnetization
M existing inside the spheroid, while outside the spheroid the magnetization drops tozero. Inside the spheroid the magnetization is related to the magnetic field and magneticintensity by
B0 = μ0 (H0 + M) (4)
where μ0 is the magnetic permeability of free space and a 0 (1) subscript is used to denotefield values inside(outside) the spheroid. Observe that the magnetization has the sameunits as the magnetic intensity. Substitution of (4) into (2) and using H0 = −∇ψ0 yields
∇2ψ0 = −∇ ·M (5)
2
where ∇2 = ∇ ·∇ is the Laplacian operator. When M is uniform as considered here∇ ·M = 0 (except at the spheroid surface) and
∇2ψ0 = 0 (6)
which is known as Laplace’s equation for magnetostatics. On the surface of the spheroidthere is a an abrupt change in M and (5) must be used. Outside of the spheroidB1 = μ0H1 since a constant magnetic permeability is assumed and we have ∇2ψ1 = 0.
1.3 Oblate spheroidal spheroidal coordinates
Problems involving the magnetic fields of an oblate spheroid are best solved by convertingto an oblate spheroidal coordinate system, such as the one shown in Figure (1). Forthis system the cartesian coordinates (x, y, z) map to the oblate spheroidal coordinates(ξ, η, φ) using the following equations
x = c√
(ξ2 + 1) (1 − η2) cos (φ) (7)
y = c√
(ξ2 + 1) (1 − η2) sin (φ) (8)
z = c ξ η, (9)
where c =√b2 − a2 is the focal length of the ellipse and a and b are the semi-minor and
semi-major radii of the spheroid with a < b. Note the formation of the oblate spheroidas the ellipse revolves around the z-axis. Through some rather tedious manipulation1
the inverse mapping is expressed as
ξ =1
2c
(√x2 + y2 + (z − jc)2 +
√x2 + y2 + (z + jc)2
)(10)
η =j
2c
(√x2 + y2 + (z − jc)2 −
√x2 + y2 + (z + jc)2
)(11)
φ = arctan(y
x
). (12)
where j is the imaginary unit. It can be shown that the oblate spheroidal coordinatesare real valued and defined on the following ranges
0 ≤ ξ <∞ − 1 ≤ η ≤ 1 0 ≤ φ ≤ 2π. (13)
When ξ = ξ0 a constant, the surface of an oblate spheroid is generated and the radii of
the oblate spheroid can be expressed as a = c ξ0 and b = c√ξ20 + 1. The volume of the
oblate spheroid is given as
v =4π
3a b2 (14)
=4π
3c3ξ0
(ξ20 + 1
). (15)
1Maple’s MapToBasis function arrives at an equivalent but different form of these equations.
3
x
y
z
a
bb
ξ
η
φ
Figure 1: Oblate spheroidal coordinates (ξ, η, φ) in relation to cartesian coordinates(x, y, z).
1.4 Transversely magnetized spheroid
We impose that the spheroid possesses a constant magnetization in the x-direction whichis transverse to the axis of revolution. This magnetization can then be described as
M = Mxex (16)
where Mx is constant and ex is the unit vector in the x-direction. Inside the spheroidbecause the magnetization is constant the scalar magnetic potential will vary linearlywith the x position and can be represented as
ψ0 = d Mx x (17)
where d is the transverse demagnetizing factor, a dimensionless constant which dependson the spheroid geometry. The magnetic field intensity inside the spheroid is using (3)given as
H0 = −d Mx ex (18)
and for the magnetic field
B0 = μ0 (1 − d) Mx ex (19)
= μ0
(1 − 1
d
)H0. (20)
Any solution must also satisfy the boundary conditions for the magnetic field and mag-netic intensity such that the component of B normal to the boundary is conserved whilethe tangential component of H is conserved.
It is also worth introducing the magnetic moment m in (A m2) which is defined asthe product of the magnetization and the volume of the magnetic material v in (m3) suchthat m = M v. Both the magnetization and dipole moment are vector quantities.
4
2 Solution of Laplace’s equation in oblate spheroidal
coordinates
Outside of the oblate spheroid the scalar magnetic potential obeys Laplace’s equationwhich in oblate spheroidal coordinates can be expanded as
(ξ2 + 1
) ∂2 ψ1
∂ ξ2+ 2ξ
∂ ψ1
∂ ξ+(1 − η2
) ∂2 ψ1
∂ η2− 2η
∂ ψ1
∂ η− ξ2 + η2
(ξ2 + 1) (1 − η2)
∂2 ψ1
∂ φ2= 0. (21)
If ψ1 is separated into the product of three functions, each a function of a single oblatespheroidal coordinate i.e ψ1 = R (ξ)T (η)P (φ), (21) will separate into the followingordinary differential equations
(ξ2 + 1
) d2 R
d ξ2+ 2ξ
d R
d ξ+
(m2
(ξ2 + 1)+ k
)R = 0 (22)
(1 − η2
) d2 T
d η2− 2η
d T
d η+
(m2
(1 − η2)+ k
)T = 0 (23)
d2 P
d φ2+ m2P = 0. (24)
Where k and m are introduced as separation constants (see [3, p420] for a reference usingprolate spheroidal coordinates). Defining k = −n (n − 1) the general solutions for theseequations are known and given as
R = am,nPmn (jξ) + bm,nQ
mn (jξ) (25)
T = cm,nPmn (η) + dm,nQ
mn (η) (26)
P = em sin (mφ) + fm cos (mφ) (27)
where Pmn and Qm
n are the associated Legendre functions of the 1st and 2nd kind2 re-spectively of order m and degree n and am,n, bm,n, cm,n, dm,n, em and fm are constants tobe determined to match the boundary conditions and satisfy finiteness of the solution.Since our solution for the magnetic potential must be finite over all space dm,n = am,n = 0and recombining the constants the general solution for the magnetic potential outsidethe spheroid is
ψ1 =∑n
∑m
bm,nQmn (jξ)Pm
n (η) (em sin (mφ) + fm cos (mφ)) . (28)
2.1 Matching the boundary conditions
First the magnetic field intensity inside the oblate spheroid is mapped onto the oblatespheroidal coordinate system yielding
H0 = −d Mx ξ√
(1 − η2) cos (φ)√(ξ2 + η2)
eξ +d Mx η
√(ξ2 + 1) cos (φ)√(ξ2 + η2)
eη +dMx sin (φ)eφ. (29)
2These definitions are in accordance with those used in Maple 9.5
5
Outside of the spheroid the magnetic intensity is computed as the negative gradient ofthe potential ψ1. To match the boundary conditions an investigation of the Legendrefunctions reveals m = 1, n = 1 and f1 = 0. For the tangential component of H
H0η = H1η (30)
and matching the normal component of B
(1 − 1
d
)H0ξ = H1ξ. (31)
Solving the simultaneous set equations yields the solution for the demagnetizing factoras
d =1
4ξ0(ξ20 + 1
)arctan
(2ξ0, ξ
20 − 1
)− 1
2ξ20 (32)
where arctan is the two-argument inverse of the tangent function and inspite of thesingularity at ξ = 1 is well behaved for all ξ > 0. The potential outside of the oblatespheroid is also determined and is given as
ψ1 =1
2ξ0(ξ20 + 1
)c Mx
(1
2
√ξ2 + 1 arctan
(2ξ, ξ2 − 1
)− ξ√
ξ2 + 1
)√1 − η2 sin (φ) .
(33)The magnetic intensity is obtained, as usual, as the negative gradient of this potential.
The reader is referred to the appendix for these equations. It is noted that the cartesiancomponents of H1 are derived in terms of the oblate spheroidal coordinates. Thus givenan (x, y, z) cartesian location the oblate spheroidal coordinates are computed using (10)-(12) and then substituted into the equations for the potential and magnetic intensity.Figure(2) provides a pictorial representation of the scalar magnetic potential and themagnetic intensity.
3 Solution verification
3.1 Uniformly magnetized sphere
As a → b the oblate spheroid will become a sphere and the solution should simplify tothat of a magnetic dipole as described by [2, p194]. We look first at how the oblatespheroidal coordinates are transformed to spherical coordinates. In the limit as a → b,c → 0 but c ξ → r remains finite. Further substitution that η = cos (θ) leads to the(standard) spherical coordinate system
x = r sin (θ) cos (φ) (34)
y = r sin (θ) sin (φ) (35)
z = r cos (θ) (36)
which justifies our original choice of the left-handed oblate spheroidal coordinate system(see Figure (1)) since (ξ, η, φ) → (r, θ, φ).
6
Figure 2: Magnetic scalar potential and magnetic intensity of a transversely magnetizedoblate spheroid on the x − z plane. The scalar magnetic potential is represented usinga colour intensity and the magnetic intensity is represented using logarithmically scaledquivers.
Inside the sphere, we can apply ξ0 → ∞ to determine that the demagnetizing factorconverges to 1/3 and that the scalar magnetic potential becomes
ψ0 =1
3Mx x (37)
as required. For the scalar magnetic potential outside the sphere, the term in ξ is ex-panded as a series with c ξ → r and with Mx = mx/v the scalar potential becomes
ψ1 =mx
4πr2sin θ cosφ, (38)
that of an x-oriented magnetic dipole in the standard spherical coordinate presentedabove. The potential outside of a sphere with uniform magnetization is precisely that ofa magnetic dipole.
3.2 Point dipole in the farfield
At distances much greater than the size of the spheroid, we have ξ � 1 and with c ξ → rand η = cos (θ) as before the oblate spheroidal coordinates are approximated by thespherical coordinates above. Applying the same expansion of the ξ term and substitutingMx = mx/v yields the result in (38) as required.
7
3.3 Transversely magnetized prolate spheroid
To obtain the solutions for the prolate spheroid the following transformation of the oblatespheroidal coordinates can be made [4, 5]
c → −jc (39)
ξ → jξ. (40)
Applying this transformation of variables the mapping from prolate spheroidal coordi-nates (ξ, η, φ) to cartesian coordinates (x, y, z) is
x = c√
(ξ2 − 1) (1 − η2) cos (φ) (41)
y = c√
(ξ2 − 1) (1 − η2) sin (φ) (42)
z = c ξη (43)
where ξ > 1 is required. The transverse demagnetizing factor transforms to
d = −1
4ξ0(ξ20 − 1
)ln
(ξ0 + 1
ξ0 − 1
)+
1
2ξ20 (44)
while the scalar magnetic potential outside the spheroid becomes
ψ1 =1
2ξ0(ξ20 − 1
)cMxξ0
(ξ√
ξ2 − 1− 1
2
√ξ2 − 1 ln
(ξ + 1
ξ − 1
))√1 − η2 cos (φ) (45)
which are the same results derived in [1].
4 Conclusions
The scalar magnetic potential, magnetic field and magnetic intensity of an oblate spher-oid with a uniform transverse magnetization were derived. The results were verified toconverge correctly for a sphere and a point dipole at large distances. The transversemagnetized prolate spheroid solution was also determined by using a transformation ofthe coordinate parameters.
The use of Maple to derive these results does not eliminate the need to understandthe underlying physics of the problem but greatly eases the algebraic burden required forthe derivation.
[1] Holtham, Peter M. and Lucas, Carmen E. (1993). New Approaches to MagneticModelling I. Prolate Spheroids II. One-Spike-at-a-Time Fitting. (Technical Report93-81). Defence Research Establishment Pacific.
8
[2] Jackson, John David (1975). Classical Electrodynamics, 2 ed. John Wiley and Sons.
[3] Stratton, Julius Adams (1941). Electromagnetic Theory, McGraw-Hill PublishingCompany.
[4] Falloon, Peter (2001). Hybrid Numeric/Symbolic Computation of the Spheroidal Har-monics and application to the Generalised Hydrogen Molecular Ion Problem. Master’sthesis. University of Western Austrailia.
[5] Barrowes, Benjamin Earl (2004). Electromagnetic Scattering and Induction Modelsfor Spheroidal Geometries. Ph.D. thesis. Massachusetts Institute of Technology.
9
eq_x := x = c ξ 2 + 1( ) -η 2 + 1( ) cos φ( )
eq_y := y = c ξ 2 + 1( ) -η 2 + 1( ) sin φ( )
eq_z := z = c η ξ
oblatespheroidal
oblatespheroidalξ, η, φ
v := ξ eξ + η eη + φ eφ
vc := c ξ 2 + 1( ) -η 2 + 1( ) cos φ( ) ex + c ξ 2 + 1( ) -η 2 + 1( ) sin φ( ) ey + c η ξ ez
vp := ξ eξ + η eη + arctan sin φ( ), cos φ( )( ) eφ
-4
-2x0
24
-4
-20
y 2
4
pde1 := 1
c 3 η 2 + ξ 2( )
⎛⎜⎜⎜⎜⎜⎝
2 c ξ ∂
∂ ξ ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
+ c ξ 2 + 1( ) ∂ 2
∂ ξ 2 ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
- 2 c η ∂
∂ η ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
+ c -η 2 + 1( ) ∂ 2
∂ η 2 ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
+
c η 2 + ξ 2( ) ∂ 2
∂ φ 2 ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
ξ 2 + 1( ) -η 2 + 1( )
⎞⎟⎟⎟⎟⎟⎠
= 0
pde1 := ξ 2 + 1( ) ∂ 2
∂ ξ 2 ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
+ 2 ξ ∂
∂ ξ ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
+ -η 2 + 1( ) ∂ 2
∂ η 2 ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
- 2 η ∂
∂ η ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
-
η 2 + ξ 2( ) ∂ 2
∂ φ 2 ψ
⎛⎜⎜⎝
⎞⎟⎟⎠
ξ 2 + 1( ) η 2 - 1( ) = 0
ψ, R, T, P
vars := ∂ 2
∂ ξ 2 R,
∂ ∂ ξ
R, ∂ 2
∂ η 2 T,
∂ ∂ η
T, ∂ 2
∂ φ 2 P
⎡⎢⎢⎣
⎤⎥⎥⎦
pde2 :=
ξ 2 + 1( ) ∂ 2
∂ ξ 2 R
⎛⎜⎜⎝
⎞⎟⎟⎠
R +
2 ξ ∂
∂ ξ R
⎛⎜⎜⎝
⎞⎟⎟⎠
R -
η 2 - 1( ) ∂ 2
∂ η 2 T
⎛⎜⎜⎝
⎞⎟⎟⎠
T -
2 η ∂
∂ η T
⎛⎜⎜⎝
⎞⎟⎟⎠
T
-
η 2 + ξ 2( ) ∂ 2
∂ φ 2 P
⎛⎜⎜⎝
⎞⎟⎟⎠
P ξ 2 + 1( ) η 2 - 1( ) = 0
kk := η 2 + ξ 2
ξ 2 + 1( ) -η 2 + 1( )
pde2 := -
ξ 2 + 1( )2 η 2 - 1( )
∂ 2
∂ ξ 2 R
⎛⎜⎜⎝
⎞⎟⎟⎠
η 2 + ξ 2( ) R -
2 ξ 2 + 1( ) η 2 - 1( ) ξ ∂
∂ ξ R
⎛⎜⎜⎝
⎞⎟⎟⎠
η 2 + ξ 2( ) R
+
ξ 2 + 1( ) η2 - 1( )
2
∂ 2
∂ η 2 T
⎛⎜⎜⎝
⎞⎟⎟⎠
η 2 + ξ 2( ) T +
2 ξ 2 + 1( ) η 2 - 1( ) η ∂
∂ η T
⎛⎜⎜⎝
⎞⎟⎟⎠
η 2 + ξ 2( ) T +
∂ 2
∂ φ 2 P
P = 0
ode1 :=
∂ 2
∂ φ 2 P
P = -m 2
pde3 := -
ξ 2 + 1( )2 η 2 - 1( )
∂ 2
∂ ξ 2 R
⎛⎜⎜⎝
⎞⎟⎟⎠
η 2 + ξ 2( ) R -
2 ξ 2 + 1( ) η 2 - 1( ) ξ ∂
∂ ξ R
⎛⎜⎜⎝
⎞⎟⎟⎠
η 2 + ξ 2( ) R
+
ξ 2 + 1( ) η2 - 1( )
2
∂ 2
∂ η 2 T
⎛⎜⎜⎝
⎞⎟⎟⎠
η 2 + ξ 2( ) T +
2 ξ 2 + 1( ) η 2 - 1( ) η ∂
∂ η T
⎛⎜⎜⎝
⎞⎟⎟⎠
η 2 + ξ 2( ) T - m 2 = 0
pde3 :=
ξ 2 + 1( ) ∂ 2
∂ ξ 2 R
⎛⎜⎜⎝
⎞⎟⎟⎠
R +
2 ξ ∂
∂ ξ R
⎛⎜⎜⎝
⎞⎟⎟⎠
R -
η 2 - 1( ) ∂ 2
∂ η 2 T
⎛⎜⎜⎝
⎞⎟⎟⎠
T -
2 η ∂
∂ η T
⎛⎜⎜⎝
⎞⎟⎟⎠
T
+ η 2 + ξ 2( ) m 2
ξ 2 + 1( ) η 2 - 1( ) = 0
pde3 :=
ξ 2 + 1( ) ∂ 2
∂ ξ 2 R
⎛⎜⎜⎝
⎞⎟⎟⎠
R +
2 ξ ∂
∂ ξ R
⎛⎜⎜⎝
⎞⎟⎟⎠
R -
η 2 - 1( ) ∂ 2
∂ η 2 T
⎛⎜⎜⎝
⎞⎟⎟⎠
T -
2 η ∂
∂ η T
⎛⎜⎜⎝
⎞⎟⎟⎠
T=
η 2 + ξ 2( ) m 2
ξ 2 + 1( ) η 2 - 1( )
pde3 :=
ξ 2 + 1( ) ∂ 2
∂ ξ 2 R
⎛⎜⎜⎝
⎞⎟⎟⎠
R +
2 ξ ∂
∂ ξ R
⎛⎜⎜⎝
⎞⎟⎟⎠
R -
η 2 - 1( ) ∂ 2
∂ η 2 T
⎛⎜⎜⎝
⎞⎟⎟⎠
T -
2 η ∂
∂ η T
⎛⎜⎜⎝
⎞⎟⎟⎠
T
+ m 2
η 2 - 1 +
m 2
ξ 2 + 1 = 0
ode2 := -
η 2 - 1( ) ∂ 2
∂ η 2 T
⎛⎜⎜⎝
⎞⎟⎟⎠
T -
2 η ∂
∂ η T
⎛⎜⎜⎝
⎞⎟⎟⎠
T +
m 2
η 2 - 1 = k
ode3 :=
ξ 2 + 1( ) ∂ 2
∂ ξ 2 R
⎛⎜⎜⎝
⎞⎟⎟⎠
R +
2 ξ ∂
∂ ξ R
⎛⎜⎜⎝
⎞⎟⎟⎠
R +
m 2
ξ 2 + 1 = -k
ode2 := -η 2 + 1( ) ∂ 2
∂ η 2 T
⎛⎜⎜⎝
⎞⎟⎟⎠
- 2 η ∂
∂ η T
⎛⎜⎜⎝
⎞⎟⎟⎠
- T k η 2 - k - m 2( )
η 2 - 1 = 0
ode3 := ξ 2 + 1( ) ∂ 2
∂ ξ 2 R
⎛⎜⎜⎝
⎞⎟⎟⎠
+ 2 ξ ∂
∂ ξ R
⎛⎜⎜⎝
⎞⎟⎟⎠
+ R k ξ 2 + k + m 2( )
ξ 2 + 1 = 0
soln1 := P = _C1 sin m φ( ) + _C2 cos m φ( )
soln2 := T = _C1 LegendreP 12
-4 k + 1 - 12
, m, η⎛⎜⎝
⎞⎟⎠
+ _C2 LegendreQ 12
-4 k + 1 - 12
, m, η⎛⎜⎝
⎞⎟⎠
soln3 := R = _C1 LegendreP 12
-4 k + 1 - 12
, m, I ξ⎛⎜⎝
⎞⎟⎠
+ _C2 LegendreQ 12
-4 k + 1 - 12
, m, I ξ⎛⎜⎝
⎞⎟⎠
soln1 := P = em sin m φ( ) + fm cos m φ( )
soln2 := T = LegendreP n, m, η( )
soln3 := R = bm, n LegendreQ n, m, I ξ( )
soln := ψ = bm, n LegendreQ n, m, I ξ( ) LegendreP n, m, η( ) em sin m φ( ) + fm cos m φ( )( )
ψ0 := d Mx x
cartesianx, y, z
H0c := -d Mx ex
H0 := - d Mx cos φ( ) -η 2 + 1 ξ
η 2 + ξ 2 eξ +
d Mx cos φ( ) η ξ 2 + 1
η 2 + ξ 2 eη + d Mx sin φ( ) eφ
soln1 := P = e1 sin φ( ) + f1 cos φ( )
soln2 := T = I -η 2 + 1
soln3 := R = b1, 1 LegendreQ 1, 1, I ξ( )
soln2 := T = -η 2 + 1
soln3 := R = LegendreQ 1, 1, I ξ( )
soln3 := R -I Ξ( ) = LegendreQ 1, 1, Ξ( )
soln3 := R -I Ξ( ) = 12
-ln Ξ - 1Ξ + 1
⎛⎜⎝
⎞⎟⎠
Ξ 2 - 2 Ξ + ln Ξ - 1Ξ + 1
⎛⎜⎝
⎞⎟⎠
⎛⎜⎝
⎞⎟⎠
1
Ξ 2 - 1
soln3 := R -I Ξ( ) = - 12
Ξ 2 - 1( ) 1
Ξ 2 - 1 ln
Ξ - 1Ξ + 1
⎛⎜⎝
⎞⎟⎠
- Ξ 1
Ξ 2 - 1
soln3 := R = - 12
-ξ 2 - 1( ) 1
-ξ 2 - 1 ln
I ξ - 1I ξ + 1
⎛⎜⎝
⎞⎟⎠
- I ξ 1
-ξ 2 - 1
soln3 := R = - 12
ξ 2 + 1 arctan 2 ξ, -1 + ξ 2( ) + ξ
ξ 2 + 1
ψ1 := K - 12
ξ 2 + 1 arctan 2 ξ, -1 + ξ 2( ) + ξ
ξ 2 + 1
⎛⎜⎜⎝
⎞⎟⎟⎠
-η 2 + 1 e1 sin φ( ) + f1 cos φ( )(
oblatespheroidalξ, η, φ
ψ1 := K - 12
ξ 2 + 1 arctan 2 ξ, -1 + ξ 2( ) + ξ
ξ 2 + 1
⎛⎜⎜⎝
⎞⎟⎟⎠
-η 2 + 1 cos φ( )
H1 := K ξ cos φ( ) -η 2 + 1 arctan 2 ξ, -1 + ξ 2( )
