On Some implications of the Poisson relation

Geophys. J. Int. (1998) 133, 207–208 R E S EA R C H N O TE On some implications of the Poisson relation F. Bocchio Dipartimento di Georisorse e T erritorio, Università di Udine, Italy Accepted 1997 October 21. Received 1997 October 15; in original form 1997 June 30 Key words: gravity, magnetics, Poisson relation. 1 we finally have, for a uniformly magnetized body, I NTR O D UC TIO N The so-called ‘Poisson relation’ (Grant & West 1965) has been frequently used in the past as a means to evaluate the magnetic field of uniformly magnetized bodies via their gravitational field (e.g. Brüggemann et al. 1973); the reason is that, at least in principle, the potential field due to a distribution of monopoles is more easily handled than the potential field due to a distribution of dipoles; nevertheless, we are not concerned here with the applications of the Poisson relation but, instead, with its potential to make clear some more intrinsic relations between these two fields. Since the magnetic field intensity H due to a magnetized body of volume V with a magnetic dipole moment per unit volume M is H(r)=V P (MΩV) V dV , |r−r | 0 (1) we can write, for the case in which the direction of magnetization a is the same throughout the body, so that in this case MΩV=M∂/∂a, H(r)=MV ∂ ∂a P 1 dV . |r−r | 0 V (2) Since the gravitational force g due to a body of constant density r is g(r)=GrV © 1998 RAS P dV , |r−r | 0 V (3) H(r)= M ∂ g, Gr ∂a (4) which is the so-called Poisson relation. 2 LO C A L A ND NO N - LO C A L IM P LI CAT IO N S OF THE PO I S S O N R EL ATI O N We may write ∂ g=U lr , rs ∂a (5) where U is the gravitational gradients tensor of the body and rs lr is the unit vector in the direction of magnetization; it thus follows that M U lr , H= s Gm rs (6) where H denotes the components of the magnetic field s intensity, M the magnetic dipole moment of the body and m its mass. We see therefore that the magnetic field of the body is closely related to the properties of its gravitational gradients tensor U ; in this connection let us consider, for example, a rs spherical body having a constant density, which is uniformly magnetized along the z-axis of a Cartesian reference. Since Gm U = rs r3 C 3x2−r2 3xy 3xz 3xy 3y2−r2 3yz 3xz 3yz 3z2−r2 D (7) 207 SU MM A RY The potentialities of the so-called ‘Poisson relation’, which holds for uniformly magnetized bodies of constant density, for showing the connections between the gravitational and the magnetic fields of such bodies are considered. In particular, it is seen that the same characteristic ratios occur among the components of the magnetic field intensity and the components of the gradient tensors of the two fields, both locally and non-locally. It is also shown that along the magnetization axis of the body the gradient tensor of the magnetic field displays a ‘tidal’ structure. 208 F. Bocchio at the points P (r, 0, 0), P (0, r, 0), P (0, 0, r), equidistant 1 2 3 from the origin along the coordinate axes, we have C C C D D D 2 0 0 Gm U (P )= 0 −1 0 , rs 1 r3 0 0 −1 Gm U (P )= rs 2 r3 Gm U (P )= rs 3 r3 it follows that −1 0 0 0 2 0 0 0 −1 −1 0 0 −1 0 0 0 3M U (0, 0, r) . H (0, 0, r)=− ij Gmr ij (15) We therefore see that, apart from the steeper decrease of H ij with distance compared with U , the gradient tensors for the ij gravitational and magnetic fields of the body display the same structure along the magnetization axis. Since the gravitational field of the body, at the points (Dr, 0, r), (0, D, r, r), (0, 0, r+Dr) on a small sphere of radius Dr centred at P , gives rise to the 3 tidal forces , 0 , 2 By comparison of eqs (7) and (14), it follows that, along the magnetization axis, (8) Dg∞=− GmDr (1, 0, 0) , r3 Dg◊=− GmDr (0, 1, 0) , r3 Dg∞∞∞= M U (P ) , H (P )= z 2 Gm zz 2 we obtain, using eq. (15), for the magnetic field at the same points, M H (P )= U (P ) . z 3 Gm zz 3 (9) 2GmDr (0, 0, 1) , r3 DH∞=− 3MDr (1, 0, 0) , r4 Since DH◊=− Gm U (P )=U (P )= H (P ) , zz 1 xx 3 M z 1 DH∞∞∞= Gm U (P )=U (P )= H (P ) , zz 2 yy 3 M z 2 (10) we obtain, taking account of the Laplace condition for U , rs the following relation: H (P )+H (P )+H (P )=0 . (11) z 1 z 2 z 3 It can also be seen, because of rotational symmetry around the z-axis, that H (P )=H (P )=−H (P )/2 . (12) z 1 z 2 z 3 The interest of the relations (11) and (12) is mainly in the way they have been obtained here and in the characteristic ratios −1, −1, 2 among the components of H at the space positions P , P , P . 1 2 3 In a Cartesian reference system we obtain from eq. (6) M U ls , H = ij Gm ij/s 3M H =− ij r5 C B 5x2 −1 z r2 5xyz r2 A 3MDr (0, 1, 0) , r4 6MDr (0, 0, 1) . r4 (17) We therefore see that the magnetic field has a ‘tidal’ structure at points on the magnetization axis; it also follows that U (P ): U (P ): U (P )=−1: −1: 2 , (18) xx 3 yy 3 zz 3 H (P ): H (P ): H (P )=−1: −1: 2 . (19) xx 3 yy 3 zz 3 Considering also eq. (12), we can thus conclude that the same characteristic ratios occur among components of the magnetic field intensity and components of the gradients tensors H and ij U , both in local and in non-local relations; this rather unusual ij behaviour seems not to take place by chance and deserves further investigation. A possible explanation could perhaps be found within the frame of a topological approach to the properties of gravitational and magnetic fields (Bocchio 1989, 1990; Hide, Barraclough & MacMillan 1997). (13) where H is the gradients tensor for the magnetic field intensity. ij If a spherical shape for the body is assumed, A (16) A B A 5z2 −1 x r2 5xyz r2 A B A B A B B B 5z2 −1 x r2 5y2 −1 z r2 5z2 −1 y r2 5z2 −1 y r2 5z2 −3 z r2 D . (14) R EF ER EN C ES Bocchio, F., 1989. Critical paths and topological structure of the Earth’s gravity field, Geophys. J. Int., 98, 623–631. Bocchio, F., 1990. Gravity field aspects from a topological invariant, Geophys. J. Int., 102, 527–530. Brüggemann, H., Grafarend, E., Kiehl, J. & Schares, N., 1973. Gravimetrische und magnetische Analyse der Anomalie des Rodderberges bei Bonn, Mitteilungen aus dem Institut für T heoretische Geodäsie der Universität Bonn, 24. Grant, F.S. & West, G.F., 1965. Interpretation T heory in Applied Geophysics, McGraw-Hill, New York, NY. Hide, R., Barraclough, D.R. & MacMillan, S., 1997. Topological characteristics of magnetic and other solenoidal vector fields, XXII EGS General Assembly, Vienna (unpublished). © 1998 RAS, GJI 133, 207–208 M H (P )= U (P ) , z 1 Gm zz 1  (...truncated)