Method for determining the electric and magnetic polarizability of arbitrarily shaped conducting bodies

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 1, FEBRUARY 1997 1

Method for Determining the Electric andMagnetic Polarizability of Arbitrarily

Shaped Conducting BodiesStanislav A. Podosenov,Member, IEEE,Alexander A. Sokolov,Member, IEEE,and Serge V. Al’betkov

Abstract—The paper deals with polarization properties ofarbitrarily shaped conducting bodies in quasistatic electric fieldsand quasisteady magnetic fields within the strong-skin-effectapproximation. Based on measuring the capacitance (inductance)of a field-forming system before and after introducing an ar-bitrary conducting body into it, a method for determining itselectric (magnetic) polarizability is proposed. The method permitsone to find eigenvalues and directions of the principal axes ofthe polarizability tensors of any conducting body. The resultsobtained are useful in solving two practical problems of theelectromagnetic compatibility: estimation of a measurement errorinduced by coupling the transducers of an electric and magneticfield to the conducting surfaces, and estimation of an additionalfield induced by coupling the objects to the field-forming systemelectrodes in electromagnetic susceptibility tests.

I. INTRODUCTION

W HEN measuring the electric and magnetic intensitiesin the neighborhood of conducting objects as well as

calibrating field transducers in a TEM cell, there arises aspecific error induced by coupling the transducer polarizedby a field to the conducting surfaces. For wavelengths muchexceeding the transducer sizes the additional field inducedby coupling may be estimated by the mirror-image method,if the transducer dipole moment induced by the field isknown. For the transducer equivalent, in polarizability, to asimple-shape conducting body (ball, ellipsoid of revolution)the polarizability is calculated by well-known equations [1]and estimation of the additional field acting on the transducerreduces to application of the mirror-image method [2], [3]. Anumerical calculation of the polarizability tensor for arbitraryconducting bodies is possible, in principle [4], but it is rathercumbersome and requires a great deal of effort. Besides,the dipole moment of capacitance transducers of electric andmagnetic field may depend on the input resistance of devicesconnected with them.

In the present paper, a solid basis is given for the experimen-tal method, proposed by the authors [5], [6], of determining theelectric (magnetic) polarizability tensor of arbitrarily shapedconducting bodies. The idea of the method is based on theassumption as follows: If a body introduced into the field-forming system distorts the electric (magnetic) field structuredue to the induced dipole moment, then there must exist a

Manuscript received November 7, 1994; revised May 23, 1996.The authors are with the All-Russian Research Institute For Optophysical

Measurements, Moscow 119361, Russia.Publisher Item Identifier S 0018-9375(97)00833-8.

relation between the body polarizability and the additional ca-pacitance (inductance) introduced by the body into the system.

II. ESSENCE OF THEMETHOD

Link the capacitance increment of a flat capacitorwith the electric polarizability tensor , considering thatthe charged capacitor of capacitanceis not connected withthe source. From the theorem on the energy of an unchargedconductor [7], it follows that introduction of the unchargedconductor into the field of charged ones diminishes the totalfield energy by the value

(1)

where and are electric field intensities before and afterintroducing the conductor into the system, is a volume ofthe body being introduced, is a volume of the surroundingspace minus the volume of the body being introduced and thatof capacitor plates. The second integral in (1) may be reducedto four surface integrals bounding the volume

(2)

where and are surfaces bounding the capacitor plates;is a surface bounding the body being introduced; is

an infinitely distant surface; and are potentials on thesurfaces before and after introducing the body; and, ,

, are unit vectors of the normals to the correspondingsurfaces (Fig. 1). Since the charges on the capacitor platesbefore and after introducing the body are invariant, and thepotentials and are constant, the integrals over and

vanish. Hence, in (2) there remains only integrals over the

0018–9375/97$10..00 1997 IEEE

2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 1, FEBRUARY 1997

Fig. 1. Integration domain.

surface , .

where is a distance from the origin of coordinates to a sur-face at infinity (Fig. 1). Since the conductor being introducedis unchanged, the integral of

vanishes. Due to the integral of becomes zeroinside .

Substituting the expressions obtained into (1), we find

(3)

where is a surface density of the conductor being intro-duced.

The last-mentioned equation is exact. On the other hand,while introducing an uncharged conductor into the field oftwo arbitrary charged conductors with chargesand , we

have a relation

where is a capacitance of the system of two conductors withcharges , is a capacitance of this system after introducingthe charged conductor. Thus

(4)

In particular, if an uncharged body is introduced intothe homogeneous domain of a flat capacitor, which is notconnected with the source, then in (4)

where is a distance between the plates,is a radius vectorfrom the origin of coordinates to the point of surface,is a dipole moment (Fig. 1).

Hence

(5)

Since in the field, which was homogeneous before intro-ducing the body, an equality [1] for the dipole momentisvalid

where —is a polarizability tensor of the body [1].

PODOSENOVet al.: METHOD FOR DETERMINING THE ELECTRIC POLARIZABILITY OF CONDUCTING BODIES 3

Then from (5) it follows that an equation for the capacitorcapacitance decrement after introducing a test body intothe capacitor is

(6)

where is a unit vector in the direction of .Equation (5) allows the dipole moment of the bodies, for whichits direction coincides with that of the field, to be calculatedat once. Among such bodies are, e.g., the arbitrary bodies ofrevolution with an axis of rotation parallel to the field as wellas the bodies having a symmetry plane and arranged so thatthe latter should be perpendicular to the field. For these bodies

(7)

Equations (6) and (7) are valid for any relationsproviding inhomogeneity of an initial (unperturbed) fieldinthe body location zone. For arbitrary one-dimensional bodiesto measure the values of is insufficient.

We link rigidly the arbitrarily shaped conductor with aunit orthonormal triad ( ) forming the Cartesiancoordinate system. If one conducts six independent measure-ments of the capacitor capacitance at different orientations ofa triad “frozen” into the conductor with respect to the fielddirection, then we obtain a system of six equations to find sixindependent components of the polarizability tensor. To have aunique solution to this system, it is necessary for the conductorto be oriented in the field so that the sixth-order determinantshould not vanish, each of its lines contains a combination ofvalues

(8)

calculated for six different orientations. In particular, a systemof unit vectors satisfies this condition

(9)

where is a Kronecker symbol. A successive substitution ofvalues (9) results in a system of six equations whose solutionis of the form

no summation over

(10)

where is a capacitor capacitance change at the fieldvector directed along the vector and is thatfor the field vector parallel to the plane passing through

and , and forming the angle . In principle,the angle is arbitrary. But at the value of thesolution to the system takes the simplest form. Equation (10)links the components of the electric polarizability tensorand the capacitor capacitance change . It is evident thatsuch a method may be developed to determine the magneticpolarizability tensor of arbitrarily conducting bodies as well.As this takes place, the inductances being introducedby a body into the system forming a homogeneous magneticfield will appear to be experimentally measurable parametersat different orientations of the body with respect to the fieldvector [5]. If the skin-layer is less than the typical dimensionsof the body, then the results obtained will be applicable toall frequencies satisfying this condition. It should also benoted that this method can be extended to dielectric andmagnetic arbitrary bodies as well. If the body has planesor axes of symmetry, the number of measurements to findthe polarizability tensor may be diminished. Consider someparticular cases.

1) Introduce a conducting ball of radius between thecapacitor plates. It is evident from symmetry considerationsthat

From (10) we find

(11)

As known [1], for the ball in a homogeneous field. Hence

(12)

where is an area of the capacitor plates. If then, which coincides with the solution obtained

in paper [1].2) Let a body of revolution be introduced into the capacitor

and the vector be directed along the axis of rotation. Fromthe symmetry it follows that


and , and we obtain an expression forthe dipole moment. On the other hand, from the symmetry itfollows that in this case , hence . From(10), taking account of the condition , wehave . Thus, if the unit vector coincides with theaxis of the body of revolution, then any orientation of thedyad and the tensor of the body of revolution isdiagonal. Thus, the above-mentioned triadsets a direction ofthe principal axes of an arbitrary body of revolution. Hence, tofind the polarizability tensor of an arbitrary body of revolution,it will suffice to conduct two measurements of the capacitorcapacitance with a body introduced into the capacitor: with theaxis parallel and perpendicular to the field. Using (10), wefind an expression for the polarizability tensor of an arbitrarybody of revolution in the principal axes

(13)

where and are changes of the capacitor capacitance,when the axis of body symmetry is perpendicular or parallelto the field , respectively. From (6) and (13), we calculatethe value of , when the axis forms the angle withthe field (Fig. 2)

(14)

3) Calculate the components of the polarizability tensor ofa parallelepiped with sides by bringing the triadinto coincidence with the corresponding edgesgoing out ofone of the vertices. If one directs the field along each ofthe unit vectors , then based on the symmetrymay be expected in these cases. Hence, for the parallelepipedthe tensor is diagonal, i.e., are principal axes of theparallelepiped. If the parallelepiped base is a square, i.e.,

, then . From linear algebra we knowthat in this case the principal axes and , remainingorthogonal to each other and to the axis, may be arbitrarilyoriented in the plane of the base. Physically, this means thatthe field directed perpendicular to the axis results inan induced dipole moment being parallel to and themoment value remaining constant independently of orientationof the vector with respect to the dyad and . In other

words, a bar of square cross section polarized by the fieldparallel to the cross section behaves as a body of revolution.If one considers a cube instead of a parallelepiped, then

. In this case the principal axes maybe arbitrarily oriented in space. Physically, this means that apolarized cube is “similar” to a sphere, i.e., by orienting thecube arbitrarily in a homogeneous field we obtain one andthe same dipole moment, parallel to the field and invariablein magnitude, independently of orientation. Thus, the cube aswell as the sphere is isotropic with respect to polarizability.In terms of the group theory, the cube and sphere belong tothe same syngony, i.e., they have to do with a system of thesame symmetry group [8]. One may show that among thesame syngony are the tetrahedron and the remaking regularpolyhedrons. For the bodies of cubic syngony, each orthogonalunit vector constructed in the body coincides with the directionof the principal axes, and the polarizability tensor is globular.The bodies of trigonal, tetragonal, and hexagonal syngonies asknown [9] have one axis each of the third, fourth, and sixthorder, respectively (an axis of symmetry of the orderiscalled the straight line, a turn about which at the anglebrings the body into coincidence with its initial position). Thetensor in the principal axes has two equal eigenvalues.Thus, these bodies are “similar” to bodies of revolution withrespect to polarizability. The above-mentioned parallelepipedwith edges in the principal axes has threedifferent eigenvalues of the polarizability tensor. It has threeaxes of the second order, which are orthogonal to one another.Therefore, it may be referred to the rhombic syngony. Themonocline system bodies have one axis of the second order,which may be chosen as a principal axis. And, finally, thetriclinic system bodies have no axes of symmetry.

4) Consider the polarization of arbitrarily shaped flat thinplates. Let two vectors of the triad and , be rigidlyconnected with the body, lie in the plate plane, and the vector

be perpendicular to the plate. It is evident that introducingsuch a plate into a flat capacitor to the capacitor plate willnot change the capacitor capacitance. From (6) atand , we find . If one transforms theplate polarizability tensor to the principal axes, then from theproblem symmetry one of the principal axes (e.g.,) mustcoincide with the vector of the initial “frozen” triad .

The directions of the other two vectors and lying inthe plate plane may be calculated. In the principal axes thepolarizability tensor components are of the form

The initial tensor of polarizability of a flat arbitrarily shapedflat thin plate is generally representable in the form

(15)

Using the well-known method of linear algebra, one canfind the directions of the principal axes of the polarizability


tensor that are set by an orthogonal normalized unit vector

(16)

From the aforementioned it follows that the polarizationproperties of arbitrarily shaped flat plates can be determinedwith the aid of three measurements, e.g.,, , and by(10).

Any flat body having an axis of symmetry of the ordercan be shown to be polarized much like a disk.

Physically, this means that the plate plane being parallel tothe field results in an induced dipole moment being unchangedin magnitude and direction at any turn of the plate in thisplane. In particular, all regular-gons possess this “similarity”property. If a given body is turned in the capacitor field so thatthe axis normal to its surface should form an anglewiththe field direction, then the capacitor change may becalculated by (14) taking account of , and ,from the aforementioned, does not change at any orientationof the body in this plane.

For an arbitrary flat figure the capacitance change in therotation is determined by two angles, and , where isan angle between the field vector projection onto the plane

, and the vector . Thereby, , , and set thedirection of the principal axes of the figure (Fig. 3).

(17)

III. RELATION BETWEEN THE ELECTRIC

AND MAGNETIC POLARIZABILITY TENSOR

COMPONENTS FORBODIES OF REVOLUTION

While measuring the magnetic polarizability tensor com-ponents of a conducting body in practice, there may arisedifficulties related to providing the requirements of skin-layersmallness as compared with the body sizes. Under theseconditions it would be appropriate to find the relation betweenthe components of the electric and magnetic polarizabilitytensors, if only for bodies of revolution—the most abundantform of transducers, permitting the magnetic measurements tobe substituted by the electric ones. To calculate the fieldsand , we choose an elliptic system of coordinates inthe meridional planes (Fig. 4) related to by

(18)

Fig. 2. Body of revolution in the electric field.

Fig. 3. Arbitrary flat figure in the electric field.

where is a focal length of the family of coordinate lines—theconfocal ellipses and hyperbolas. If one assumes

then

(19)

It is evident that the problem of finding the magnetic fieldpotential satisfying the Laplace equation outside the bodyof revolution in a strong skin-effect is equivalent to thehydrodynamic problem of a longitudinal liquid flow about the


Fig. 4. Problem geometry.

bodies of revolution [10]. Making use of the result [10], weobtain in our designations

(20)

where and are Legendre’s functions of the1st and 2nd kind, respectively. The coefficients maybe determined if one takes into account that the normalcomponent of the magnetic field on the conductorsurface, which is equivalent to the equality

(21)

determining a zero surface of the current in the longitudinalflow about the bodies of revolution [10]. It should be addedby a given equation of a body-of-revolution profile

(22)

Of course, from the calculational viewpoint finding thecoefficients from system (21), (22) is rather a complicatedproblem (simply solved, e.g., for the ellipsoids withconst). To find the electric field potential outside the body,when the field direction (Fig. 4) is perpendicular to the axisof rotation, we use the well-known general solution to theLaplace equation in the elliptic coordinates [10]

(23)

where

(24)

are Legendre’s associated functions, . Assumingthat in the plane , we find . From thesymmetry it follows that the dependence of the potentialon

contains only . Hence, using (23) and (24), we have

(25)

To find the coefficients , we take advantage of thepotential being constant on the conductor surface and, inparticular, equal to zero. From (25) it follows that

(26)

at . Comparing (21), (22), and (26), we find

(27)

Equation (27) links solutions (20) and (25).Consider (20) and (25) in the dipole approximation at the

distances much larger then the body sizes. It isevident that the greater corresponds to the asymmetricallygreater , both being of the same order.

To conduct a further analysis, we expand the functionin a power series of [11]

(28)

where is the gamma-function, and is thehypergeometric function. The dipole approximation allows oneterm of the expansion only to be retained in (20) and (25).From (28) it follows that

(29)

From (27) it follows that . As seen from Fig. 4,(29) is representable in the form

(30)

where and are induced magnetic and electric dipolemoments, respectively.

On the other hand, in the principal axes of the electricpolarizability and magnetic one , we have by definition

where , , and the orientation of theprincipal axes of the tensors is , , and .Comparing with (30), we obtain

(31)


Fig. 5. Field-forming system for forming a homogeneous electric field.

Equation (31) admits a simple analytic verification in someparticular cases. For example, the problems on an ellipsoidof revolution in an electrostatic field and on a conductingellipsoid in a variable magnetic field are solved in paper[1]. For the conducting ellipsoid of revolution whose axisis perpendicular to , we have , where

are lengths of the ellipsoid semiaxes and is adepolarization coefficient. In the case of a strong skin-effect wehave for the same ellipsoid whose axisis parallel to the variable magnetic field. Taking into accountthat , we obtain . In particular,for a ball we have

IV. EXPERIMENTAL RESULTS

While conducting the measurements, the problems posedwere as follows:

1) to give experimental evidence for the validity of themethod proposed for determining the polarizability ten-sor. It is evident that the method is verifiable on thebodies that can be calculated (ball, ellipsoid, and disk),as well as on others of different degrees of symmetry;

2) to expand experimentally the validity of the conclusionpresented in [12] on the inscribed body polarizabilitybeing less than that of a circumscribed one. In [12] thisconclusion is proved theoretically only for bodies in-scribed in the sphere. The proof of this assertion for othershapes of the circumscribed body allows its universality(i.e., correctness for any shape of the circumscribed andinscribed body) to be spoken about almost plausibly.There is no doubt of this statement being of practi-cal importance, which permits the upper estimate ofthe polarizability of any complex body inscribed in ageometrically simpler body of the known polarizability;

Fig. 6. Field-forming system for forming a homogeneous magnetic field.

3) to determine quantitatively values of the polarizabilitytensor components for simplest geometry bodies not cal-culated yet (cube, cylinder, etc.). This is of importancefor estimating the polarizability of objects close in shapeto the given bodies.

4) To provide an experimental evidence in favor of thevalidity of the relation between the electric and magneticpolarizability of the conducting bodies of revolutionderived in the third section on the present paper.

Determining the electric polarizability tensor componentswas carried out in a flat capacitor (see Fig. 5), and measuringthe magnetic polarizability tensor components was done ina three-coil field-forming system (see Fig. 6) similar to thatdescribed in [13]. To mount the bodies under investigation inthe middle of the homogeneous zone, a foam plastic supportand dielectric fibers were used. A digital instrument was usedto measure capacitance and inductance of the system. The flatcapacitor capacitance from 20 to 100 pF was measured, andthe value of did not exceed 2%. The magnetic field-forming system inductance amounted to about 6 mcH, and thevalue of did not exceed 1%. Thus, in the course of theexperiment, the relations and weresatisfied.

While determining and , readings of the capacitanceand inductance values of the field-forming system before andafter introducing the body were taken in the same limit ofmeasurements and in small time intervals, therefore the unitof a less significant digit of the instrument indicator wasadopted as an absolute error of measuring and . Whileanalyzing and representing the results, it is most convenientto operate on the electric and magnetic polarizability tensorcomponents normalized to the calculated measure polarizabil-ities (e.g., for the ball or disk) and , respectively (butnot their absolute values and ). Under the condition

, , from (10) for the normalized valuesof the polarizability components it follows that the expressions

(32)


Fig. 7. Relation of the electric and magnetic polarizability tensor components of a conducting cylinder to its geometry;�0 and 0 are electric andmagnetic polarizability of the ball circumscribed about the cylinder.

are valid, where and are a capacitance and aninductance, respectively, introduced into the system by ameasure placed at the same point of the working zone of thefield-forming system.

An aluminum ball of diameter mm wasused as a measure of the electric and magnetic polarizability.As known, the quantity of the electric polarizability of theball is . Hence, for the chosen value of wehave F m , with the relative errorof determining %. Since the condition

and was satisfied under themeasurements, then the total relative error of determining

may be estimated as follows

(33)

The experiment has shown that at mm the value ofthe capacitance introduced by the ball amounted to 0.29 pF.In this case the quantities and differs by 1.5%,which lies within the estimate % corresponding tothe value of pF, mm. Thus, the methodproposed for determining the polarizability tensor componentsis efficient.

Consider the results obtained. Fig. 7 depicts dependence ofthe components , , and , of the electric and mag-netic polarizability tensors on the geometry of the conductingcircular cylinder inscribed in the sphere. The anglebetweenthe generatrix and the diagonal of the axial cross section of thecylinder is chosen as a parameter characterizing the cylindershape. From the figure it is seen that the polarizability of anycylinder inscribed in the ball is less than the ball polarizability.Coincidence within the accuracy of measuring the curves

and indicates that for any cylinder there takesplace the same relation between and as for the ball,i.e., (31) obtained in the third part of the article is valid forthe cylinder of a circular cross section.

Of utmost interest, from the viewpoint of proving thestatement on the polarizability of an inscribed body being lessthan that of a circumscribed one, are the objects consistingof elements with a small radius of curvature. Fig. 8 presentsresults of measuring the polarizability of a complex body ofrevolution composed of identical conducting disks and a diskcircumscribed in the cylinder with the parameter .The values of and in the figure are normalized to thepolarizability of the ball circumscribed about the cylinder. Forthe cylinder with the angle , as follows from Fig. 7

. The dependences obtained confirmonce again that the polarizability of an inscribed body does notexceed that of a circumscribed one, and for the bodies of revo-lution (31) is valid. Fig. 9 presents results of measuring the po-larizability of the cylinder whose lateral is made of the increas-ing number of parallel wires. From the plot it follows that thepolarizability of such a body is less than that of the cylinder.

The polarizability tensor components of two bodies ofrevolution inscribed in the sphere were measured: a circularcone of angle 60 and a body having a regular octagonalcross section in the axial plane. The results presented in TableI support the validity of (31), within the accuracy of themeasurements, and the relation between the polarizabilities ofinscribed and circumscribed bodies.

Measurements of the electric and magnetic polarizabilitiesof the cube and tetrahedron, inscribed in the ball at differentorientations of these bodies with respect to the field vector,confirm the globular structure of the cubic syngony body polar-izability tensors. Table II presents values of the polarizabilitytensor components obtained as a result of measurements. Thedependences of introduced capacitance on the anglebetween the body rotation axis and the field vector weremeasured for four bodies of revolution inscribed in the sphereof diameter 120 mm. The experimental results presented inFig. 10 give evidence for the validity calculated in (14) (thecontinuous lines in Fig. 10).


Fig. 8. Polarizability of a body of revolution ofn identical conducting disksinscribed in the cylinder of� = 45

�; �0 and 0 are polarizabilities of theconducting ball circumscribed about the cylinder;�=L = 0:11.

Fig. 9. Polarizability of a conducting body composed ofn parallel wiresuniformly arranged on the lateral area of the cylinder at� = 45

�; �0 is anelectric polarizability of the ball circumscribed about the cylinder.

TABLE IRELATION BETWEEN THE POLARIZABILITIES

OF INSCRIBED AND CIRCUMSCRIBED BODIES

�11=�) �+=�0 += 0 += 0Circular cone of angle

60 �

0.33 0.40 0.41 0.28

Octahedral axial crosssection body of

revoultion0.86 0.86 0.88 0.88

Of particular interest is a comparison of the measured com-ponents of the polarizability tensor with their calculated values.The measurements have been performed of the components ofthe electric polarizability tensor for an ellipsoid of revolutionand for a thin disc (Fig. 10, curves 1 and 4), whose exactanalytic expressions are known [1]. The ellipsoid of revolution

Fig. 10. Dependence of the capacitance introduced into the field-formingsystem by a body of revolution on the angle between the axis of rotation andthe field vector: (a) 1 is an ellipsoid of revolution with semiaxes ratio 1/3,(b) 2 is a cylinder of� = 25

�, (c) 3 is a cylinder of� = 65�, and (d) 4

is a thin disk.

TABLE IIRESULTS OFMEASUREMENTS OF THEPOLARIZABILITY TENSOR COMPONENTS

�=�0 = 0Tetrahedron 0.23 0.19

Cube 0.43 0.41

was chosen to have axes mm mm.The capacitance of a flat capacitor with a gap mmamounted to pF. The coupled capacitance of theellipsoid oriented parallel and perpendicular to the electric fieldvector amounted to pF, pF (Fig. 10

, ). From (13), we findFm , Fm .

From [1] we find

(34)

From (34), we have Fm ,Fm . The related deviations of the measured and

calculated values amounted to 6.3% for and 9% for .


Similar comparisons have been carried out for the thin disc aswell (Fig. 10, curve 4).

The coincidence of the experimental results with theoreticalones [1] for the ellipsoid of rotation and the thin disc indicatesthat the method proposed is effective.

V. CONCLUSION

The present paper substantiates the method allowing oneto determine experimentally the electric (magnetic) polariz-ability tensor of arbitrary conducting bodies by measuringan additional capacitance (inductance) being introduced by abody into the field-forming system at different orientationsof the body with respect to the field vector. The method isextendible to the bodies made of different materials (magnetic,dielectric) and having complex structure (e.g., containingelectron components). The polarizability tensor defines thebody dipole moment induced by the acting field and anadditional one acting on the body due to a mirror image of thedipole being a reflection from the conducting surfaces locatedin the neighborhood.

The polarizability of simplest geometry bodies has beenmeasured. The polarizability of an inscribed body is shownto be less than that of a circumscribed one.

A relation between the tensors of the electric and magneticpolarizability is found for conducting bodies of revolution. Thepaper results may be used to estimate errors of measuring theelectric and magnetic field intensities in the neighborhood ofthe conducting surfaces for any field transducers. The resultsof even greater importance have been obtained to estimatethe additional field acting on the object being tested in stripsimulators of the electromagnetic field. Since the polarizabilityof such bodies is proportional to their volume, then in this casethe polarizability tensor may be determined in the laboratoryusing the object scale model. The paper results may be alsouseful to solve the problem of diffraction on small arbitrarilyshaped bodies.

REFERENCES

[1] L. D. Landau and E. M. Lifshitz,Electrodynamics of Continuous Media.London, U.K.: Pergamon, Addison-Wesley, 1960.

[2] R. P. Harrison and E. A. Levis, “A method for accurately measuring thevertical electric field strength of propagating VLF wave,”IEEE Trans.Instrum. Measure.,vol. 14, pp. 87–97, Dec. 1965.

[3] Y. A. Pivovarov, S. A. Podosenov, V. I. Sachkov, and A. A. Sokolov,“Calibration of a spherical antenna in a flat capacitor,” inAbstr. Rep. 4thAll-Union Conf., Metrol. Antenna Measure.,Yerevan, 1987, pp. 68–69.

[4] R. E. Kleinman and T. B. A. Senior, “Low frequency scattering by spaceobjects,” IEEE Trans. Aerosp. Electron. Syst.,vol. 11, pp. 672–675,1975.

[5] A. A. Sokolov, “Coupling of transducers of electromagnetic field tothe field-forming system,” inAbstr. Repts. 4th All-Union Conf., Metrol.Antenna Measure.,Yerevan, 1987, pp. 70–72.

[6] S. V. Al’betkov, Y. A. Pivovarov, S. A. Podosenov, V. I. Sachkov,and A. A. Sokolov, “Method of determining the polarizability tensor ofarbitrary bodies,”Izmeritel’naya Tekhnika,no. 12, pp. 27–29, 1989.

[7] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill,1941.

[8] G. Y. Lyubarsky, Group Theory and Its Application to Physics.Moscow, Russia: Gostekhteorizdat, 1957, in Russian.

[9] M. A. Akivis and V. V. Goldberg, Tensor Calculation. Moscow,Russia: Nauka, 1969, in Russian.

[10] L. G. Loitsyansky,Fluid and Gas Mechanics. Moscow, Russia: Nauka,1973, in Russian.

[11] E. T. Whittaker and G. N. Watson,A Course of Modern Analysis.Cambridge, U.K.: Cambridge Univ. Press, 1927.

[12] S. A. Podosenov, V. I. Sachkov, and A. A. Sokolov, “Measurement ofthe electromagnetic field intensity in the neighborhood of conductingsurfaces,” inMeasurements of Pulse Electromagnetic Fields.Moscow,Russia: VNIIFTRI, pp. 10–19, 1986, in Russian.

[13] K. Y. Sakharov and A. A. Sokolov, “Investigation of the properties offield-forming systems to create calculated pulse electric and magneticfields,” in Measurements of Pulsed Electromagnetic Fields.Moscow,Russia: VNIIFTRI, pp. 33–45, 1986, in Russian.

Stanislav A. Podosenov(M’96) was born in 1937in Arkhangelsk, Russia. He received the degree intheoretical physics in 1963 and the Ph.D. degreein relativistic mechanics of deformable media in1972, both from Moscow State University PhysicalFaculty, Russia.

From 1963 to 1970, he lectured physicsat Moscow State Technical University, wherehe completed his postgraduate study. Since1970, he has been working at the All-RussianResearch Institute of Optophysical Measurements

on problems of relativistic mechanics and electrodynamics, noninertial framesof reference in special and general relativity, and nonstationary neutrontransfer theory. Presently, he is working in the field of EM pulses radiationtheory and their interaction with conducting bodies.

Alexander A. Sokolov (M’96) was born in 1935in Kashira, Russia. In 1958, he graduated fromMoscow State University Physical Faculty. He re-ceived the Ph.D. degree from the Institute of NuclearPhysics, Novosibirsk, Russia, in 1965 and the Dr.Sc.degree in research results of pulse EM field gener-ation and measurement in 1992.

From 1958 to 1968, he worked at the Insti-tute of Nuclear Physics in Novosibirsk, where heelaborated pulse electron synchrotrons. Since 1968,he has been working at the All-Russian Research

Institute of Optophysical Measurements State Standards of Russia, where heis presently a head of the laboratory for electromagnetic compatibility. He isa Professor in information measurement systems and Assistant Chairman ofthe scientific council on the EMC problem, under the Russian Ministry ofScience and Engineering Policy.

Dr. Sokolov is a full member of the International Information Academy.

Serge V. Al’betkov was born in 1964 in Penza,Russia. In 1987, he graduated from MoscowEngineering-Physical Institute specializing inelectrophysical plants.

Since 1987, he has been working at the laboratoryfor electromagnetic compatibility in the All-RussianResearch Institute of Optophysical Measurements,State Standards of Russia. Presently, he is workingin the field of short EM-pulse radiation andmeasurement.

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Method for determining the electric and magnetic polarizability of arbitrarily shaped conducting bodies