Bioelectromagnetics Supplement 1:209-235 (1992)

Dosimetry of Extreme1y-Low- Frequency Magnetic Fields Charles Polk Department Island

of Electrical Engineering University of Rhode Island, Kingston, Rhode

Extrapolation of quantitative measurements across biological systems requires knowledge of field-organism interaction mechanisms. In the absence of such knowledge, one can only indicate which parameters would be important under some plausible assumptions that still lack experimental proof. In the first part of the paper it is assumed that biological effects of low intensity, extremely low frequency magnetic fields are caused by the electric fields which they induce. It is shown that detailed knowledge of electrical properties on a microscale is important to predict effects that may be due to local current density, electric field strength, surface charge distribution, and mechanical forces. I n the second part of the paper, it is shown that all proposed mechanisms for direct interaction between alternating magnetic fields and cells involve also the magnitude and direction of a simultaneously present static magnetic field. Reviewed are “cyclotron resonance,” quantum mechanical effects on ions weakly bound to proteins, nuclear magnetic resonance, and recent progress i n magneto chemistry dealing with effects of magnetic fields of a few hundred microtesla on chemical reactions that involve free radicals. 0 1992 Wiley-Liss, Inc.

Key words: ELF, magnetic fields, dosimetry, ELF bio-effects


The scaling problem of dosimetry involves the proper quantitative modification of exposure parameters-or interpretation of given exposure conditions-when comparing different tissue cultures or different animal species. Reliable scaling is only possible when we know the mechanism by which a particular environmental variable, in our case the time varying magnetic field, influences the biological system. In the absence of a known and well understood interaction mechanism it is only possible to identify the different variables, which are of importance, by assuming plausible, but nevertheless still hypothetical, field-biota interaction scenarios. This paper therefore consists of two parts. In the first we shall assume that biological effects of time varying magnetic fields are due to the electric fields, currents, Address reprint requests to Charles Polk, Department of Electrical Engineering, University of Rhode Island, Kingston, RI 02881,

01992 Wiley-Liss, Inc.



and charge distributions which they induce. Such biological effects would presumably depend upon ion paths around or through cells, field forces on membrane surfaces or protein strands protruding from membranes, on mechanisms for time averaging or rectification and on ion diffusion. Details of induced field structure and current paths at the cellular level are important as a clue to the understanding of field effects. This part of the paper therefore outlines how electric fields, current, and charge distributions induced into inhomogeneous and anisotropic biological substances by extremely-low-frequency (ELF) magnetic fields must differ from fields and currents that are due to an externally applied electric field. Consequences of these differences for the selection of dosimetric measurements are then discussed. The second part of the paper considers possible direct physiological effects of ELF magnetic fields and indicates the magnetic, chemical, mechanical and physiological parameters that must be known to “scale” such effects among different experimental systems.

MAGNETICALLY INDUCED ELECTRIC FIELDS AND CURRENTS Fundamental Considerations To a very high degree of approximation, the extremely low frequency magnetic flux density [B (t)], inside living tissues or cell cultures, is equal to that applied externally. This is a consequence of two conditions: First, the magnetic permeability of tissue and cells is, to within a fraction of one percent, equal to that of free space; secondly, the relatively low electrical conductivity (at most of the order of 1 S/m) of living matter, in comparison with that of metallic structures ( = 107S/m),guarantees that the magnitude of the “secondary” magnetic field produced by the induced “eddy currents” is negligible [Polk, 19901. Therefore the applied magnetic field can be measured or calculated (see Appendix) from known external current distributions (such as in power frequency transmission or distribution lines, or in specially designed coil configurations) without need to correct for any modification due to the insertion of an animal or tissue preparation. This is very different from the situation existing when the applied field is electric [Kaune and Miller, 19841. Nevertheless the “scaling problem” for magnetic fields is far from trivial. The principal problem (if one assumes that biological effects are due to the induced electric fields or currents) is the prediction of current paths, current magnitudes and charge distributions at boundary surfaces. These depend not only upon the applied magnetic field, but on the electrical properties of media which are highly inhomogeneous. The nature, size and location of inhomogeneities-such as membranes, entire organs, density gradients of cells in tissue culture, etc.-is clearly different in various systems which may be exposed to a magnetic field of the same magnitude. Different effects can therefore be expected from the same “dose” in different systems. The electric fields [El] and current densities [J] induced by B(t) follow different paths than those produced in tissue by an externally applied electric field [E,]. This is illustrated by the lumped circuit approximation, shown in Figure 1, of a rectangular “body” where discrete circuit elements (which are principally resistive and capacitive) have been substituted for the continuous distribution of different cells, inter-cellular matrix and fluid that make up a real biologic entity. For an applied EiL,which is much smaller inside than outside the body, [Kaune and Gillis, 1981;

Magnetic Field Scaling

21 1

Fig. I . Lumped circuit approximation of tissue mass: left-for externally applied electric field; rightfor externally applied time varying magnetic field.

Polk, 1986”], the generator or generators are arranged linearly as indicated. Furthermore, since Ea at ELF is “quasi-static” [Wait, 19851, the electric field distributions inside the body over any closed path must be such that

even in the presence of inhomogeneities or anisotropic structures. If the “body” at the left side of Figure 1 is electrically conductive, currents will primarily flow in the direction of the applied field. Internal structures which are electrically insulated, for example by membranes, will simply be bypassed [for details of such situations see, for example Kaune and Forsythe, 1985; Deford and Gandhi, 19851. On the other hand, B(t) will produce circulating currents in electrically isolated structures that are proportional to their electrical conductivity [ 0 1 , their cross-sectional areas [ai] perpendicular to B(t), and the time rate of change of B(t)

For the lumped circuits on the right side of Figure 1 , Equation 3 becomes

In the above equations V, is the emf applied over a closed path, d i is a directed element of length, dF or ai are elements of area and d E / d t is the time derivative of the magnetic flux density.



Since electrical properties of biologic substances can change substantially over a scale of nanometers (for example when a membrane is present), Equation 3 does not predict the electric field at every point. Any numerical computation which is based on it [Orcutt and Gandhi, 19881 will give correct answers on a micro-scale only when selected areas, or the corresponding mesh sizes, as in Figure I , are sufficiently small to correctly represent inhomogeneities on a micro-scale. Conversely, Maxwell’s curl equation for the magnetic flux density gives only spatial derivatives of E I ,rather than E, itself (the “curl” of E , , written V x E , is the circulation of E , per unit area): -


V X E , =---


A complete equation for the induced electric field at any point is

where is the magnetic vector potential (see Appendix) and V is an electrostatic potential that satisfies Laplace’s equation. V will appear even if the only source is a time varying magnetic field, because Gauss’s law and the law of conservation of electric charge lead to the formation of surface charge distributions at the boundaries between electrically dissimilar media. These quasi-static charge distributions will in turn produce V [see Appendix; also, Polk 1990; Polk and Song, 19901. Evaluation of the surface charge distributions may turn out to be important for “dosimetry”, because they could conceivably affect the action of the transmembrane channels [Blank and Britten, 19781, of receptors, or of other protein strands protruding from cell membranes and from membranes that surround many different organs. Calculation of Field and Current Distributions-Microscale Data Needed for Scaling

For a sinusoidally time varying magnetic flux density B(t) = Bo cos wt of constant magnitude over a circular cross-section of radius r, Equation 3 gives

wr E , =?B,,



where E, is the component of the induced electrical field in the circumferential (“ q”) direction at the radial distance r. This result is based on the assumption that the electrical properties are constant over the circular path of radius r. Furthermore, E, is equal to the total electric field El, only if the particular cylindrical body of radius r I a is completely electrically insulated from its surroundings. If it is not insulated, Gauss’s law and the requirement for conservation of electric charge at boundaries between materials of different electrical conductivity or dielectric permittivity, will lead to the existence of El components which are not circumferentially directed [Polk and Song, 19901, in addition to E, terms.

Magnetic Field Scaling


The size of electrically insulated structures, for example of organs contained within insulating membranes, is thus important. The value of E may however be considerably larger than indicated by Equation 7 if the electricaf properties of the cylindrical structure are not uniform over the entire circumference. Two conditions are illustrated by Figure 2. It represents a transverse cross-section through a long cylindrical body consisting of n sections, each having a mean length of [email protected], along the circumference. If the electrical resistance of the first section R extending from $ = 0 to $ = is very much larger than that of all the other sections Equation 3 gives in place of Equation 7 (R I >> R



Or, alternatively, if the sections of Figure 2 represent individual biological cells, whose cytoplasms have low electrical resistance, but whose membranes (i.e., the radially directed section boundaries in Fig. 2) have high resistance, the $-directed electric field in each membrane, of thickness A becomes

wB,, n r 2 En,



sin wt


When r = 1 cm, A = 5 nm, and the mean circumferential length of each cell is 20 pm, so that n = 1,000 n, Equation 9 would give Em=wB,,(20)sin cot. While Figure 2 obviously represents a highly idealized example, biological systems may present similar situations of tightly packed cells embedded in water containing inter-cellular matrix. As an example, Figure 3 illustrates an arrangement of tightly packed mesenchymal (undifferentiated) cells (the small cells at the right of the figure), located subcutaneously at the thoracic musculature of an immature

Fig. 2. Cross section through multisection cylindrical body. Magnetic field is assumed to be perpendicular to the plane of the paper.



rat, together with developing chondrocytes (cartilage cells) which are the larger, clearly nucleated cells at the center and left side of the illustration (the large dark sections are pieces of implanted decalcified bone matrix). The challenge for both experimental and analytical work is to determine induced electric fields, currents, and charge distributions in situations such as those illustrated by this figure. It should also be pointed out that i n such systems, where new cell development and growth are taking place (in this case cartilage development after implant of decalcified bone matrix), cell size and cell packing can change significantly over a period of a few days. Therefore it is to be expected that field magnitudes, current paths, and charge accumulations at boundaries will change likewise over such periods, even when the applied B(t) remains completely unchanged. Rigorous analytical methods have been employed to calculate magnitudes and directions of electric fields and currents magnetically induced into multilayer, electrically anisotropic fat-muscle-bone structures [Van Amelsfort, 19901. Various approximate methods that are well adapted to calculations of field distributions or field and current distributions in tissue at low frequencies are also available [Armitage et al., 1983; Polk and Song, 1990; Orcutt and Gandhi, 19881. However, they have thus far not been applied to situations such as those illustrated by Figure 3. Two difficulties are encountered here: first the electrical properties of such complex physiological structures are not known well enough on a micro-scale for mathematical modeling, and secondly, the mesh-size in the circuit cell methods of Candhi et al. [ 19841 would have to be very small and would therefore lead to very lengthy computations.

Fig. 3. Early chondrogenesis after subcutaneous implant of demineralized bone matrix at the thoracic musculature o f an immature rat.

Magnetic Field Scaling


The response to B(t) of structures such as those illustrated by Figures 2 and 3 will also be substantially influenced by the presence or absence of gap junctions, which are prevalent in many biological systems [Peracchia and Bernardini, 19841. The electrical conductance of a typical open gap junction is to S [Spray et al., 1984; Brink et al., 19881, while below 100 Hz the total conductance across the membrane that encloses a 10 pm diameter cell is between lo-’ and 10.’”S [based on data by Takashima, 1989; Schanne, 19781. A s a consequence, multiple cells connected by gap junctions would form continuous systems in which B(t) could induce currents flowing through the cell interior. This is important, because Equation 7 chows that the field (and resulting circulating current) induced into a typical cell of 10 or 20 p n diameter would be extremely small, even for B, as large as 0.1 mT (at 60 Hz E,= 0.2 pVlm for r = 1 1 pm). On the other hand, a large number of cells, electrically connected by gap junctions and enclosing an area significantly larger than the individual cell cross-section, would support a much larger magnetically induced electric field. The result is that a small B(t) could produce a net voltage over the circumference of the cell arrangement of Figure 2 that is substantially above thermal noise. If Figure 2 were to represent a circular ring resistor of mean radius r, radial width w, height h (perpendicular to the plane of the page), and conductivity 0,its resistance would be R = 2nr/(ohw), from Equation 3 the RMS value of the magnetically induced voltage would be V, = wB, m 2la,and the RMS thermal noise voltage [Bennett, 19701 is V n = ( 4 k TR where k is Boltzman’s constant, T is the absolute temperature, and the width of the admitted frequency band. The “signal-to-noise ratio” is then

where n is the number of cell boundaries (= 2ndcell length) and G is the gap junction conductance. If one assumes a current path with r = 1 cm and only single gap junctions between cells, with G = S, one needs for F = 1 at 60 Hz a not unreasonable value of the cell cross-section hw = l 0-’rnm’when the cell length along the circumference is 20 pm, B,, = 0.1 mT, 6f = 10 Hz, and (T = 1 S/m. With multiple gap junctions between cells the hw needed for F = 1 would become even smaller; alternatively, the value of B, could be reduced. Dielectric Permittivity and Ion Diffusion at ELF: Should the Effects of Magnetically Induced Electric Fields Increase Linearly With Frequency?

Some experimental work suggests that ELF alternating electric field effects at the cellular level are related to the field itself rather than the associated current [Miller, 19861. Also, there is evidence that receptors or trans-membrane proteins, in particular “G-protein,” which transmit information to the cell interior, are affected by the electric field at the exterior of the cell membrane [Luben, 19911. Furthermore, some histological data indicate that B(t) effects on bone start at the bonemuscle boundary [Rubin et al., 19891. It is therefore of interest to investigate how



B(t) and the resulting E, affect electric charge distributions at the boundaries between electrically different media, such as the cell-boundary, the boundary between bone and muscle, or between any other two tissues or organs that have different electrical conductivity or dielectric permittivity. As reviewed in the Appendix, Gauss’s law and electric charge conservation require the formation of a time varying surface charge density [p,] at such boundaries. This p, depends upon the shape of the boundary, the electric properties of the media on each side, and the magnitude and phase of the components of the electric field which are perpendicular to the boundary surface. As an illustration we consider the idealized situation depicted on Figure 4, which represents the cross-section through circular cylinders that are long compared with their diameters. The magnetic flux density B(t) = B,,cos Ot is perpendicular to the plane of the paper. The magnetically induced electric fields for this case have been evaluated [Polk and Song, 19901. From these results and the equations for p, in the Appendix, one finds that at the boundary between media 1 and 2 (the “bone-muscle” boundary)

The quantities (T and E represent, respectively, the electrical conductivities and dielectric permittivities. Each dielectric permittivity is equal to the dielectric permittivity of free space E, multiplied by the relative dielectric permittivity E ~ thus , E,=E,,E, ,. All other symbols are defined by Figure 4. Equation 11 gives the surface charge density at a sharp boundary between dissimilar media. It does not give the value









s 2--

0 +jo& 2

Fig. 4. Nonconccntric cylinders of “bone” (medium 2) and “muscle” (No. 1 ) surrounded by air ( N o . 0). The magnetic field is assumed to be perpendicular to the plane of the paper.

Magnetic Field Scaling


of the sinusoidally time varying counterion density at the surface of individual cells, which is [Schwarz, 1962; Polk, 19901

where q = ionic charge, no = counterion number density in the absence of an applied field, d = cell radius, k = Boltzman's constant, T = absolute temperature, and z = counterion relaxation time given for spherical cells by Equation 44 in Polk [ 19901 or Equation 5.33 in Pethig [1979]. Equation 12 is valid for long cylindrical cells (when E l is perpendicular to the cylinder axis) as well as for spheres. Through E l the value of p , will also be a function, such as sin 8 cos @, of the angular coordinates 8 and 4 [Polk, 19901. We propose to examine p , and p: as a function of frequency. For a sinusoidally time varying magnetic flux density Equation 7 gives Em,, = wB,,r/2. It then follows from Equation 12 that the absolute value of p: at its maximum position on the cell surface will remain within *lo% of its value p,:, at the relaxation frequency 1 /(2nZ) over a 1.5/1 frequency range. p: then approaches the truly constant value f i p : , as the frequency increases further (so that 07 >> 1) until at some very high frequency Z is no longer a constant and begins to increase as a result of decreasing counterion mobility. The value of p , also remains virtually constant over a wide frequency range at low frequencies. For example, referring to Figure 4 and using 0, = 0.5 S/m for muscle and 0,=. 0.01 S/m for bone [Foster and Schwan, 19861, it appears that below 1 kHz p, =: JWK(CT,E, - CT2&,) sin 4, where K combines all the constant quantities in Equation 11, including E,. Using the values of &,,and &, , given on Table 1, one obtains p, = 10hKat both 10 Hz and 100 Hz and p , = 1.7 (lo6) K at 1 kHz. Still largely unexplained are phenomena such as constancy of an increase in DNA synthesis [Liboff et al., 19841 when the frequency of a constant magnitude magnetic field varies over a wide frequency range. By Equation 7 one would expect linear variation with frequency, rather than constancy, of any effect that is directly proportional to the magnetically induced E. The constancy with frequency of p , and p: , indicated above, would offer an explanation if very small changes in surface charge distribution at cell boundaries could influence receptor action or enzymatic processes and thereby cell metabolism. The frequency range over which p , remains roughly constant could be wide enough to explain the 15 Hz to 4 kHz data of Liboff et al. [ 19841. However, both p , and p: are very small, except when Bo is large. From Equations 7 and 12 the ratio of the number density n=( p:/q) to the normally existing n, is (qdcoBc;)/(2kT 4 2 ) at the relaxation frequency f = 1/2X2. TABLE 1. Typical Values of Relative Dielectric Permittivity of Skeletal Muscle and Bone* Frequency 10 Hz 100 Hz 1 kHz ~~~~~~~

*From Foster and Schwan [I0861

E rl

E r2

107 1.1 (106) 2.2 10')

lo4 (estimate) 3,800 1 .000



Thus one obtains (n/n,]) = 1.7 ( at 300 OK even if BOis as large as 1 mT, r = 10 cm, d = 10 pm, 0 = 2K(100) s - ' ,and ions are doubly charged ( q = 3.2 x 10 C ). A similar evaluation of p , at a muscle-bone boundary with the values of o,,02,&, , and E2 given above, a = c = 1 cm and b = 5 cm gives lo-' C/m' between 10 Hz and 100 Hz. This corresponds only to 3.1 (lo-') no if n,, = 10'' m-? [Pethig, 19791. In an ionized fluid medium, the magnetically induced steady-state alternating current will also be modified by a transient, i.e., exponentially decaying, quasiDC current due to ion diffusion in the presence of any spatial variation of charge density. It is shown in the Appendix that diffusion currents decay with a relaxation time of the order of 0.1 s in cylindrical systems of cellular dimensions when currents are constrained to flow along circular paths, but can take hours to decay in systems which have radii of the order of cm. Forces Due to Interaction of Induced Currents and Applied Magnetic Fields

Wires carrying current in the same direction attract. Similarly the current filaments induced in the intercellular space around electrically insulating cells will attract and therefore exert a pressure on the cell wall. However, in view of the relatively low conductivity of tissue-at most of the order of a few S/m-the induced currents are small and the resulting pressures are estimated to be negligible. We use an expression for the force between two filamentary conductors [Smythe, 19501 to calculate the pressure on a cell wall when the induced electric field has the relatively high value of 1 mV/m (that value corresponds to Ei induced by a 60 Hz magnetic flux density of 0.1 mT in a uniform region at a radius of 5 cm). Assuming that the distance between adjacent cells is about equal to the cell diameter ( = 10 pm ) and using 0 = 1 S/m for the conductivity of the inter-cellular medium, we obtain a value of 10." P (= N/m?). To place this number in perspective, we note that the minimum sound pressure that can be detected by the human ear is approximately 1 pP [Beranek, 19541 and that significant deformation of erythrocytes seems to require pressures of the order of lo3 P [Ashe et al., 19881. McLeod and Rubin [I9921 have pointed out that another source of pressure at the cell wall could be the Kelvin polarization force density [Haus and Melcher, 19891 due to the high gradient in dielectric permittivity at that location. They indicate that for large dielectric constants the resulting pressure would be approximately equal to E E ,,,, where E, is the tangential electric field outside the cell and &, the dielectric permittivity of the immediate cell environment. Assuming again an electric field of 1 mV/m, and E,, = E , (1.13) l o 7 one obtains a peak pressure of approximately lO-'"P. The pressure to be considered here in more detail is that due to the interaction of the induced current with the magnetic field that caused it. At first it would seem that such a force should have a zero time average, because i t follows from Equations 2 and 7 that the current density induced into a homogeneous conductor by a magnetic density B = i B, cos cot (selecting cylindcical coordinates with the z axis perpendicular to the plane of the page) would be 7 = @ ( OWB r/2) sin Wt . Since t h e volume force density is given by -

f = J x B N/m3


the magnitude of the resulting radially directed force density would be proportional to (sin mt cos mt) = sin 2 cot which has zero time average.

Magnetic Field Scaling


Before we show under what conditions this double frequency variation can be replaced by cos2Wt,we evaluate the magnitude of the resulting pressure on a rigid cylindrical surface which surrounds a medium whose electrical properties are independent of the radial distance r. This situation might be a crude first approximation for the inside surface of solid bone surrounding bone marrow. For the case of complete homogeneity, the integration shown in the Appendix gives

where a is the radius of the cylindrical boundary and f(t) I 1 is the appropriate function of time, such as (t)sin 2 cot, above. Using the same frequency-i.e., 60 Hz, as above-and the same flux density of Bo = 0.1 mT, a conductivity 0 = 1 S/m, and selecting for the radius a = 1 cm, we obtain a maximum value (in time) p = 0.62 (lO-‘O) P. While this value is about equal to what is obtained from the polarization force, the effect of the J x B force, which is everywhere in the radial direction, would be restricted to cells located near a rigid boundary. This is apparent from Equation 14 and Figure 2: Unless r = a represents a rigid boundary and no current flows at r > a, a “cell” of diameter w, located between r = a - w and r = a, would experience only a pressure p ( a ) - p ( a - w ) = o B % ( 2 a w - w 2 ) 0 / 6 , which would be negligibly small for w = 10 p m . We now consider the conditions under which the time average of f(t) in Equation 14 can become non-zero. This, of course, would be equivalent to “rectification” of the input “signal” and would produce a steadily continuing effect of the applied time varying field. Such a situation could occur when B,, cos a t induces an electric field over a path where current must flow through a material which behaves primarily as a dielectric, connected in series with a good conductor. The resulting current in the conductor can then be nearly 90” out of phase with the applied voltage and consequently, nearly in phase with the applied magnetic field. Figure 5 provides an illustration of this principle. If the magnetic flux density B in that figure is uniform over the entire cross-sectional area of the cylinder, it follows from the boundary condition, A-9 in the Appendix, that the circumferentially directed field in the conductor, E,, is related to the electric field in the dielectric ED,by


For illustration we assume that the conductive “gap,” extending from - 4, to @, , is small (4, > O I , ,. The resulting pressure will therefore have a unidirectional average value and will continue as long as the application of the sinusoidally time varying magnetic field continues. DIRECT EFFECTS OF TIME-VARYING MAGNETIC FIELDS

We shall consider several proposed theories which imply direct interaction of B(t) with tissue and cells. They have been advanced to explain some puzzling experimental results [Smith et al., 1987; Blackman et a]., 19881 and suggest, in some cases, critical experiments for their verification or rejection. Existence of direct magnetic field effects would, of course, profoundly affect dosimetric extrapolation across biological systems. Cyclotron Resonance

As a consequence of the Lorentz force F = q (V x B), a charge q of mass m, which moves at velocity V in a constant magnetic flux density B,. will follow, in a vacuum, a circular path of radius Rc

Magnetic Field Scaling


the angular velocity, w~,,will be given by qB m

m c =-


If an electric field is applied as in a cyclotron [Sears, 19461, or induced by an alt ) , that has the same direction as B\, the ternating magnetic field B = B exp (jo, tangential velocity v of the charge will increase, as will correspondingly, its orbital radius Rc. It has been noted [Polk, 19841 that the “cyclotron frequencies”, f , = (o,/ 2z), of many physiologically important ions fall below 100 Hz in the geomagnetic field ( =50 pT depending on latitude). For example f, = 38.4 Hz for the T a + + ion at 50 pT. Liboff [ 19851 and Chiabrera et al. [ 19851 have therefore proposed various mechanisms which require cyclotron motion of ions along circular or helical paths, either in transmembrane channels or at receptors on the cell surface [Chiabrera et al., 19851. However, it has been pointed out [Polk, 1986b; Durney et al., 1988; Halle, 1988; Sandweiss, 19901 that such motion is clearly impossible in the dense, collision dominated fluids of biological materials. Furthermore, ions in biological fluids are normally hydrated and the frequency given by Equation 19 would depend on the number of water molecules (and their total mass) in the hydration sheath of each ion [Koryta, 19821. “Parametric Resonance” of Ca-ions in Ca-Binding Proteins

Another attempt to explain results such as those reported by Thomas et al. [ 19861 or Smith et al. [1987] is due to Lednev [1989, 19911. He proposes that an ion weakly bound within a protein, notably Ca++within calmodulin, can be modeled as a charged oscillator. The oscillations are thermally excited and are at infrared frequencies. An ambient, steady (“DC”) magnetic field, B\, such as that of the earth, would cause Zeeman splitting [Haken and Wolf, 19841 of each vibrational level into two levels, 0, and 0,, separated by the ion cyclotron frequency. Thus W , = 0, - 0,. When an alternating field, B , , is then applied in parallel with B,, the resulting frequency modulation of levels o,and o,will change the probability of ion transition, P, from levels to some ground state of hequency 0,. Applying earlier work by Podgoretskii and Khrustalev [ 19641, Lednev then predicts that P = A : + A : + 2 A , A , (-1)” J,,( ~ ) c o s ~


where A , and A, are the amplitudes of IR radiation corresponding to transitions from 0, and 0,to W , ,6 is the difference in phase of the radiation from two sublevels, and the argument of the Bessel function Jn(x) is



The term in brackets in Equation 20 is zero, except when the frequency of B exp( j u t ) is f=-= 0-,


f;. n

n = integer




Lednev’s model thus predicts not only “resonances” at the cyclotron frequency and its subharmonics, but also prescribes the relative amplitudes of B , and BI for which the “resonance” effects should be maximized. These are given by the values of the argument x which corresponds to extreme values of Jn(x). For example, J,(x) has its first maximum at x = 1.84. Lednev explains the appearance of “resonances” at harmonics 2f,, 3f,, etc. of the cyclotron frequency, in addition to the predicted effects at the subharmonics (f / 2 .If two degrees of freedom correspond to the rotational motion of the ion (since two independent angles are needed to specify the orientation of the axis of rotation), the rotational energy is kT and the magnitude o f the magnetic moment becomes

where r is the orbital radius and m is the mass of the ion. The maximum value of the potential energy of M in a static magnetic flux density Bpwould be

_ _ W=M.B,


J if q is equal to two elementary charges, m = 40 which is equal to B (1.2) proton masses for Ca, and r = 0.1 nm. Since M experiences a torque x B,, it will tend to precess at the Larmor frequency f, = (fi/2) about the direction of B,,. Such precession would require displacement of the orbit 0 in Figure 6 from its original position and, according to Zhadin and Fesenko, would lead to deformation of the ion-protein bonds. However even a very weak ionic bond would have an energy of about 0.1 -eV - = 1.6 (10 2") J, which is about 10 orders of magnitude larger than the energy M . B = (10 'I ) J in the earth's field of about = 50 pT. A further assumption in this model is that either an electric field of frequency f, = (fi/2), perpendicular to B,, or an alternating magnetic flux density B,, in parallel with B\ and of frequency f,, would produce resonance conditions that would provide power to the system via a torque ?; = x El cos cot. Over a sufficiently long period of time chemical bonds could be broken. As shown in the Appendix, it appears that the alternating field B, could be perpendicular, as well as parallel to E\ to provide


Fig. 6. Zhadin-Fesenko model for precession of chemically bound ion.



resonance conditions. However for the conditions assumed above, which give f, = 19.17 Hz, an alternating flux density with a peak amplitude of 0.1 mT would have to act on the system for 7 years to supply an energy l T d 6 equal to 0.01 eV. Nuclear Magnetic Resonance

Nuclear magnetic resonance is widely used in chemistry to obtain information on chemical structure, and in medicine-thus far-to obtain images that reflect the influence of surrounding tissue on hydrogen nuclei. The technique is based on the fact that many, but not all, atomic nuclei appear to rotate about their axis. This “spin” can be described by the angular momentum quantum number I, which can have integral or half-integral values between -12.5 and +12.5 [see Weast et al., 19861. Since nuclei also carry electric charge, the spin provides a magnetic moment M. In the presence of a static magnetic field B\, that moment can orient itself along preferred directions which are determined by the nucleus’ angular momentum. Transitions between the various discrete orientations correspond to changes in energy. The energy change is

Transfer from one discrete energy level to the next higher one requires irradiation by a signal at the nuclear precession frequency f n such that A W = f,, h


where h is the Planck constant. Conversely, transition to a lower energy level will be accompanied by radiation of energy at the precession (Larmor) frequency f,. From Equations 25 and 26 it follows that

The usefulness of NMR spectroscopy lies in the fact that the exact value of the frequency f, is not only determined by the externally applied field B\, but depends also on the magnetic field generated by the surrounding orbiting electrons (which form diamagnetic moments). This orbital motion, which depends on Bs,will not only affect the parent nucleus, but also the Larmor frequency of neighboring nuclei [Paudler, 197 I ]. In addition, multiple peaks are observed in the NMR spectrum whose separations are independent of Bs.They are due to the interaction between neighboring spins. When the thermal equilibrium of a molecular system is disturbed, nuclei will change their spin state. Thereafter, as a result of magnetic field fluctuations, which reflect molecular thermal motion, the nuclei will be induced to return to their equilibrium state. The exponential law which describes this return is characterized by the spin-lattice relaxation time, T , , which may have values between 10.’ and 10’ seconds in liquids and can reach days in solids. In addition, a generally shorter period, the spin-spin relaxation time, T1, characterizes return to equilibrium due to the influence of neighboring nuclear spins. T, is influenced by such factors as molecular

Magnetic Field Scaling


geometry and the nature of the particular chemical bond. Values of T, are normally between a few milliseconds and at most one second [Ando and Webb, 19831. As a possible mechanism for biological effects of low intensity, low frequency alternating magnetic fields, in the presence of the earth's static field, NMR has both very attractive and very unattractive features. First, the characteristic precession frequencies, fn,are below 1,000Hz for many biologically important elements at earth strength fields. Thus, with B, = 50 pT,f, is equal to 827 Hz for 'Li, 653 Hz for 23Na, and 99.3 Hz for 3yK.Secondly, stability of the resonance lines (or "high Q" of the resonance peaks), reflected in the relatively long relaxation times, should make it possible to observe resonance excitation by weak alternating magnetic fields of the appropriate frequencies. Unattractive features, which make NMR unlikely as an interaction mechanism are, first, the small probability of spin alignment in a weak applied field. Thus the ratio of the maximum energy of M in a field B, to the thermal energy is


For the hydrogen nucleus, which is employed in medical diagnosis at fields of the order of 1 T, = 3.3 while = 3.8 (lo-'") for 'Li at 100 pT . Whether this difference by 4 orders of magnitude makes biological effects of NMR at such levels of magnetic field completely impossible is still debatable. Effects could only occur if the resonance phenomenon would influence extremely critical processessuch as enzyme mediated reactions-where the effect of a very minute change can be greatly amplified. It is also questionable how a change in spin state could significantly influence chemical structure-although the reverse is true, as indicated by the spin-lattice and spin-spin relaxation phenomena. The spin-spin relaxation phenomenon, in particular, is an indication that changes in the spin state of one nucleus do affect its surroundings. However strong interaction with the nuclear environment, such as by I4N nuclei which have an electric quadruple moment [Haken and Wolf, 19841, would lead to significant line broadening [Paudler, 19711 and thereby to the loss of observable frequency sensitivity at ELF. NMR effects have been suggested as an explanation of relatively low field intensity effects, notably changes in dielectrophoretic yield and dielectric permittivity, that were observed at a few kHz [Aarholt et al., 19881. They have also been mentioned as a possible explanation of field effects on the efflux of radio-active calcium ions (45Ca++) from chick brain tissues [Blackman et a]., 19881. The NMR phenomenon is attractive in the latter context, because the experiments showed sensitivity to steady and alternating magnetic fields which were mutually perpendicular, as required for NMR excitation. However, it should be noted that the effect would only occur with radioactive Ca, and not with the non-radioactive '"Ca isotope, which accounts for 97 percent of all Ca occurring in nature. 40Cahas zero nuclear spin and therefore no magnetic moment. (Some other biologically important elements with zero spin are '*C, IhO,56Fe,24Mgand 33S.)One way of differentiating between an NMR effect and any type of whole ion-resonance phenomenon, involving Ca, would be to repeat experiments, which require a radioactive isotope, with both 43Caand 45Ca.These isotopes have the same spin (-7/2) and practically the same M (- 1.3 172 and - 1.316 in nuclear rnagneton units) and therefore the same





NMR frequencies fn. However the ionic masses are sufficiently different (43/45) to give a 4.5 percent change in the cyclotron resonance frequency f,. Magnetic Field Effects on Free Radical Reactions Free radicals are important in many biological processes [Freifelder, 19821. For example, free radicals are formed as intermediate products when light is incident upon the visual pigment Rhodopsin. Chemical reactions that involve free radicals are strongly influenced by static magnetic fields. This has been known for some time [Hoff et al., 1977; Blankenship et al., 1977; Werner et al., 19781. However, more recent work [Hamilton et al., 1988; McLauchlan, 19891 suggests that DC fields as low as 10 Gauss may affect such chemical processes. By implication, as will be shown below, alternating fields of the same order of magnitude, acting in concert with steady fields should also influence reaction yields. A number of recent observations i n this area is based on Pauli’s exclusion principle, which states that the electronic states of an atom can only be occupied in such a way that no two electrons have exactly the same set of quantum numbers. Thus if there are, for example, two valence electrons in the same orbital, characterized by the same set of orbital quantum numbers, their individual spin quantum numbers must be + 112 and -1/2; i.e., their spins must be in opposite directions. If two electrons in a chemical bond are paired in this manner and if this bond is broken, for example, by incident light, resulting in two free radicals, subsequent recombination is only possible if the two electrons preserve this oppositely directed spin. Interaction with the local magnetic field-due to nuclear magnetic moments or nearby other spinning and orbiting electrons-can, depending upon details of the particular molecular structure, either favor or destroy opposite spins. In the latter case, of now equally oriented spins, recombination of the radicals becomes impossible. Conversely, the radical pair may have been formed from excited atomic states with the unpaired electrons coming from different atomic shells. In that case they may already have equal spin preventing chemical combination of intermediate products. As long as the electrons have opposite spin, the products have “singlet” character, i.e. the total quantum number J, which characterizes the electron states, is equal to the orbital quantum number L, since the spin quantum number S = (1/2) - (1/2) = 0 and J = S + L. However, as the products diffuse, some fraction will acquire “triplet” character (i.e., the electron spins may become parallel; then S = k (1/2 + 1/2) = k 1 and J can have 3 values, L + I , L - I and L, in view of quantum rules for the combination of angular momenta. If the products were initially in the triplet state, diffusion will have the opposite effect, i.e., partial conversion from triplet to singlet character. The singlet and triplet states have, in general, different energies, as indicated at B = 0 on Figure 7. Any magnetic field, including that of nearby magnetic nuclei, will cause triplet states, T+,and T-,, which have electron spin in the direction of the field, to gain or lose energy. Therefore the energy levels of these states will separate with increasing B, as indicated on Figure 7. Interconversion between the singlet and triplet states can occur either by external energy input, or between To and S through a distance dependent “electron exchange interaction” at any level of B. However, at some critical level of B = Bc interconversion between the T-, and

Magnetic Field Scaling



TIME Fig. 7. Effect of magnetic field on radical pair energy levels.

S is also possible without external energy input. Hamilton et al. [ 19881 have shown experimentally that this level is approximately 1 mT in their pyrene-dimethylaniline system. Since interconversion between singlet and triplet states in the direction of greater singlet product will make possible chemical combination, application of the correct value Bc will obviously affect the rate of chemical reaction. However it is possible that an ambient flux density (from external and internal sources) may have a value B,, which is either slightly larger or smaller than the required B,. In that case addition of an alternating field B,,cos o t (with a period 2 a l o longer than necessary for T-, + S ) will periodically establish optimum conditions for conversion. On the basis of theoretical and experimental results [Hamilton et al., 19881 it is therefore at least possible that combination of DC and AC magnetic fields of the order of 0.5 mT (= 1/2 of the measured B,) can affect chemical reactions in biological systems that involve free radicals as intermediate products.



CONCLUSIONS FOR DOSIMETRY Assuming That B(t) Acts Only via the Induced Electric Fields, Currents, and Charge Distributions

We have shown that prediction of the electric field from the applied magnetic field in cellular systems requires detailed knowledge of their electrical properties on a microscale. Furthermore, the orientation of structures, such as organs and groups of cells in vivo, or organ cultures and confluent cell cultures in vitro, in relation to the direction of the applied magnetic field must be known. Transfer, with appropriate scaling, of information obtained on one in vivo or in vitro system to another one requires comparison of the size and electrical properties of corresponding body parts, cells or cell assemblies that support circulating currents. A further consideration must be that cell configurations, including their size and internal electrical connections, can change radically over a period of a few days in developing systems such as a healing wound or bone fracture. The electromotive force (emf) applied to closed paths is directly proportional to the time derivative of the applied magnetic flux density. Consequently one might expect, for a sinusoidally time varying B(t), that effects caused by induced electric fields and currents would increase linearly with the frequency of B(t). However, such a variation will not occur at ELF whenever the modification of electric charge distribution at boundaries is physiologically important. Information obtained on one system can be used to predict field effects on another system only if the variation of electrical conductivity and dielectric permittivity with frequency of both systems is known. Mechanical properties of cell systems, including the viscosity of inter-cellular tluids and the existence of rigid boundaries, such as bone surfaces, may also play a role in field-cell interaction. Assuming Direct Magnetic Field Effects

The mechanisms which are presently of most interest for direct interaction between alternating magnetic fields and tissue have two features in common: (1) they deal ultimately with kinetic, non-equilibrium phenomena, such as the effectiveness of enzymes that determine chemical reaction rates; and (2), the effects of alternating magnetic fields depend upon the simultaneous presence of a non-time varying (“DC”) magnetic field. The “parametric resonance” model of Lednev requires presence of DC and AC magnetic fields that are directed inparullel and have amplitudes within the same order of magnitude. Frequencies at which effects can be expected are the cyclotron frequency, fc, of physiologically important ions, its subharmonics and possibly also harmonics. Since the model is particularly applicable to ions which are weakly bound to a protein, and since the ubiquitous Ca-binding calmodium affects numerous physiological processes, the field combinations which correspond to 4”Ca++ “resonances” are of particular interest. The model of Zhadin and Fesenko also considers ion binding within a protein. Thermally excited ion motion is assumed to take place along some constant orbit which, upon application of a DC magnetic field, Bs, will tend to precess about the direction of Bsat the Larmor frequency f,- = (fJ2). Addition of an alternating electric field, perpendicular to B\, or of an alternating magnetic flux density B (,cos at

Magnetic Field Scaling


of the same frequency, is then said to produce resonance conditions. The orientations of B, and Bocan be either parallel or perpendicular to each other; however it appears that significant effects on ion-protein binding, with B5and Bo of the order of 50 pT, would require years of continuous field application. Excitation of nuclear magnetic resonance (NMR) by the combination of earth strength DC and low intensity ELF magnetic fields is possible, i n principle, since the NMR frequencies of some physiologically important elements are below 1,000 Hz when the DC field is of the order of 50 pT. However, the ratio of the nuclear magnetic moment’s energy in such a field to the thermal energy is so low that only a minute fraction of nuclear spins would align with the applied field. Furthermore, it is not clear how a change in nuclear spin state could affect chemical binding without very substantial “line broadening,” implying destruction of resonance. Chemical reactions that involve free radicals as intermediate products are common in living systems. Recent theoretical and experimental results suggest that such reactions can be sensitive to combinations of DC and AC fields of less than 1 mT. The investigation of alternating magnetic field effects on living systems requires more critical experiments to either confirm or eliminate suggested interaction mechanisms. It is clear, however, that the action of alternating fields can be profoundly affected by the presence of static fields which, alone, would have biological consequence only in organisms which contain ferromagnetic material [Frankel, 19861. Experiments with AC fields should therefore always include measurement of the magnitude of the ambient DC field. The orientation of the DC field in relation to that of the AC field vector also appears to be an important variable. REFERENCES Aarholt E, Jaberansari M, Jafari-As1 AH, Marsh PN, Smith CW (1988): NMR conditions and biological systems. In Marino AA (ed): “Modern Bioelectricity.” New York, Basel: Marcel Dekker, Inc., pp 75-104. Ando I, Webb GA (1983): “Theory of NMR Parameters.” London, New York: Academic Press, pp 83-105. Armitage DW, LeVeen HH, Pethig R ( 1 983): Radiofrequency induced hyperthermia: Computer simulation of specific absorption rate distribution using realistic anatomical models. Phys Med Biol 28 (1):3142.

Ashe JW, Bogen DK, Takashima S (1988): Deformation of biological cells by electric fields: theoretical prediction of the deformed shape. Ferroelectrics 86:3 1 1-324. Bennett WR (1970): “Introduction to Signal Transmission.” New York: McGraw-Hill, p 47. Beranek LL (1954): “Acoustics.” New York: McGraw-Hill, p 395. Blackman CF, Benane SG, Elliott DJ, House DE, Pollock MM (1988): Influence of electromagnetic fields on the effux of calcium ions from brain tissue in vitro: A three-model analysis consistent with the frequency response up to 510 Hz. Bioelectromagnetics 9:215-227. Blank M, Britten JS (1978): The surface compartment model of the steady state excitable membrane. Bioelectrochemistry and Bioenergetics 5:535-547. Blankenship RE, Schaafsma TJ, Parson WW (1977): Magnetic field effects on radical pair intermediates in bacterial photosynthesis. Biochim Biophys Acta 461 :297-305. Brink PR, Mathias RT, Jaslove SW, Baldo GJ (1988): Steady-state current flow through gap junctions. Biophys JI 53:795-807. Bruckner-Lea C, Durney CH, Janata J, Rappaport C , Kaminski M (1992): Calcium binding to metallochromic dyes and calmodulin i n presence of combined AC-DC magnetic fields. Bioelectromagnetics 13: 147-1 62.



Cheung WY (1982): Calmodulin. Sci Am 246:62-70. Chiabrera A, Bianco B, Caratozzolo F, Gianetti G, Grattarola M, Viviani R (1985): Electric and magnetic field effects on ligand binding on the cell membrane. In Chiabrera A, Nicolini C, Schwan HP (eds): “Interaction Between Electromagnetic Fields and Cells.” New York, London:Plenum. Cussler EL ( 1984): “Diffusion-Mass Transfer i n Fluid Systems.” Cambridge, New York: Cambridge University Prcss, pp 148-150. Deford JF, Gandhi OMP (1985): An impedance method to calculate currents induced in biological bodies cxposed to quasi-static electromagnetic fields. IEEE Trans Electrom Compatibility EMC-7 (3): 168173. Durney CH, Rushforth CK, Anderson A A (1988): Resonant AC-DC magnetic fields: calculated response. Bioelectromagnetics es of tissue. I n Polk C, Postow E (eds): “CRC Handbook Foster KR, Schwan HP ( 1986): Dielec of Biological Effects of Electromagnetic Fields.” Boca Raton, Florida: CRC Prcss, Inc., pp 74, 88-89. Frankel RB (1986): Biological effects of static magnetic fields. I n Polk C. Postow E (eds.): ”CRC Handbook of Biological Effects of Electromagnetic Fields.” Boca Raton, Florida: CRC Press. Inc., pp. 169-196. Freifelder D (1982): “Principles of Physical Chemistry with Applications to the Biological Sciences.” 2nd ed. Boston:Jones and Bartlett Publishers, Inc. p 379. Gandhi, OMP, DeFord JF. Kanai H (1984): Impedance method for calculation of power deposition patterns in magnetically induced hyperthermia. IEEE Trans. Biomed Engr BME-3 I ( 10): 644651. Haken H, Wolf HC (1984): “Atomic and Quantum Physics.” Berlin, Heidclberg, New York. Toronto: Springer-Verlag, pp 202-208. Halle B (1988): On the cyclotron resonance mechanism for magnetic field effects on transmembrane conductivity. Bioelectromagnetics 9:38 1-386. Hamilton CA, Hewitt JP, McLauchlan KA, Steiner UE (1988): High resolution studies of the effects of magnetic fields on chemical reactions. Molecular Physics 65:423-438. Haus HA, Melcher JR ( 1989): “Electromagnetic Fields and Energy.” Englewood Cliffs, New Jersey: Prentice Hall, pp 5 14-5 16. Hoff AJ, Radcniaker H, Van Grondelle R, Duysens LNM (1977): On the magnetic field dependence of t h e yield of the triplet state in reaction centers of photosynthetic bacteria. Biochim Biophys Acta 460347-554. Johnson CC (1965): “Field and Wave Electrodynamics.” New York: McGraw-Hill, pp 36-40. Kaune WT, Gillis MF [ 198 I ) : General Properties of the interaction between animals and ELF electric fields. Bioelcctromagnetics 2: 1-1 I . Kaune WT, Miller MC (1984): Short-circuit currents, surface electric fields, and axial current densities for guinea pigs exposed to ELF electric fields. Bioelectromagnetics 5:361-364. Kaune WT, Forsythe WC (1985): Current densities measured in human models exposed to 60 Hr. electric fields. Bioelectromagnetics 6: 13-32. Koryta J (1982): ‘‘Ions, Electrodes and Membranes.” New York: John Wiley and Sons, pp 14 ff. Lakshminarayanaiah N ( 1984): “Equations of Membrane Biophysics.” Orlando, Florida: Academic Press, Inc., pp 50-54. Lednev VV (1989): Possible mechanism of the effcct of weak magnetic fields on biosystems (In Russian). Pushino: Nauchnyi Tsentr Biologicheskikh Issledovanii. Preprint. Lednev VV (1991): Possiblc mechanism for influence of weak magnetic fields on hiosystems. Bioelectromagnetics 12:7 1-75. Liboff AR, Williams T, Strong DM, Wistar R (1984): Time-varying magnetic fields: Effect on DNA synthesis. Science 223:818. Liboff AR (1985): Cyclotron resonance in membrane transport. In Chiabrera A, Nicolini C, Schwan HP (eds): “Interactions Between Electromagnetic Fields and Cell.” New York, London: Plenum. Luben R ( 1991): Effects of low energy electromagnetic fields on membrane transduction in biological systems. Health Phys 61 : 15. Markov MS, Ryaby JT, Wang S, Pilla AA (1992): Modulation of myosin phosphorylation rates by weak (near ambient) DC magnetic fields. Proc IEEE/EMBS 18th Northeast Bioengineering Conference. McLauchlan KA ( 1 9893: Magnetokinetics, niechanistics and sysnthesis. Chemistry in Britain. Sept. 895-898.

Magnetic Field Scaling


McLeod BR, Liboff AR (1986): Dynamical characteristics of membrane ions in multifield configurations at low frequencies. Bioelectromagnetics 7: 177-1 89. McLeod K, Rubin C (1992): The role of polarization forces in mediating the interaction of low frequency electric fields with living tissue. In Blank M (ed.): “Electricity and Magnetism in Biology and Medicine”. San Francisco Press. Miller MW ( 1986): Extremely low frequency (ELF) electric fields: Experimental work on biological effects. In Polk C, Postow E (eds) “CRC Handbook of Biological Effects of Electroniagnetic Fields.” Boca Raton, Florida: CRC Press, Inc., pp. 148-149. Orcutt N, Gandhi OMP (1988): A 3-D impedance method to calculate power deposition in biologic bodies subjected to time varying magnetic fields. IEEE Trans. Biomed Engr 35:577-583. Panofsky WKH, Phillips W (1955): “Classical Electricity and Magnetism.” Cambridge, Massachusetts: Addison-Wesley, pp 162, 163. Paudler WW (1971): “Nuclear Magnetic Resonance.” Boston: Allyn and Bacon, Inc., pp 1-15; 81. Peracchia C, Bernardini G (1984): Gap junction structure and cell-ro-cell coupling regulation: Is there a calmodulin involvement‘! Federation Proceedings (FASEP) 43 ( I 2):268 1-269 I . Pethig R (1979): “Dielectric and Electronic Properties of Biological Materials.” New York: John Wiley and Son, pp 107, 172. Podgoretskii MI, Khrustalev O A ( 1964): Interference Effects in Quantum Transitions. Soviet Physics Uspekhi (in translation) 6:682-700. Polk C ( 1984): Time varying magnetic fields and DNA synthesis: Relative magnitudes of Lorentz forces and induced emf’s in counter-ion sheets. Abstracts, URSl XXIst General Assembly, OS1: Interaction of Electromagnetic Fields with Biological Systems, p. 25. Polk C ( I986a): “CRC Handbook of Biological Effects of Electromagnetic Fields.” Introduction. Boca Raton, Florida: CRC Press, Inc. Polk C (1986b): Physical mechanisms by which low-frequency magnetic fields can affect the distribution of counterions on cylindrical biological cell surfaces. J Biol Phys. 14:3-8. Polk C ( 1990):Electric fields and surface charges induced by ELF magnetic fields. Bioelectromagnetics 11:189-201. Polk C, Song JH (1990): Electric fields induced by low frequency magnetic fields in inhomogeneous biological Structures that are surrounded by an electric insulator. Bioelectromagnetics 1 I :235249.

Rainteau D, Wolf C, Lavialle F (1989): Effects of calcium and calcium analogs on calmodulin: A Fourier transform infrared and electronic spin resonance investigation. Biochim Biophys Acta 101 1.8 I 87. Rubin CT, McLeod KJ, Lanyo LE (1989): Prevention of osteoporosis by pulsed electromagnetic fields. J Bone Joint Surgery 71A:41 1-416. Sandweiss J (1990): On the cyclotron resonance model of ion transport. Bioelectromagnetics 11:203205.

Schanne O F (1978): “Impedance Measurements i n Biological Cells.” New York: John Wilcy and Sons. Schwarz G (1962): A theory o f the low frequency dielectric dispersion of colloidal particles in electrolyte solution. J Phys Chem 66:2636-2642. Sears FW (1946): “Electricity and Magnetism.” Reading, Massachusetts: Addison-Wesley Publishing Co., pp 231-235. Shuvalova LA, Ostrovskaya MV, Sosunov EA, Lednev VV (1991): Influence of weak magnetic. field under conditions of paramagnetic resonance on the rate of calmodulin-dependent phosphorylation of myosin in solution. Proc Natl Acad Sci (USSR), Vol. 3 17, no. I , UDK 577.3, Biophysics, pp 227-230. Silva RS, Amital Y, Connors BW (1991): Intrinsic oscillations of neocortex generated by layer 5 pyramidal neurons. Science 25 1 :432-435. Smith SD, McLeod B, Liboff AR, Cooksey K (1987): Calcium Cyclotron Resonance and Diatom Mobility. Bioelectromagnetics 8:2 15-227. Smythe WR (1950): “Static and Dynamic Electricity.” 2nd ed. New York: McGraw Hill, Inc., p 281. Spray DC, White RL, Campos de Carvalho A. Harris AL, Bennett MVL (1984): Gating of gap junction channels. Biophys JI 45:219-230. Steiner FR, Marshall L, Needleman D ( 1986): The properties of calmodulin at physiological temperature. Biopolymers 25:35 1-37 I . Takashima S ( 1989): “Electrical Properties of Biopolymers and Membranes.” Bristol, Philadelphia: Adam Hilger.



Thomas JR, Schrot J, Liboff AR (1986): Low-intensity magnetic fields alter operant behavior in rats. Bioelectromagnetics 7:349-357. Van Ainclsfort AMJ ( 1990): “An Analytic Algorithm for Solving Inhomogeneous Electromagnetic Boundary-Value Problems for a Set of Coaxial Circular Cylinders.” Eindhoven, Netherlands: James Clerk Maxwell Foundation (ISBN 90-73532-01-9). Wait JR (1985): “Electromagnetic Wave Theory.” New York: Harper and Row. Weast RC. Aqtle M J , Beyer WII (eds) (1986): “CRC Handbook of Chemistry and Physics.’. 67th edition. Boca Raton, Florida: CRC Press, Inc., pp B-219 to B-439. Werner HJ. Schulten K, Weller A (1978): Electron transfer and spin exchange contributing to the magnetic field dependence of the primary photochemical reaction of bacterial photosynthesis. Biochim Biophys Acta 502:255-268. Zahn M (1979): “Electromagnetic Field Theory.’‘ New York: John Wiley and Sons, p 156. Zhadin MN. Fesenko EE (1990): Ion cyclotron resonance in hiomolecules. Biomedical Science 1:245250. (Biomedical Science is published in English i n thc USSR).


All equations given below are for “quasi-static’’ fields: the dimensions of all physical structures are assumed to be very small compared with the wavelength of electromagnetic fields: time retardation effects are disregarded. Magnetic Flux Density

B Due to Linear Current Distribution

From Biot-Savart law:

where R = unit vector in direction from current element I dT to point where is evaluated; and R = distance from current element to point where B is evaluated. From vector potential A : -



Electric Field

Electrostatic potential due to electric charges

p = volume charge density C/m p = surface charge density C/m ’


Magnetic Field Scaling


V must satisfy Laplace’s equation:



Important Boundary Conditions Relations for the electric field components which are perpendicular to the boundary between electrically different media (Nos. 1 and 2). From Gauss’ law:

To satisfy conservation of electric charge:

= - j w p , for sinusoidal variation in time

Combining A-7 and A-8 gives S,E: -S2El = O


S=o+ j m

(A- 10)


Diffusion in Unbounded Fluid Media in the Presence of an Applied Magnetic Flux Density Bocos OA Nernst-Planck equation for current density [Zahn, 1979; Cussler, 19841 with electrochemical potential term omitted [Lakshminarayanaiah, 19841:

(A-11) where p = R + p , R = volume charge density independent of position and time, p = volume charge density which is a function of position and time; u = electrical mobility of ions (a typical value in free solution is 10-7m2/Vs[see Pethig, 1979]), k = Boltzman’s constant, T = absolute temperature, q = ionic charge. Conservation of charge equation: ~

V . -J = - -aP dt

(A- 12)

Combination of A- 11 and A- 12 gives in cylindrical coordinates, when p is independent of r, when B = Bo cos ot,and when E, is given by Equation 8:





P = T




(A-15) A-13 has the solution (A-16) which shows that the relaxation time for diffusion is r la: [See also Schwarz, 1962; Polk, 1986b] when r is either fixed or when the variable r > 1 cm (in those cases the d p 1h and dp / dr terms, which appear in the derivation of A- 13, become negligible).

Attractive Force Between Two Wires Carrying Currents in Same Direction (A-17)

p = magnetic permeability, d = distance between wires. Pressure on a Cylindrical Surface, Volume Force Density and Maxwell’s Stress Tensor Total force in volume that carries current density J and is subjected to B: (A-18) -

Defining Maxwell’s stress tensor and Phillips, 19551 by

for quasi-static fields [Johnson, 1965; Panofsky

it follows from the divergence theorem that (A-20) -


in A-20 is uniform over the entire surface and perpendicular to it, one obtains

(A-21) JJ

Magnetic Field Scaling


For a cylindrical surface, when j= ( oB r/2) sin ot and 14 follows from A-21.


= i B cos at,Equation

Orbital Precession Model of Zhadin and Fesenko

With reference to Figure 6: If the alternating magnetic field is in parallel with B, so that B I = 2B (cos Rt + a ) (where Q = 2 nf and a = arbitrary phase angle), the torque TI,is



T,, = M x B , = M [ - ~ s i n 8 c o s ~ 2 t + ~ s i n 8 s i n Q t ] Bclo, s ( Q t + a ) (A-22)

If the alternating magnetic field is perpendicular to


T I = M x El = M

Bsso that B, = 2B

[ - 2sin8sin Qr + jkos8] B ,

(cos Qt

cos( Qt + a )



If 8 is a random variable, the expected values !€(sin 8 ) and !€(cos 8 ) are !€(sine) =

-In o sinode 1 .

= 0.636


(A-25) For any value of a in A-22 the average magnitude of the torque over the duration ( l/fl.) of one precessional orbit, is found by integration: ( T , ,) = O.318MB0


( T , ) = 0.318MBl,


Likewise for a = +$ in A-23:

With Bs= 50 pT, so that f, = 19.2 Hz (lo-”)Jh-evolution. For the energy W = number of revolutions would be 4.18

Ca++,and for Bo = 100 pT, (T,,) = 3.82 = 1.6 (10 -*‘ ) J = 0.01 eV , the required requiring 6.9 years.

Dosimetry of extremely-low-frequency magnetic fields.

Extrapolation of quantitative measurements across biological systems requires knowledge of field-organism interaction mechanisms. In the absence of su...
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