SCIENCE

Monte-Carlo calculation of electrically inducedhuman-body currents

J.H. Pickles, MA, PhD, FIMA

Indexing terms: Biomedical applications, Biological effects, Electromagnetic theory, Mathematical techniques

Abstract: A method is described for calculatingthe charges and currents induced in the humanbody by high-voltage alternating electric fields.This relates the surface charge induced on thebody to the potential in a reciprocal Laplaceproblem, which is then calculated by a Monte-Carlo random-walk technique. The method isapplied to an experimental geometry used tostudy the effect of electric fields on cardiac pace-makers. Induced charges and currents for severaldifferent body configurations are calculated. Cali-bration factors are also derived to correct themeasured currents for the effect of the walls of thetest area and the nonuniformity of the appliedelectric field. The solutions show good agreementwith the magnitude of the total induced chargeand its distribution over the body surface, as esti-mated in other experimental and computationalwork.

rf = termination of random walk i in Monte-Carlocalculation

r0 = position of point electrode carrying trialcharge

(r, 9, x) = spherical polar co-ordinate systems = standard deviation of Monte-Carlo estimate

V = high-voltage electrode(x, y, z) = rectangular co-ordinate systemYlm = spherical harmonice0 = permittivity of free spacea{r) = charge density on body surface4>{r) = Laplacian potential describing high-voltage

field £(*•)\l/{r) = Laplacian solution of reciprocal potential

problemil/A(r) = approximation to i//(r) derived from known

distributions\j/(r) = Monte-Carlo estimate of \j/{r)(o = power-system frequencyV = gradient operator

List of symbols

alm = coefficient in spherical harmonic expansionB = body surfaceB, = subarea of body surfaceE{r) = electric field pattern generated by high-

voltage electrodeEo = magnitude of uniform vertical fieldG = ground planeG(r, r') = Green's function for reciprocal Laplace

problemh = height of body BI = alternating current flow corresponding to

induced charge qN = number of random walks in Monte-Carlo cal-

culationnB, nt = number of random walks terminating on

body surface B, or on subsurface B,Q = trial charge on point electrode4B> <?c> 4 . = t o t a l charge induced on body B; charge

induced on body surfaces with current pathto earth via the chest; charge induced onbody surface B,

r = position vector

Paper 5568A (S9), first received 11th June 1985 and in revised form 25thFebruary 1986

The author is with the CEGB Technology Planning and Research Divi-sion, Central Electricity Research Laboratories, Kelvin Avenue,Leatherhead, Surrey KT22 7SE, United Kingdom

1 Introduction

This paper is concerned with the calculation of thecurrent induced in the human body when exposed to ahigh-voltage alternating electric field. Calculations ormeasurements of this current form the starting point for arange of investigations which seek to assess theenvironmental impact of high-voltage transmission lines[1-4].

The complex geometry of the human body can lead tosubstantial difficulty in the current calculation. Finite-difference methods [5], for example, may need a largenumber of grid points for an accurate solution. For thisreason, several approximate solutions have been derived,with the body represented by a hemisphere, hemi-spheroid, or cylinder [5-7]. This allows analytic calcu-lation of the electric field around the body and theinduced current. Moment method techniques [3] assumea body made up from a combination of thin cylindricalsections. A self-consistent solution is then obtained,which fixes the current flow down each section from thesum of the unperturbed and scattered electric fields.

The aim here is to describe a Monte-Carlo solutiontechnique which lends itself to more detailed geometricalmodelling. Some sample calculations are also presentedwhich relate to an experiment to assess the effect ofpower-frequency fields on cardiac pacemakers [8].

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 9, NOVEMBER 1987 705

2 Mathematical methods

2.1 The Quasi-static ApproximationAt power-system frequencies, the body tissues can beregarded as good electrical conductors, and the inter-action of the body with an applied alternating electro-magnetic field can be treated within a quasistaticapproximation. An earthed human body thus remains atapproximately zero potential, and the surface chargeinduced on the body is determined by a time-independentelectrostatic calculation. The induced body current canthen be derived as that current which keeps the surfacecharge matched to the instantaneous value of the alter-nating electric field. Reference 4 gives a more detailed dis-cussion of the assumptions underlying this physicalpicture.

For an explicit mathematical formulation, consider ahuman body B of height h standing on the ground G andexposed to an alternating electric field (see Fig. 1). The

trial charge Q at r_—r

The surface charge o{r) induced on B is determined bythe discontinuity in the normal component of electricfield, and, because the fields inside the body are extremelysmall,

o{r)= -son • E(r) (2)

where n is the outward normal to B at r. Integrationgives the total charge induced on B as

qB = | a{r') dS' (3)

and the total current flowing through the feet is

dqB

/„ = dt

or(4)

Similar arguments apply to subsections of the body B.For example, the total induced charge on the surface Bucomprising the head and neck, is

) dS' (5)

This leads to a total current passing through the neck

lx =jcoql (6)

body heighth

on body surface B

on ground plane G

Fig. 1trade

Boundary conditions on potential <f>(r) with idealised point elec-

field is produced, in practice, by the charge on the surfaceof a high-voltage electrode V, although, for the applica-tions considered here, it is sufficient to consider fieldsproduced by a single point charge Q a t r = r0. Withinthe quasistatic approximation, the time dependence ofthe field is expressed, in phasor notation, by a factor ejwt,while its variation in the airspace above B and G isdescribed by an electrostatic potential 4>(r). Thus

E(r)= -

where <$> satisfies Laplace's equation

The boundary conditions on <j> are then

<f> = 0 on the body surface B

0 = 0 on G

and, for r close to r0,

Q

(1)

4ne01 r — r01+ smoothly varying terms

2.2 Reciprocity relationshipThe direct calculation of induced body charges and cur-rents via eqns. 2, 3 and 4 involves an integration over thecomplicated surface B, and this may prove difficult inpractical applications. A transformation to an equivalent,but more convenient, form may be effected by the use ofGreen's theorem.

Consider the Laplace problem for a potential \l/(r)which satisfies boundary conditions:

for r on B= 1

= 0 for r on G

as I r\ -*• oo

(7)

where there is now no high-voltage electrode. The generalsolution can be expressed in terms of the Green's func-tion G(r, r') as an integral over the boundary surfaces:

1JB

= \n VG(r, r')JB

or because

\j/(r') = 1 on B

Mr)JB

• VG(r, r') dS' (8)

The contribution from the boundary surface G vanishesbecause of the boundary condition \j/ = 0 on G. Now theGreen's function G(r, r') satisfies the conditions

G(r, r') = 0 for r' on B or G

and

G(r, r')1

4n\r' -r\+ smoothly varying terms

when r' approaches r. Comparison with the boundaryconditions 1 shows that the Green's function for thisreciprocal problem is related to the electrostatic potential

706 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 9, NOVEMBER 1987

of the physical Laplace problem. Thus ary points r = rf where the walks terminate. Thus,

and

By virtue of eqns. 3, 8 and 9, therefore

(9)

(10)

The solution of the reciprocal Laplace problem at thesingle point r = r0, thus, gives the ratio of the inducedbody charge qB to the test charge Q in the physicalproblem.

Again similar arguments apply to subsections of thebody B. For example, calculations of the charge q1

induced on the surfaces Bx of the head and neck, asrequired for eqns. 5 and 6, depend on the use of a Lapla-cian reciprocal potential ^ ( r ) . This satisfies boundaryconditions:

(11)

Y lv) =

•Al(**) =

Y" lif) ~*

and gives

1

0

0

r on # !

r elsewhere on

as | r | -» oo

B, or r on G

(12)

Eqns. 10, with its analogies such as eqn. 12, forms thebasis for the calculations of induced body charges andcurrents presented in Section 3. It can be regarded as aparticular form of Green's reciprocity relationship [9],which applies when the high-voltage electrode V is verysmall and becomes simply a vehicle for a point charge.This modelling of V is the main assumption involved inthe use of eqn. 10, but there is one other approximationwhich should also be noted. This applies when a com-parison is made between body charges qB induced in dif-ferent body configurations. These are calculated viaeqn. 10 as fractions of a test charge Q, Q being taken asconstant over the range of cases considered. In practice,however, the electrode V is maintained at the samepotential for each case, rather than given the samecharge. Variations in the body charge qB will inducesmall variations in the charge on V. These are of secondorder in the ratio [qB/Q]. Eqn- 10 can therefore be usedas a direct measure of the induced body charge onlywhen [qB/Q~\ is small.

2.3 Monte-Carlo solution techniqueTo solve for the potential ^(r) in the reciprocal Laplaceproblem, it is convenient to use a Monte-Carlo solutiontechnique. This is well suited to problems with complexboundary geometries, and is economical to use when, ashere, a solution ij/(ro) is needed at only one point r — r0.Full details are available elsewhere [10, 11], but a briefoutline of the method is given below.

The distinguishing feature of the method is the use ofrandom walks. The unknown potential \l/(r0) is estimatedas the average, taken over a set of random walks i = 1,. . . , N each starting at r0, of the potentials at the bound-

(13)

The same random walks can be used to estimate poten-tials tj/(r) at points in the neighbourhood of r0 as aweighted average:

lv ;=i(14)

The weighting factors w(r, r0, i) here depend on r and r0

and on the path of the random walk i. The random walksused in these estimates are floating random walks, withvariable steps in freely chosen directions, rather thanwalks on a fixed grid. As shown (in 2-dimensional form)in Fig. 2, the walks move at each step in a randomly sel-ected direction with a step length equal to the distance to

starting point _r

777Y/777///7/////,

Fig. 2 Floating random walks starting at r0

the nearest point on the boundary. They need relativelyfew steps to approach very close to the boundary, and soreduce the computational effort needed.

With any finite set of random walks, the Monte-Carloestimates, eqns. 13 and 14 are subject to a statistical sam-pling error. The estimate of eqn. 13, for example, hasstandard deviation s, where the variance s2 is given bythe standard formula

^ u" w-l.t'!In any given application, the sampling error may bereduced, either by taking more random walks, or by con-structing a known Laplacian potential \j/A(r) which willapproximately match the boundary conditions for ij/(r).The random-walk process can then, in effect, be used tosolve for the small difference ij/(r) — \j/A(r), giving

Wo) = (15)

with a corresponding estimate for the variance.The computer program MOCFIE used for the Monte-

Carlo calculation takes a representation of the boundary

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 9, NOVEMBER 1987 707

surface as a combination of certain elementary geometri-cal shapes: spheres, cylinders, boxes etc. These are juxta-posed or superposed as required.

The ground surface G in Fig. 1 was represented by aninfinite plane, while the body surface B was made upfrom a total of nine separate elementary shapes (threeround-ended cylinders, four cones, and two round-edgedboxes) as shown in Fig. 3.

Fig. 3 Representation of experimental subject in terms of elementarygeometrical shapesBody dimensions are given in millimetres

In the calculations which follow, the computerprogram was used in two ways. Calculations of the totalinduced charge on the body, or of the charge on somespecified subsection B{, were based on eqn. 15, or, insome cases, the generalisation of eqn. 15 which usesweighting functions as in eqn. 14. Here, the accuracy ofthe Monte-Carlo calculation was enhanced by the use ofan approximate solution ij/A(r). The required \j/A(r) wasconstructed by combining the Laplacian potential dis-tributions of appropriately placed point, ring or linecharges. A total of up to ten charges were used for thispurpose.

The second type of calculation aimed at determining,simultaneously, the proportions of the total charge qB onthe various different subsections of the body. Heresimpler, although less precise, results are available on thebasis of eqn. 13, without the use of an approximate solu-tion if/A(r). Note first that the particular form of theboundary conditions 7, for if/(r), allow eqn. 13 to beexpressed in the alternative form

where nB is the number of random walks which terminateon the body surface B. Combining with eqn. 10 thengives

Q(16)

The same principle applies to the individual subsurfacesB, which together make up B. For example, the chargeinduced on the surface Bl is fixed by the reciprocal

potential tj/^r) which satisfies the boundary conditions11, and is in turn estimated via eqn. 12 as

N(17)

«! is here the number of random walks terminating onthe surface Bv In this way, the distribution of totalcharge

over the different surfaces Blt B2, B3i ... of the body canbe modelled (within the inherent Monte-Carlo samplingerror) by the distribution of random walks

terminating on B.

2.4 The case of uniform applied fieldMuch computational and experimental work is con-cerned with a uniform applied field, with a potentialsatisfying the boundary condition

V0(r)~(O, 0, £ 0 ) f o r | f | > / j (18)

It is assumed here that the z-axis is vertical and that theground G lies in the plane z = 0. This case can be model-led within the approach of Section 2.2 and a point chargeat r0 = (0, 0, z0), by considering the limit of large z0. Acharge

Q = 2ne0 Eo z% (19)

at r0 (together with its image in the ground plane G) thensatisfies condition 18 over a sufficiently large domain,h 4 \r\ <̂ z0. Eqn. 19 can be combined with eqn. 10 toexpress the induced body charge qB in terms of theapplied field Eo. This gives

qB = 2n£0 EO lim \ZQ-+ 00

(20)

The Monte-Carlo procedure for calculating <A(**0)becomes less accurate for large z0, when if/(r0) is verysmall. Direct numerical estimation of eqn. 20 is thereforedifficult, and it is better to use an extrapolation pro-cedure based on a multipole expression of \j/(r) [9]. Thisis an expansion in powers of l/ |r | which is valid for| r | > h. In spherical polar co-ordinates (r, 9, x), with0 = 0 along the line of the z-axis:

= l alm YJLO,lm

(21)

where the Ylm are spherical harmonics. For the particularcase of r = r0 on the z-axis, the expression 21 reduces to

(22)

for constants a, / ? , . . . . Eqn. 22 contains no constant termbecause of the boundary condition 7 on \j/{r) at infinity,while the terms in odd powers of (h/z0) vanish because ofthe symmetry constraint on the coefficients alm in eqn. 21,imposed by the condition i// = 0 for 0 = n/2 on theground plane G. The expansion 22 shows that, for largez0, (zo/h)2\j/(0, 0, zo) varies approximately linearly with(h/z0)

2. Extrapolation to the limit (h/z0) -* 0, or z0 -> oo,then gives the induced body charge qB via eqn. 20 asrequired.

708 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 9, NOVEMBER 1987

3 Results

3.1 Ex peri men tal workThe experimental work, towards which these calculationsare aimed, was a study of the effect of electromagneticfields on cardiac pacemakers. This has been reported indetail elsewhere [8]. Each experimental subject stood inan alternating field produced by a high-voltage electrodeformed from two toroids, each of ring diameter 1.28 mand cross-sectional diameter 0.33 m. These were sus-pended close together, with their common axis horizontaland their centres 6.14 m above the floor of a large testbay, so that in the neighbourhood of a subject standingon the ground below the field pattern produced approx-imated to that of a single point charge. The unperturbedvertical field intensity, measured (in the absence of asubject) at a point 1.8 m above the ground and directlybelow the electrode, was allowed to vary in the range1-20 kV/m. The performance of the pacemakers wasmonitored while the subject adopted different body posi-tions. The total body current IB was also measured andused as an index of the strength of the interference to thepacemaker from the imposed electric field.

32 Geometry of test facilityWhile the experimental conditions approximated to thegeometric ideal where the subject stands on an infiniteground plane, the walls of the test bay and the apparatuspresent in it do create perturbations in the field pattern.Furthermore, the quasi-point-charge field produced bythe high-voltage electrode does not precisely match theideal uniform applied field condition which is morenearly approached in exposures under transmission linesat heights of 10 m or more. The first aim of the calcu-lations, therefore, was to 'normalise' the experimentalresults to the standard infinite plane-uniform field case.

As a starting point consider a 'baseline' calculation ofthe total induced body charge qB for a subject standingerect on an infinite ground plane. The high-voltage elec-trode was here, as described in Section 2.2, representedby a point charge Q, positioned 6.14 m above the grounddirectly above the subject's body B. The Monte-Carlocalculations for the corresponding reciprocal potentialil/(r0) used 5000 random walks, in conjunction with anapproximate solution \ffjr) as described in Section 2.3.This leads to the result

7? = (2.35 ± 0.02) x 10 - 2 (23)

where the error figure quoted is the RMS statistical sam-pling error of the Monte-Carlo calculation. The CPUtime needed on an IBM 370 computer was 25 seconds.

To assess the effect of the walls of the test bay, thiscalculation was repeated with an additional verticalearthed plane. This was placed 6 m behind the subject,approximating the position of the nearest bay wall. Cal-culations using a total of 4000 random walks gave theresult

^ = (2.17 ± 0.02) x 10" 2 (24)

The presence of the wall thus reduces the induced bodycharge and current by about 8%. After a simple imagecharge calculation to relate the trial charge Q to the ver-tical fields £z(0, 0, h) measured at head height, eqn. 24predicts a total body current of 11.8 /iA/kV/m. This com-pares well with regression analysis of the experimental

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 9, NOVEMBER 1987

data which, for a subject of height 1.8 m, gives a currentofll.7//A/kV/m.

To assess the effect of the nonuniform point-chargefield in the baseline geometry, further calculations weredone with the trial charge placed at heights of 4.5, 5.5,7.0, 10.0, 14.0 and 20.0 m. The uniform field result thenfollows by extrapolation, as described in Section 2.4. Theleast-squares plot of the calculated (zo//i)

2^(0, 0, z0)values against {h/z0)

2 intercepts the axis (h/z0)2 = 0 at

(zo/h)2ij/(0, 0, z0) = 0.260 ± 0.001

and so, from eqn. 20,

(25)

Comparison with eqns. 19 and 23 shows that, for fields ofthe same ground-level intensity, the baseline geometrywith the trial charge at a height of 6.14 m producesinduced charges and currents 5% greater than in theuniform-field case.

This result is in close agreement with that derivedfrom an argument by Deno and Zaffanella [12]. Theysuggest that the total induced body current for a personof height 1.8 m is determined broadly by the inductionpotential at a height of about 1.5 m, which, in turn, canbe estimated by the electric field at about half this height,or 0.75 m. Scaling in this way would give a differencebetween the cases z0 = 6.14 m and z0 -• oo of 4.6%. Theagreement between Deno and Zaffanella's scaling and theMonte-Carlo results persists over a wide range of modelelectrode heights z0, as shown in Fig. 4.

o§2 1.10c ^<u _en 0)

*> 1.05

>, c -

o o 3

0.05 0.10(h/z0) , dimensionless unitsi i i i

0.15

oo201512 10 9 8 7 6 5 4.5electrode height zo, m

Fig. 4 Scaling relationship for changes in induced body charge withvarying electrode height z0

The auxiliary scale for z0 is drawn for a subject of height h = 1.8 mleast squares fit to Monte Carlo runspredictions according to Deno and Zaffanella [12]

• experimental results

The uniform field limit of eqn. 25 can also be checkeddirectly against other computational and experimentalwork. Di Placido et al. [13] used the method of spheres[14] to calculate the induced body current for an axisym-metric model of the human body. For the body heighth = 1.8 m assumed here, their result is equivalent to

qB = UUoEoh2

Similarly, Deno [15] made experimental measurementson a realistic manikin figure. In conditions approx-imating to a uniform field exposure, these gave

qB= 1.62e0 Eoh2

Both of these results show good agreement with the esti-mate qB = 1.63e0 Eo h

2 given in eqn. 25.

709

3.3 Effect of body configurationThe same calculation procedure can be applied to findthe changes in the total induced charge qB which occurwith the body in different positions. Three additionalcases were considered, as shown in Fig. 5: one arm hori-

Table 2: Distribution of induced charge over body surfaces

a b c

Fig. 5 Body configurations for induced charge calculations

zontal, one arm raised vertically and both arms raisedvertically. The trial charge was maintained at a height of6.14 m. In each Monte-Carlo calculation, 2000 randomwalks were used in conjunction with a suitable initialapproximation \j/A{r). The results together with their esti-mated RMS sampling errors are shown in Table 1. As

Table 1 : Total induced charges for different body configu-rations

Body configuration Ratio qB/Q

Arms at side (Fig. 3)One arm horizontal(Fig. 5a)One arm vertical(Fig. 5b)Both arms vertical(Fig. 5c)

(2.35 ±0.02) * 1 O - 2

(2.74 ±0.02) x 10" 2

(2.87 ±0.03) x 10 - 2

(3.26 ±0.02) * 1 O - 2

would be expected, there is a steady increase in the ratio(qB/Q): the value (qB/Q) ^ 2.35 x 10 ~2 given in eqn. 23for a subject standing normally, as in Fig. 3, reaches(3.26 ± 0.02) x 10~2 for the same subject with both armsraised vertically.

3.4 Charge and current distributionsAs shown in Section 2.3, the Monte-Carlo procedure alsoprovides an estimate of the distribution of the totalinduced charge qB over the body surface. For thispurpose, a series of calculations totalling 14500 randomwalks was done for the standard body configuration ofFig. 3. The height of the trial charge Q was again fixed at6.14 m. In all, 360 random walks terminated on the bodysurface B gave

^ = (2.48 ±0.13) x 10 - 2 (26)

from eqn. 16. This is consistent, to within the statisticalsampling uncertainty, with the estimate 23 of Section 3.2.The distribution of the walks over the separate surfaces(head and neck, arms, legs and shoulders, and trunk) ofthe body can be matched to the distribution of qB as ineqn. 17. The results are subject to an RMS samplingerror in the range 0.02qB — 0.03qB, or between ± 0.02and ± 0.03 in the proportions given in Table 2. Neverthe-less, there is as shown good agreement with empiricaldata given by Deno [15] for the uniform field case.

710

Body surface

Head and neckEach armShoulders andtrunkEach leg

Proportion of total charge ofsurface as estimated by(a) Monte Carlo calculation

0.290.14

0.330.05

(6) Deno [15]

0.300.14

0.320.05

An important physical parameter in the experimentalwork on pacemakers in the total vertical current Icpassing through the chest of the subject at the level of theheart. This can be estimated using an equation analogousto eqn. 6 for the charge qc induced on the head, neck,arms and shoulders. One route to the calculation of qc isto combine the charges on the individual body surfaces,in the manner of Table 2 and eqn. 17, giving

= (1.60 ±0.11) x 10 - 2

It is, however, again more accurate to compute the totalqc via eqn. 15, using an appropriately defined reciprocalpotential \jfc{r) and initial approximation. This gives

^ = (1.57 ± 0.05) x 10~2 (27)

from a calculation in which 3000 random walks wereused. These results indicate that the chest current Ic isabout two thirds of the total current IB passing throughthe feet, as determined by eqns. 23 or 26.

Calculations similar to eqn. 27 are also possible fordifferent body configurations. For a subject with botharms raised vertically, as in Fig. 5c,

- ~ (2.18 ±0.06) x 10"2

In this configuration, there is an increase, both in thecharge induced on the arms, and also (because of areduction in the shielding effect of the arms) in the chargeinduced on the trunk, and the chest current Ic remainsabout two thirds of the total body current IB, as derivedfrom Table 1. The broadly constant relationship betweenchest current and total current thus supports the use oftotal body current IB as a measure of interference to thepacemaker from the applied field [8]. A more completeanalysis, however, would have to take into account thepossibily nonuniform distribution of the current Icthrough the cross-section of the chest.

4 Discussion and conclusion

The Monte-Carlo calculation procedure described hereoffers a convenient and simple method for estimatinginduced body charges and currents. It has been applied inSection 3 to calibrate some of the experimental measure-ments made [8] in a study of the performance of cardiacpacemakers. Estimates of the electrode proximity andwall effects in the experimental geometry are derived, andthe changes in total induced current, for varying bodyposture, are calculated. Increases of up to 50%, as com-pared with a normal standing posture, are found. Thedistribution of induced charge over the surface of thebody is also estimated. The vertical current through thechest, which controls the response of the pacemaker tothe applied field, is found to be about two thirds of thetotal current passing through the feet. For the ideal

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 9, NOVEMBER 1987

infinite-plane uniform-field geometry, the Monte-Carloresults are in good agreement with other experimentaland computational work [13, 15].

The particular strength of the Monte-Carlo method inthis application is its ability to allow a detailed represen-tation of the shape and posture of the human body.Results accurate to within about 1% in total inducedcurrent can readily be obtained for a range of differentgeometries, without extensive computation or elaboratepreparation. Similar calculations would also be possible,for example, with animal experiments to supplementcapacitance estimates made with physical models [16].

One of the limitations of the method, as presentedhere, is the restriction to geometries where the high-voltage electrode can be represented as a point charge.Extensions to more general cases, and in particular toline or plane electrodes, are however also possible. Thesewould require an integration of the reciprocal potentialof Section 2 over the surface of the electrode in question,rather than its evaluation at a single point.

5 Acknowledgment

The work described in this paper was done at the CentralElectricity Research Laboratories and is published bypermission of the Central Electricity Generating Board.

6 References

1 BRACKEN, T.D.: 'Field measurement and calculations of electro-static effects of overhead transmission lines', IEEE Trans., 1976,PAS-95, pp. 494-504

2 SHEPPARD, A.R., and EISENBUD, M.: 'Biologic effects of electricand magnetic fields of extremely low frequency' (New York Uni-versity Press, 1977)

3 SPIEGEL, R.J.: 'High voltage electric field coupling to humansusing moment method techniques', IEEE Trans., 1977, BME-24, pp.466-472

4 KAUNE, W.T., and GILLIS, M.F.: 'General properties of the inter-action between animals and ELF electric fields', Bioelectromagnetics,1981, 2, pp. 1-11

5 BARNES, H.C., McELROY, A.J., and CHARKOW, J.H.: 'Rationalanalysis of electric fields in live line working', IEEE Trans., 1967,PAS-86, pp. 482^192

6 SPIEGEL, R.J.: 'ELF coupling to spherical models of man andanimals', ibid., 1976, BME-23, pp. 387-391

7 REILLY, J.P.: 'Discussion contribution to paper by Deno', ibid.,1977, PAS-96, p. 1526

8 BUTROUS, G.S., MALE, J.C., WEBBER, R. S., BARTON, D. G.,MELDRUM, S.J., BONNELL, J.A., and CAMM, A.J.: 'The effectof power frequency high intensity electric fields on cardiac pace-makers', Pace, 1983,6, pp. 1282-1292

9 PANOFSKY, W.H., and PHILLIPS, M.: 'Classical electricity andmagnetism' (Addison-Wesley, 1962)

10 PICKLES, J.H.: 'Monte Carlo field calculations', Proc. IEE, 1977,124, pp.1271-1275

11 BEASLEY, M.D.R., PICKLES, J.H., D'AMICO, G., BERETTA, L.,FANELLI, M., GIUSEPPETTI, G., DI MONACO, A., GALLET,G., GREGOIRE, J.-P. and MORIN, M.: 'Comparison of threemethods for computing electric fields', Proc. IEE, 1979, 126, pp.126-134

12 DENO, D.W., and ZAFFANELLA, L.E.: 'Discussion contributionto paper by DiPlacido, Shih and Ware', IEEE Trans., 1978, PAS-97,p. 2175

13 DI PLACIDO, J., SHIH, C.H., and WARE, B.J.: 'Analysis of theproximity effect in electric field measurements', ibid., 1978, PAS-97,pp. 2167-2175

14 TRANEN, J.D., and WILSON, G.L.: 'Electrostatically inducedvoltages and currents on conducting objects under EHV transmis-sion lines', ibid., 1971, PAS-90, pp. 768-775

15 DENO, D.W.: 'Currents induced in the human body by highvoltage transmission line electric fields — measurement and calcu-lation of distribution and dose', ibid., 1977, PAS-96, pp. 1517-1527

16 KAUNE, W.T., and PHILLIPS, R.D.: 'Comparison of the couplingof grounded humans, swine and rats to vertical 60 Hz electric fields',Biolectromagnetics, 1980,1, pp. 117-129

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