Current density and forces for a current loop moving parallel over a thin conducting sheet

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1 INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 5 (4) EUROPEAN JOURNAL OF PHYSICS PII: S43-87(4) Current density and fores for a urrent loop moving parallel over a thin onduting sheet BSPalmer Laboratory for Physial Sienes, College Park, MD 74, USA bpalmer@lps.umd.edu Reeived Marh 4 Published 3 July 4 Online at staks.iop.org/ejp/5/655 doi:.88/43-87/5/5/8 Abstrat An analytial expression for the eddy urrent fores on a irular urrent loop moving with onstant veloity over a thin onduting sheet is derived in this paper. This alulation is based on using the boundary onditions aross the onduting sheet to solve for the total magneti vetor potential. The eddy urrents in the sheet and the fores on the sheet are first presented in the quasistati limit and the perfet ondutor limit. Finally, an analytial expression for both the drag and lift fore is derived.. Introdution There is a popular leture demonstration in undergraduate physis ourses that involves a metalli sheet swinging in a magneti field. In the frame of referene of the sheet the magneti field is hanging; this hanging magneti field indues urrents (alled eddy urrents) in the sheet. A Lorentz fore ats on the eddy urrents, ausing a drag fore (alled magneti drag) on the onduting sheet. This drag fore an be enhaned by inreasing the ondutivity of the metal or by inreasing the magneti field. For example, plaing the sheet in liquid nitrogen and repeating the demonstration after it has equilibrated to 77 K dramatially inreases the eddy urrent damping fore due to the inrease in ondutivity of the metalli sheet. In this paper the indued urrents and fores are analytially alulated for a irular d urrent loop moving with a onstant non-relativisti veloity in a parallel plane over a thin onduting sheet as shown in figure. To simplify the alulations in this paper we will assume that the thikness of the onduting sheet approahes the infinitesimal limit suh that the D ondutane of the sheet remains finite; i.e. σ = lim t (σ t) > where σ represents the bulk ondutivity of the metalli film and t the thikness of the film. While eddy urrents have theoretially been understood sine the late 8s (Maxwell 87), a omplete solution of the problem solved in this paper has not been found by the author. Our interest in 43-87/4/5655+$3. 4 IOP Publishing Ltd Printed in the UK 655

2 656 BSPalmer y Drag fore on loop x Lift fore on loop v Lift fore on film Conduting film Drag fore on film Figure. Pitorial setup for eddy urrent alulation. A irular urrent loop with a d urrent I flowing in the lokwise diretion is moving with a onstant veloity v ˆx over a thin onduting sheet. understanding the urrents and fores on the sheet is a result of our reent measurements of the eddy urrent drag fore using high-q mehanial osillators with a spatial resolution of µm(palmer et al ). A more typial appliation of the analysis of eddy urrent fores is for magneti levitation systems (MAGLEV). The magnitude of the eddy urrent drag fore an be estimated by using some simple arguments. For example, if the onduting sheet is stationary and the urrent loop is moving with a veloity v = v ˆx, as shown in figure, using a relativisti argument there will be an eletri field in the sheet s frame of referene given by E(r) = (v/) B loop (r) where B loop (r) is the magneti field from the urrent loop at point r. The eletri field in the sheet gives rise to a urrent density j(r) = σ E(r) whih using the above estimate for the eletri field has a y omponent given by j y (r) = σ (v/)(b loop (r) ẑ). The magneti field, from the urrent loop, exerts a Lorentz fore on the urrent density in the sheet. The differential fore at a point r on the sheet is given by df(r) = j(r) B loop (r) ds. The longitudinal fore, or drag fore, is F x = j y (B loop ẑ) ds and the lift fore is given by F z = [j x (B loop ŷ) j y (B loop ˆx)]ds where the integral is a surfae integral over the entire film. For example, the drag fore on the sheet using the urrent density from the estimation above is F x (r) = σ v (B loop (r) ẑ) ds. () Provided that the veloity of the urrent loop is suh that v /(πσ ), equation () is a good estimate of the fore, as will be shown in the next setion. This limit is just the quasi-stati limit for whih the effets of sreening urrents an be negleted. The estimate that produed equation () negleted onservation of urrent density and the sreening response of the ondutor. In the next setion the total magneti vetor potential is solved. The outline of this alulation is as follows. The stati magneti vetor potential of a irular urrent loop is the starting point to alulate the urrents in the onduting sheet and the fores on the sheet. A Galilean transformation will be performed on the stati magneti vetor potential to alulate the magneti vetor potential of the urrent loop in the sheet s frame of referene. Boundary onditions aross the sheet will be used to alulate the magneti vetor potential assoiated with the urrents formed in the sheet due to the hanging magneti field. After solving for the total magneti vetor potential, the urrent density in the sheet and the fores on the sheet will be investigated in two limits. Finally, in setions 3 and 4 an analytial expression for the drag fore and lift fore is analytially solved. In this paper, Gaussian units are used. See appendix A in Jakson (975) for the onversion to SI.

3 Fores on a urrent loop moving parallel over a thin onduting sheet 657. Calulation of A sheet The magneti vetor potential for a urrent loop of radius R, lokwise urrent I and with the entre of the loop loated at a height z = b and at the origin in the (x,y ) plane is given by the real part of A loop (r,t)= ˆx IR k J [kr]e ix sin(y ) exp[ k z b ]dd + ŷ IR i k J [kr]e ix os(y ) exp[ k z b ]dd () where k = + and J is the Bessel funtion of the first order (see appendix A for a derivation of A loop ). In the frame of referene of the onduting sheet, a Galilean transformation on equation () an be performed for v by replaing x = x vt (y = y and z = z) in equation (). The response of the onduting sheet will be alulated using a method originally devised by Maxwell (87) and more reently applied by Reitz (97). In this method, the ontinuous motion of the loop is broken up into an infinite number of instantaneous infinitesimal steps. The infinite number of steps are added up and a limit is taken to get the ontinuous motion of the loop. Before alulating the infinite series, first onsider one instantaneous step of the entre of the urrent loop from x = vδt to x = vδt at t =. For t <,x in equation () is replaed by x x + vδt and for t>,x x vδt. At t =, urrents are produed in the sheet to maintain the initial magneti field and hene the initial magneti vetor potential (Smythe 95). Therefore the magneti vetor potential assoiated with the response of the sheet, whih we denote as A sheet, in the plane of the sheet (z = ) and at t =, is the magneti vetor potential of the loop at x = vδt minus at x = vδt A sheet (x, y) = ˆx IR k eix J [kr](e ivδt e ivδt ) sin(y) e kb d d + ŷi IR k eix J [kr](e ivδt e ivδt ) os(y) e kb d d. (3) Boundary onditions determine how the urrents in the sheet evolve in time. From the disontinuity of the H field aross the sheet, the following boundary ondition for the magneti vetor potential an be derived (Smythe 95) A sheet z ɛ z= ɛ = 4πµσ d dt (A sheet + A loop ). (4) z= For the ase of a single step at t = the magneti vetor potential assoiated with the loop is not hanging for t. In this ase, for t>, equation (4) redues to A sheet z ɛ z= ɛ = 4πµσ d dt A sheet. (5) The urrents assoiated with A sheet are onfined within the sheet or the z = plane so that A sheet e k z, therefore d dt A sheet = kwa sheet (6) where w = /πµσ has dimensions of veloity. Solving this differential equation yields A sheet = A sheet (z =,t = ) e k z e kwt where A sheet (z =,t = ) is given by equation (3). In MKS units, w = /µ σ. We follow the same notation that Reitz (97) used in defining this parameter as w. Saslow (99) used the notation v for this parameter. For the rest of this paper we set µ =.

4 658 BSPalmer This implies that an instantaneous hange in the magneti field produes urrents in the sheet that deay with a deay rate inversely proportional to σ. As the ondutivity inreases, the urrents take a longer time to deay. Another way to interpret this result is that a hange in the urrent loop produes an image on the opposite side of the sheet that reedes away from the z = plane. This is how Maxwell (87) envisioned eddy urrents as disussed in Saslow s artile (99). The exat solution of A sheet an be derived by using the initial onditions at t =. For t>the urrents in the sheet deay so that A sheet (r) = ˆx IR k eix e kwt J [kr]i sin(vδt) sin(y) e k( z +b) d d + ŷi IR k eix e kwt J [kr]i sin(vδt) os(y) e k( z +b) d d. (7) Equation (7) onsiders only one disrete jump in the exiting magneti field. To alulate the eddy urrent distribution when the loop has moved ontinuously from x =to the origin (x = ), the sum of disrete steps from t =to t = is alulated followed by the limit of an infinite number of infinitesimal steps. Taking the terms from equation (7) that depend on x and t the nth step would be e ix[ e iv(n+/)δt e iv(n /)δt] e kw(nδt) = e ix e iv(nδt) isin(vδt/) e kw(nδt). (8) Adding up all of the steps and taking the limit δt yields lim δt n= e iv(nδt) sin(vδt/) e kw(nδt) = v = v e ivt e kwt dt kw + iv (v) + (kw). (9) Therefore A sheet at t = when the loop has moved ontinuously from x =to x = is A sheet (r) = ˆx IR k J [kr]e ix sin(y) e k( z +b) kw +iv iv d d (v) + (kw) + ŷi IR k J [kr]e ix os(y) e k( z +b) kw +iv iv (v) + (kw) d d. () For t,x in equation () is replaed by x x vt. The total magneti vetor potential is A total = A loop + A sheet where A loop is given by equation () in the frame of referene of the onduting sheet and A sheet is given by equation (). Sine the total magneti vetor potential is, in priniple, solved, the urrent density in the sheet and the total magneti field an be alulated... Quasi-stati limit The first limit that shall be investigated is when the ondutivity is small suh that v w, whih is the quasi-stati limit. In this limit, kw v and equation () beomes A sheet (r,t)= ˆxi IR k J [kr]e i(x vt) k( z +b) v sin(y) e d d kw ŷ IR k J [kr]e i(x vt) k( z +b) v os(y) e d d. () kw

5 Fores on a urrent loop moving parallel over a thin onduting sheet 659 A sheet is v/w smaller than A loop, therefore A total A loop. The shielding urrents from the onduting sheet in this limit deay instantly sine the ondutivity of the sheet is small, and hene ontribute negligibly to the urrents in the sheet. The total urrent density in the sheet is given by j σ d dt A loop z= and at t = is j(x, y) t= = ˆx IRσ v k J [kr]sin(x) sin(y) e kb d d ŷ IRσ v k J [kr] os(x) os(y) e kb d d. () The dimensionality of this integral an be redued by swithing the and integrals to polar omponents. Making the substitution = k os(θ) and = k sin(θ), j(x, y) t= = ˆx IRσ π v kj [kr]e kb os(θ) sin(θ) sin[xk os(θ)]sin[yk sin(θ)]dθ dk ŷ IRσ v π kj [kr]e kb os (θ) os[xk os(θ)] os[yk sin(θ)]dθ dk. (3) Most of the azimuthal integrals in this paper have this form. To assist the reader, the integrals of this nature relevant to this paper have been tabulated in appendix B. Using equations (B.4) and (B.3), equation (3) beomes j(x, y) t= = ˆx πirσ v xy ρ ŷ πirσ v kj [kr]j [kρ]e kb dk kj [kr] (J [kρ]+ y x ) J ρ [kρ] e kb dk. (4) Figure is a vetor plot of the urrent density (equation (4)) in the sheet in the limit v w for b = R/. The last things to alulate in this limit are the fores on the sheet. The three omponents of the magneti field from the loop in the plane of the sheet (z = ) and at t = are B loop ˆx t= = πir x ρ kj [kr]j [kρ]e kb dk (5) B loop ŷ t= = πir y ρ kj [kr]j [kρ]e kb dk (6) B loop ẑ t= = πir kj [kr]j [kρ]e kb dk. (7) The drag fore on the sheet in the quasi-stati limit using equations (7) and (4) an now be alulated. The ŷ omponent of the urrent density has two omponents; the first term has spatial polar symmetry whereas the seond term depends on x and y. When alulating the total fore by integrating over the entire sheet, this seond term goes to zero sine the z omponent of the magneti field has spatial polar symmetry. Investigation of the remaining integral shows that the total longitudinal fore on the sheet is F x = σ v π(b loop (ρ) ẑ) ρ dρ = v (πir) w kj [kr]e kb dk (8) where the losure relation for the Bessel funtion ( ρj [k ρ]j [kρ]dρ = k δ(k k ) ) has been used in the seond equality. This fore is half of the fore that was originally estimated

6 66 BSPalmer y x Figure. Eddy urrent density in the limit v w (quasi-stati limit). The irle denotes the loation of the urrent loop with a d lokwise urrent flowing in the loop. The arrow denotes the diretion of the veloity of the loop. The height of the loop was b = R/ for this alulation. in setion (equation ()) 3. The lift fore due to the symmetry of both the urrent density (equation (4)) and the magneti field (equations (5) and (6)) is zero in this limit. The question arises: why does the fore that has been alulated differ from the bak of the envelope alulation by a fator of two in the limit when the veloity is small? Investigation of this fator of two differene in the fore initially started out in the alulation of the eletri field, i.e. the eletri field (urrent density) used in equation () is a fator of two larger than the eletri field (urrent density) used in equation (8). When the magneti vetor potential of the urrent loop was transformed to the moving oordinate system, this transformation negleted the salar potential beause that term does not play a role in the urrent density. Let us investigate this term further. The magneti vetor potential forms a 4-vetor with the eletri salar potential. When the magneti vetor potential was transformed to a moving oordinate system, the orret transformation should have yielded an eletri salar potential as well as the magneti vetor potential. For v, the orret transformation of the magneti vetor potential to the moving oordinate system yields the following salar potential: ϕ(r) = 4IRv k J [kr] os[(x vt)]sin[y]e k z b d d. (9) The eletri field from the salar potential is given by E ϕ (r) = ˆx 4IRv k J [kr]sin[(x vt)]sin(y) e k z b d d ŷ 4IRv k J [kr] os[(x vt)] os(y) e k z b d d ẑ 4IRv k k J [kr] os[(x vt)]sin(y) e k z b d d. () 3 The urrents and the fore were alulated in a different manner by Salzman et al ().

7 Fores on a urrent loop moving parallel over a thin onduting sheet 66 This eletri field is similar in form and magnitude to the eletri field from the magneti vetor potential of the loop. There are two differenes between this eletri field and the form in equation (). The first differene is that the x-omponent in E ϕ has the opposite sign from equation () so that instead of the eletri field pointing outward, away from x = asshown in the vetor plot of the urrent density in figure, this eletri field points inward (towards x = ). The eletri field also points perpendiular to the sheet (i.e. E ϕ has a z-omponent) whih is the seond differene. The eletri field perpendiular to the sheet produes surfae harges on the sheet and hene an eletrostati field that anels this eletri field. Beause of this anellation, the salar potential does not produe any urrent that ontributes to the eddy urrent fores. Thus the drag fore in the quasi-stati limit is / of what is expeted from the bak of the envelope alulation (equation ())... Perfet ondutor The other limit that shall be investigated is the perfet ondutor limit. In this limit w (i.e. the urrents do not deay), v w, and the seond part of the integrand in equation () an be approximated as the following: ( ) kw +iv ikw iv (v) + (kw) (v) v (v) Substituting this result into equation () A sheet (r,t)= ˆx IR k J [kr]e i(x vt) sin(y) e k( z +b) [ i kw ]. () v [ i kw ] d d v ŷi IR [ k J [kr]e i(x vt) os(y) e k( z +b) i kw ] d d. v () Comparing equation () to the vetor potential for the moving urrent loop (equation ()) it is noted that the first term in the square brakets anels with the vetor potential assoiated with the moving urrent loop so that the total vetor potential in the plane of the sheet is A total (x, y) = ˆxi IRw v k J [kr]e i(x vt) sin(y) e kb d d ŷ IRw v k J [kr]e i(x vt) os(y) e kb d d. (3) The urrent density in the sheet at t = is j(x, y) t= = ˆx IR π k J [kr] os(x) sin(y) e kb d d + ŷ IR π k J [kr]sin(x) os(y) e kb d d. (4) Swithing the and integrals to polar oordinates and performing the azimuthal integral yields j(x, y) t= = IRy ρ ˆx kj [kρ]j [kr]e kb dk + IRx ρ ŷ kj [kρ]j [kr]e kb dk. (5) Figure 3 is a vetor plot of the urrent density in the limit that the ondutane approahes infinity for b = R/. In this limit the urrent density in the sheet ats as an image loop whih shields the onduting sheet from hanges in the magneti field.

8 66 BSPalmer y x Figure 3. Eddy urrent density in the limit that v w (perfet ondutor). The irle denotes the loation of the urrent loop with a d lokwise urrent flowing in the loop. The arrow denotes the diretion of the veloity of the loop and the height of the loop was b = R/ for this alulation. To demonstrate that the magneti field does not penetrate the onduting sheet in this limit, the z omponent of the magneti field between the sheet and the loop will be alulated next. The total magneti vetor potential between the sheet and the loop <z<bis given by A total (r) t= = IR ˆx k J [kr] os(x) sin(y) e kb sinh(kz) d d IR ŷ The z omponent of the total magneti field is given by B total (r) ẑ = IR = IR = 4πIR k J [kr]sin(x) os(y) e kb sinh(kz) d d. (6) J [kr] os(x) os(y) e kb sinh(kz) d d kj [kr]e kb sinh(kz) π os[kx os(θ)] os[ky sin(θ)]dθ dk kj [kr]j [kρ]e kb sinh(kz) dk. (7) As z, the z omponent of the magneti field goes to zero. This is the expeted result for a perfet ondutor. Sine the indued urrents in the sheet have the same symmetry as the urrent loop the drag fore is zero in this limit. The lift fore given by F z = [j x (B loop ŷ) j y (B loop ˆx)]ds, on the other hand, is not zero. Using equations (5), (5) and (6) and swithing the spatial oordinates in the integral over the entire sheet to ylindrial oordinates allows us to perform both the azimuthal integral (yielding a π) and radial integral (using the losure relation) as

9 Fores on a urrent loop moving parallel over a thin onduting sheet 663 shown here: σ F z = π (IR) 3. Drag fore π k J [k R]e k b dφ dρ dk dk = (πir) kj [kr]e kb ρj [k ρ]j [kρ] kj [kr]e kb dk. (8) In this setion an analytial expression for the drag fore (F x = j y (B loop ẑ) ds) is derived. The drag fore an be written as F x = F x (QS) + F x (R) where F x (QS) is the quasi-stati drag fore given by equation (8) and F x (R) is the drag fore due to the response of the onduting sheet to the hanging magneti field (i.e. the urrent density due to A sheet : equation ()), whih was negligibly small in the quasi-stati limit. To begin the alulation of F x (R),the azimuthal integral an be alulated using equation (B.) after swithing to spatial ylindrial oordinates (x = ρ os(φ) and y = ρ sin(φ)). Swithing the and integrals to polar oordinates allows us to integrate the azimuthal dependene: π 4 ( ) π dθ = k os 4 (θ) dθ = πk w + k w v os (θ) + w v + (w/v) 3. (9) v +(w/v) Using the losure relation for the Bessel funtion yields the following result for the drag fore due to the response of the ondutor: ( ) F x (R) = v ( w ) (w/v) 3 (πir) + kj w v +(w/v) [kr]e kb dk. (3) The total drag fore on the onduting sheet using equation (8) for the drag fore in the quasi-stati limit and equation (3)is F x = w ( ) w (πir) kj v v + w [kr]e kb dk. (3) 4. Lift fore Finally, an analytial expression for the lift fore (F z = [j x (B loop ŷ) j y (B loop ˆx)]ds) on the sheet is derived. To alulate j x and j y we take the time derivatives of equations () and () and use equations (5) and (6) for the x and y omponents of the magneti field. We start this alulation by swithing the spatial oordinates in the integral over all of spae to ylindrial oordinates, whih allows us to integrate the azimuthal (φ) integral using equation (B.). Swithing the and integrals to polar oordinates allows us to alulate the azimuthal (θ)integral: π j x (B loop ŷ)ds and π j y (B loop ˆx)ds ( os (θ) sin (θ) ( w os (θ) + (w/v) dθ = π + v ) ) w ( w ) + v v os 4 (θ) ( w ) os (θ) + (w/v) dθ = π (w/v) 3 + v + ( w v ) (3). (33)

10 664 BSPalmer..8 Lift fore.6 F/Fo.4. Drag fore v/w Figure 4. Drag (equation (3)) and lift fore (equation (36)) as a funtion of v/w. The fore (y axis) has been normalized to F = (πir) kj [kr]e kb dk. Using the losure relation for the Bessel funtion yields j x (B loop ŷ)ds = (πir) ( ( w ) ) kj w ( w ) [kr]e kb dk + + v v v and ( j y (B loop ˆx)ds = (πir) w ) kj (w/v) [kr]e kb dk 3 + v + ( w v Subtrating equation (35) from equation (34), the total lift fore on the sheet is ( ) w (πir) F z = v + w ) (34). (35) kj [kr]e kb dk. (36) Figure 4 shows the magnitude of both the drag and lift fore on the sheet as a funtion of v/w normalized to F = (πir) kj [kr]e kb dk. Aside from the normalization fator (F ), equations (3) and (36) have the same funtional dependene on v/w as the lift and drag fore alulated for a moving magneti monopole over a thin onduting sheet (Maxwell 954, Reitz 97 and Saslow 99). This agreement is reasonable sine the fields from the loop appear to be dipole in nature when far from the sheet and a magneti dipole ould be onsidered to onsist of two magneti monopoles. 5. Conlusions An analytial expression for the fores on a urrent loop moving with onstant veloity in the non-relativisti limit (v ) over a onduting sheet has been derived in this paper. The ratio of the drag fore (equation (3)) to the lift fore (equation (36)) is F drag = (w/v)f lift. This result agrees with Davis s result (97) in whih it was derived using the Poynting vetor and is a general result whih does not depend on field geometry.

11 Fores on a urrent loop moving parallel over a thin onduting sheet 665 Under most experimental onditions, the quasi-stati limit is the valid limit. For example, assume the veloity of the loop is v = 5 m s and the onduting sheet has a resistivity equal to µ m, whih is approximately the resistivity of opper at room temperature. For a thikness of mm, σ 9 6 m s making w 6 m s. Using these numbers the deviation from the quasi-stati limit is on the order of.% for the drag fore. Aknowledgments I am grateful to both David Griffiths (Reed College) and Wayne Saslow (Texas A&M University) for providing valuable suggestions and keen insight in understanding the fator of two differene between the simple relativisti argument and the quasi-stati limit. Both H D Drew (University of Maryland) and R S Dea (IUPUI) motivated this alulation and provided initial suggestions. Appendix A. Magneti vetor potential To alulate the magneti vetor potential for the urrent loop in Cartesian oordinates (equation ()) we start with the magneti vetor potential in ylindrial oordinates provided by Jakson (975, problem 5.4): A loop (ρ, z) = ˆϕ πir J [kr]j [kρ]e k z b dk. (A.) Transforming equation (A.) to Cartesian oordinates yields A loop (r) = ˆx πiry J [kr]j [kρ]e k z b dk ρ ŷ πirx J [kr]j [kρ]e k z b dk. (A.) ρ The Bessel funtion with the spatial dependene (e.g. (y/ρ)j [kρ] intheˆx term) an be onverted into an integral representation using equation (B.), A loop (r) = ˆx IR π sin(θ) J k [kr] os[kx os(θ)]sin[ky sin(θ)]e k z b dθ dk ŷ IR π os(θ) J k [kr]sin[kx sin(θ)] os[ky os(θ)]e k z b dθ dk. (A.3) Finally we an onvert the integrals into their Cartesian ounterparts using = k os(θ), = k sin(θ), and k = + : A loop (r) = ˆx IR k J [kr] os(x) sin(y) e k z b d d ŷ IR k J [kr]sin(x) os(y) e k z b d d. (A.4) Appendix B. Definite integrals The following integrals an be alulated from integral of Gradshteyn and Ryzhik (98): π [ os[δ os(θ)] os[γ sin(θ)]dθ = πj δ + γ ] (B.)

12 666 BSPalmer π sin(θ) os[δ os(θ)]sin[γ sin(θ)]dθ = πγ δ + γ J [ δ + γ ] (B.) π π os (θ) os[δ os(θ)] os[γ sin(θ)]dθ [ = πj δ + γ ] + π(γ δ ) [ J δ + γ δ + γ ] (B.3) os(θ) sin(θ) sin[δ os(θ)]sin[γ sin(θ)]dθ = πδγ δ + γ J [ δ + γ ]. (B.4) Referenes Davis L C 97 J. Appl. Phys Gradshteyn I S and Ryzhik I M 98 Table of Integrals, Series, and Produts 4th edn (San Diego, CA: Aademi) Jakson J D 975 Classial Eletrodynamis nd edn (New York: Wiley) Maxwell J C 87 Pro. R. So. London 6 Maxwell J C 954 A Treatise on Eletriity and Magnetism 3rd edn (New York: Dover) Palmer B S, Drew H D and Dea R S Rev. Si. Instrum Reitz J R 97 J. Appl. Phys Salzman P J, Burke J R and Lea S M Am. J. Phys Saslow W M 99 Am.J.Phys Smythe W R 95 Stati and Dynami Eletriity (New York: MGraw-Hill)