E.C. Caparelli and D. Tomasi 85 the curl can be carried out, and the Eq. expressed as B(r) = iμ 8ß 3 (4) can be j(k) k k e ik r d 3 k: (6) If the curr

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1 84 Revista Brasileira de Ensino de F sica, vol. 3, no. 3, Setembro, An Analytical Calculation of the Magnetic Field Using the Biot Savart Law Um Cálculo Anal tico do Campo Magnético Usando a Lei de Biot Savart E.C. Caparelli and D. Tomasi Escuela de Ciencia y Tecnologia, Universidad Nacional de Gral. San Martin, Alem 39, 65, San Andres, Buenos Aires, Argentina Recebido em de Junho. Aceito em 6 de Agosto. This work presents an analytical method to calculate the magnetic eld at any point of the space, by solving the Biot Savart equation in the reciprocal space. This is applied to express the magnetic eld due to a circular current distributions as a convergent series. The comparison between the proposed method with the standard numerical integration of the Biot Savart law has shown a good agreement. Neste trabalho apresentamos um método anal tico para calcular o campo magnético em qualquer ponto do espaοco, resolvendo a equaοc~ao de Biot Savart no espaοco rec proco. O mesmo é aplicado para expressar em termos de uma série convergente o campo magético gerado por uma distribuiοc~ao de corrente circular. A comparaοc~ao entre o método proposto e a integraοc~ao numérica usual da lei de Biot Savart mostra uma satisfatória concord^ancia. I Introduction II The method The Biot Savart (BS) law is a well known equation used to calculate the magnetic eld vector at any point of the space for a given current distribution. Although it has extensive application in many branches of physics and engineering, analytical solutions of this integral equation are available only for some current distributions, which have specials symmetry conditions. Usually numerical methods are used to calculate magnetic elds, for example, to design coils and magnets that produce high homogeneous magnetic elds for Magnetic Resonance Imaging(MRI) [], to optimie toroidal magnetic systems [] or to evaluate the stress in fusion devices [3]. However, when the calculation of the magnetic eld is performed over a large set of points, the increase in the computing time requires the use of semi-analytic approaches [4]. In this work we are showing that the BS law can be solved in the reciprocal space for a planar current distribution. As an example, we derived analytically an expression to calculate the magnetic eld at any point of the space for a circular turn of wire carrying a current. The magnetic eld vector B at a given point in the space, r, is related to the current distribution j(r ) through the Biot Savart law [5] B(r) = μ 4ß j(r ) jr r j jr r j 3 d 3 r ; () where μ is the permeability of the free space and the integral is performed over the whole space. This law can also be expressed as a function of the curl of the vector potential, A(r), as B(r) = μ 4ß r j(r ) jr r j d3 r = r A(r); () that, using the Green function expansion [6] G(r; r )= can be written as B(r) = μ 8ß 3 r jr r = e ik (r r ) d 3 k (3) j ß k j(r eik (r r ) ) d 3 kd 3 r : (4) k Now, considering the Fourier transform of j(r ) j(k) = j(r )e ik r d 3 r (5)

2 E.C. Caparelli and D. Tomasi 85 the curl can be carried out, and the Eq. expressed as B(r) = iμ 8ß 3 (4) can be j(k) k k e ik r d 3 k: (6) If the current distribution is in the (x; y; ) plane, the Fourier transform of this distribution can be written as j(k) =j x (k x ;k y )^i + jy (k x ;k y )^j (7) and the continuity equation, expressed in terms of the Fourier components, is given by [7] k x j x (k x ;k y )+k y j y (k x ;k y )=: (8) Then the magnetic eld components due to this current distribution can be expressed, using Eq. (6), as where B ff (r) = B (r) = k ff k f(k; r) d 3 k; ff = x; y (9) (k x + k y)f(k; r) d 3 k; () f(k; r) = iμ j x (k x ;k y ) 8ß 3 k y e ik r k : () Therefore, upon integrating over k,we nally have B (r) = e i(kxx+kyy) k y c j x (k x ;k y )e jj p k x +k y dk x dk y ; = x; y; () d which are analytical expressions for the magnetic eld components produced by a planar current distribution. III Application Now we will consider a current distribution dened by a single loop of radius R, which carries a current I, asit is illustrated by Fig.. In this case the Fourier transform of the complex current density can be written as [7] j x (k x ;k y )+ij y (k x ;k y )=ßIR e if J (Rq); (3) where k x + ik y = qe q if is a complex number with modulus q = kx + ky and J (Rq) is the Bessel function of order. Using the Eq. (8), the expression (3) can be reduced to j x (k x ;k y )=ißi k y q RJ (qr): (4) The magnetic eld components are then obtained substituting this current distribution in Eq. () Figure : The circular current distribution diagram and the cylindric reference frame. B (r) = μ IR 4ß ß J (qr)e where the angular integral can be written as the convergent series [8] ß e iq(x cos(f)+y sin(f)) df =ß e iq(x cos f+y sin f) df dq; (5) ( ) k (qr)k ( k k!) ; r = x + y : (6)

3 86 Revista Brasileira de Ensino de F sica, vol. 3, no. 3, Setembro, Because of the axial symmetry of this problem we will use the cylindrical representation of the magnetic eld vector. The radial component B r (r) =B x(r)+b y(r) and the axial component B (r) of the magnetic eld can be obtained from Eq. (5) and can be written as the series expansion B fl (r) = The integral over q is related to the hypergeometric function, F as [8] which is dened as [8] ( ) k+ r k q k e jjq J ( k k!) (qr) dq ; fl = r;: (7) q k e jjq J (qr) dq = R jj(k+) [(k + )] F k +;k+ 3 R ; ; ; (8) F (a; b; c; w) = (d) n = (d + n) (d) n= (a) n (b) n (c) n w n n! (9) for d = a; b; c: () Finally the magnetic eld components can be written as the series B r (r) = μ IR k+ krk ( ) ( k k!) (k+) [(k +)] F k +;k+ 3 B (r) = μ IR k (k +)rk ( ) ( k k!) jj (k+3) [(k + )] F k + 3 ; ; R ; () ;k+; ; R : () In table, we show some expressions of the function F : d k = k = k = Table : Values of Hypergeometric function F F k +;k+ 3 R ; ; R F k + 3 R ;k+; ; = + 5= + 7= = 7= = Note that, in the Ref. [5], the magnetic eld for a single loop is calculated analytically using the vector potential, whose solutions are valid only near the axis, near the center of the loop and far from the loop, but here we got analytical expressions that are valid at any region. Analysis To study the quality of the above result as a function of the number of terms N, considered in the Eqs. () and (), we calculated the magnetic eld generated by a current ofi =Aflowing in a loop with R =min two ways: on one hand we used the MATHEMATICA 3. software to carry out the operations involved in MATHEMATICA is a registered trademark of Wolfram Research, Inc. Eqs. () and (), and on the other hand we wrote a double precision FORTRAN code performing the integral in Eq. (). In order to maximie the accuracy of the calculation accomplished by the numerical algorithm, we divided the wire in 89 elements. Figure compares the component of the magnetic eld obtained from Eq. (), for different values of N, with that achieved by the numerical integration of Eq. () (solid line), along the radial direction at =. As can be noticed the increase in the number of terms in the series expansion () results in an improvement of the matching between the analytical solution and the numerical calculation of the BS law. Whereas for r < :8R both calculations give similar results, for r

4 E.C. Caparelli and D. Tomasi 87 approaching R a mismatch between the two methods is observed. This mismatch corresponds to the cut off in the series expansion (), while for r>rthis behavior is due to the non cenvergence of the series (). The difference between the analytical method, B (An) fl (r;), and the standard BS method, B (BS) fl (r;), can be analyed using the relative difference between the magnetic elds (in part per million - ppm) obtained from both methods, B fl (r;)= 6 B(An) fl B (BS) fl (r;) (r;) : (3) Figure 3 shows B (r; = ) as a function of the radial position in the central transverse plane. While for N = 5 both methods differ less than ppm at r» :4R, forn =thematching region is extended to r» :6R, showing the importance of the number of terms in the B (r;) series. Figure 3: B(r; = ) as a function of the radial position. Figure 4: Contour maps of Br;(r;). (A) Radial component; (B) Axial component. Scale: logarithmic. Figure : B(r; = ) calculated by the standard Biot Savart method and by the proposed analytical method for different values of N, as function of the radial position. The contour plots in Fig. 4 are R R logarithmic maps of the relative differences B (r;) and B r (r;), respectively, in the (r;) plane. In these

5 88 Revista Brasileira de Ensino de F sica, vol. 3, no. 3, Setembro, gures the black-white transitions edges dene contours where the magnetic eld, calculated with the BS method and with the proposed method using N =, present a constant difference according to k = B fl, with k = ; ; 5. Additionally a logarithmic grey scale was introduced in these gures to increase resolution, and the wire position is marked by solid circles. The extended white regions at the center of both gures, which represent differences in magnetic eld that are less than ppm, demonstrates that the vality region of this method, at =, is a disk of radius R d» :6R but for 6= we have diskswith radius that increase with. IV Conclusion In this work we proposed an analytical method to calculate the magnetic eld from the BS law. This was applied to get expressions for calculating the magnetic eld components due to a circular current distributions. This components can be expressed as a series expansion that converges over the extended region. The accuracy of the method inside the convergence region depends on the number of terms considered in the series expansion. Acknowledgement This research was partially supported by FAPESP project 96/ References [] S. Croier and D.M. Doddrell, J.Mag.Reson. 7, 33 (997). [] E. Thomas, Jr., G.E. Sasser, S.F. Knowlton, et al. Comp.Phys.Comm., 3 (997). [3] De-man Wang, Guang-mei Wei, Hui-cai Xie, et al., IEEE Trans. on Magn. 3, 8 (995). [4] D. Tomasi, E.C. Caparelli, H. Panepucci and B. Foerster, J.Mag.Reson. 4, 35 (999). [5] J.D. Jackson, Classical Electrodynamics, (Wiley, New York, 975). [6] K.Yoda, J. Appl Phys. 67, 4349 (99). [7] E.C. Caparelli, D. Tomasi and H. Panepucci, J.Mag.Reson.Imag. 9, 75 (999). [8] I.S. Gradshteyn and I.M. Ryhik, Table of Integrals, Series, and Products, Academic Press, Inc. (98).