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ALTERNATIVE METHOD to CALCULATE the MAGNETIC FIELD of PERMANENT MAGNETS with AZIMUTHAL SYMMETRY 9 1 ~M · Nˆ Φ (~X) = , 2.1

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ALTERNATIVE METHOD to CALCULATE the MAGNETIC FIELD of PERMANENT MAGNETS with AZIMUTHAL SYMMETRY 9 1 ~M · Nˆ Φ (~X) = , 2.1 EDUCATION Revista Mexicana de F´ısica E 59 (2013) 8–17 JANUARY–JUNE 2013 Alternative method to calculate the magnetic field of permanent magnets with azimuthal symmetry J. M. Camacho and V. Sosaa Facultad de Ingenier´ıa, Universidad Autonoma´ de Yucatan,´ Av. Industrias no contaminantes por Periferico´ Norte, A.P. 150 Cordemex, Merida,´ Yuc., Mexico.´ aPermanent address: Departamento de F´ısica Aplicada, CINVESTAV-IPN, Unidad Merida,´ A.P. 73 Cordemex, Merida,´ Yuc., C.P. 97310, Mexico.´ Phone: (+52-999) 9429445; Fax: (+52-999) 9812917. e-mail: [email protected]. Received 6 September 2012; accepted 8 January 2013 The magnetic field of a permanent magnet is calculated analytically for different geometries. The cases of a sphere, cone, cylinder, ring and rectangular prism are studied. The calculation on the axis of symmetry is presented in every case. For magnets with cylindrical symmetry, we propose an approach based on an expansion in Legendre polynomials to obtain the field at points off the axis. The case of a cylinder magnet was analyzed with this method by calculating the force between two magnets of this shape. Experimental results are presented too, showing a nice agreement with theory. Keywords: Permanent magnet; calculation; analytical; symmetry. PACS: 75.50 Ww; 41.20 Gz. 1. Introduction develop an alternative method of calculation and find ana- lytical expressions for these fields. This will allow estima- tion of many important variables in certain applications, such Since ancient times, magnetism has captured the interest of as the force between magnets. We will see that the results human beings. The feeling of an unseen force (but no less obtained theoretically and experimentally are consistent with invisible than the force of gravity) acting with great inten- each other. sity, is often astonishing. The properties of magnetite, an iron mineral used as the needle in a compass, were known since the eleventh century A. D. Over years, knowledge of 2. Theory magnetic phenomena has opened new perspectives and has encouraged the development of new technologies. For ex- Typically, the magnetic field of a permanent magnet can be ample we now know that some living organisms have small calculated from the vector potential: B~ = r £ A~. For a amounts of magnetite in their tissues and use it to orient in the body whose magnetization is constant inside its volume and Earth’s magnetic field. Bees, salmons and some turtles are drops abruptly to zero outside it, as is the case of a magnet, a few examples of such species [1]. The effect of magnetic this potential at point ~x is given by the following surface in- fields on water is a topic of current research too [2-4]. For ex- tegral [9]: ample, it has been found that these fields inhibit scale forma- I ¹ M~ (x~0) £ n^0 tion, by inducing changes in the crystal structure of carbonate A~(~x) = 0 da0 in the liquid. On the other hand, other researchers investi- 4¼ j~x ¡ x~0j gate to what extent cell phones and magnetism from electric power lines affect living things. The development of technol- where M~ is the volume magnetization of the magnet, n^0 is 0 ogy has brought many important magnetic applications like the unit vector normal to the surface at point x~ , and ¹0 is the electric motors, generators and storage of information. How- vacuum magnetic susceptibility. The integral is performed ever, it is not easy to find in the literature information about over the entire surface of the magnet. In this work we adopt the magnetic fields produced by these magnets. Generally, a different method. It is assumed that the magnet is a contin- these fields are calculated by numerical methods, such as the uous distribution of dipoles, which occupy a volume dV and finite-element method, [5,6] and it is rare to find analytical have a magnetic dipole moment d~m = MdV~ . Again, here expressions. The most known cases are the spherical and the we assume that the magnetization is constant, and it remains rectangular prism magnets. The field of a ring-shaped mag- unchanged for any external magnetic field. That is, the mag- net has been discussed recently [7,8]. This paper is focused nets are supposed to be hard. To facilitate the calculations, on the analytical calculation of the field produced by perma- we always choose M~ = Mk^. As a starting point, we recall nent magnets with different shapes. We studied Nd2F e14B that the magnetic scalar potential and the induction field at permanent magnets with the following geometries: sphere, point ~x produced by a magnetic dipole located at the origin cylinder, ring, cone and rectangular prism. The aim was to are given by: ALTERNATIVE METHOD TO CALCULATE THE MAGNETIC FIELD OF PERMANENT MAGNETS WITH AZIMUTHAL SYMMETRY 9 1 ~m ¢ n^ © (~x) = ; 2.1. Sphere dipole 4¼ j~xj2 According to Fig. 1, we write the contribution of each in- ~ ¹0 3^n(^n ¢ ~m) ¡ ~m Bdipole(~x) = (1) finitesimal dipole as: 4¼ j~xj3 1 dm ¢ cos ® where ~m is the magnetic dipole moment and n^ is the unit vec- d©axis(z) = 4¼ j~r ¡ r~0j2 tor in the direction of ~x. To calculate the magnetic field of the whole magnet, we integrate over its volume the contributions 1 MdV (z ¡ z0) = from the infinitesimal dipoles, either of the scalar potential or 4¼ (z2 + r02 ¡ 2zr0 cos θ0)3=2 of the field. In the first case, we take finally B~ = ¡¹0r©. The decision of which way to go depends on the ease of cal- The total scalar potential on the z axis is then given by: culation in each case. In particular, the configurations with azimuthal symmetry offer the use of interesting mathemati- Z2¼ Z¼ M cal properties. The basic idea is to find the scalar potential © (z) = dÁ0 sin θ0dθ0 axis 4¼ on the symmetry axis (z axis in our case), which is relatively 0 0 easy to calculate, and from this function the solution is built ZR for off-axis points. For example, Jackson [10] used succes- r02(z ¡ r0 cos θ0) £ dr0 sive derivatives of the axial function to find the magnetic field (z2 + r02 ¡ 2zr0 cos θ0)3=2 due to a solenoid. In this work we use a property of the solu- 0 tions of the Laplace equation. This property, which is often ZR Z1 used in the case of the electrostatic potential, [5,9] states that M z ¡ r0x = r02dr0 dx if you have the solution on the symmetry axis expressed as a 2 (z2 + r02 ¡ 2zr0x)3=2 series: 0 ¡1 1 µ ¶ By using Mathematica, we find the indefinite integral in X V © (z) = U z` + ` (2) x results axis ` z`+1 `=0 ¡r0 + xz then the solution at any point in space (r; θ) is given by: z2(r02 ¡ 2r0xz + z2)1=2 X1 µ ¶ Evaluating in the limits and simplifying, we obtain 2=z2, V` ©(r; θ) = U r` + P (cos θ) which does not depend on r0. Finally, performing the inte- ` r`+1 ` `=0 gration on this variable, we obtain 3 where P`(cos θ) is the Legendre polynomial of order `. MR 1 ©axis(z) = (3) Therefore, all we have to do is to calculate ©axis(z) and ex- 3 z2 pand it in powers of z. Now, we illustrate this approach with the case of a spherical magnet. The last expression is a particular case of Eq. (2), with 3 coefficients U` = 0 8` and V1 = MR =3, V` = 0 8` 6= 1. Applying the property described above, we see that the po- tential at any point in space is given by: MR3 1 ©(r; θ) = P (cos θ) 3 r2 1 MR3 cos θ 1 m cos θ = = 3 r2 4¼ r2 where m = 4¼MR3=3 is the magnetic dipole moment of the sphere. This result is well known and is equiva- lent to that produced by a point dipole with moment m placed at the center of the sphere, as can be seen from Eq. (1). As mentioned above, the induction field is given by B~ (r; θ) = ¡¹0r©(r; θ). Next, we present the results of the magnetic field on the axis of symmetry of the other magnets, FIGURE 1. Scheme for calculating the field produced by a spheri- and compare them with measurements made in the labora- cal magnet. tory. Rev. Mex. Fis. E 59 (2013) 8–17 10 J. M. CAMACHO AND V. SOSA The additive constant is irrelevant and can be suppressed. With the change of variable x = z ¡ z0, we can write zZ¡h M xdx © (z) = axis 2 [Dx2 + Ex + F ]1=2 z where R2 2R2 ³ z ´ ³ z ´2 D = 1 + ;E = 1 ¡ ;F = R2 1 ¡ : h2 h h h After integrating, evaluating in the limits and simplifying, we finally obtain: Mh2 h p i © (z) = §(z ¡ h) ¡ R2 + z2 axis 2(h2 + R2) FIGURE 2. Scheme for calculating the field produced by a conical MhR2(z ¡ h) magnet. + 2(h2 + R2)3=2 p 2.2. Cone (h § h2 + R2)(z ¡ h) £ ln p p (4) According to Fig. 2, each infinitesimal dipole contributes to R2 + hz + h2 + R2 R2 + z2 the potential on the z axis with: 1 dm cos ® 1 M½0d½0dz0dÁ0(z ¡ z0) Then, the induction field has the following dependence d©axis(z) = = 4¼ j~r ¡ r~0j2 4¼ [(z ¡ z0)2 + ½02]3=2 on the z axis: Therefore, the potential on the axis is given by " µ 2 ¶ Z2¼ Zh ¹0Mh 1 zh¡R M B(z)= p ¨ h © (z) = dÁ0 (z ¡ z0)dz0 2 h2+R2 R2+z2 axis 4¼ p # 0 0 R2 (h § h2+R2)(z¡h) ½0 (z0) maxZ ¡ ln p p (5) ½0d½0 (h2+R2)3=2 R2+hz+ h2+R2 R2+z2 £ [(z ¡ z0)2 + ½02]3=2 0 0 R(1¡ z ) Zh Z h The double sign in the Eqs.
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