Measuring the magnetization of a permanent magnet B. Barman Department of Computer Science, Engineering and Physics, University of Michigan-Flint, Flint, Michigan 48502 A. Petrou Department of Physics, University at Buffalo, The State University of New York, Buffalo, New York 14260 (Received 1 July 2018; accepted 10 February 2019) The effect of an external magnetic field B on magnetic materials is a subject of immense importance. The simplest and oldest manifestation of such effects is the behavior of the magnetic compass. Magnetization M plays a key role in studying the response of magnetic materials to B. In this paper, an experimental technique for the determination of M of a permanent magnet will be presented. The proposed method discusses the effect of B (produced by a pair of Helmholtz coils) on a permanent magnet, suspended by two strings and allowed to oscillate under the influence of the torque that the magnetic field exerts on the magnet. The arrangement used Newton’s second law for rotational motion to measure M via graphical analysis. VC 2019 American Association of Physics Teachers. https://doi.org/10.1119/1.5092452 1,5–7 I. INTRODUCTION of the loop. The magnitude l is given by the following equation: Magnetism has been intriguing mankind for centuries now. A magnet’s ability to influence magnetic materials, l ¼ iA: (1) from a distance, mesmerized numerous inquisitive minds of the past. The magnetic compass, used across all continents, If we place this loop in a uniform magnetic field B~ at an is such a device, acting under the influence of Earth’s mag- angle h with ~l, as shown in Fig. 1(a), we have a torque act- netic field.1 With the advancement of science, particularly ing on the loop whose magnitude s is given by the following after the discovery of the atomic model, magnetism at the equation: microscopic scale came to the fore. Hans-Christian Oersted’s discovery of a current-carrying wire producing a magnetic s ¼ lB sin h: (2) field added a whole new dimension, leading to the advent of a technological revolution in the field of electromagnetism.2 If the loop is allowed to move, it will rotate under the action With the exploration of newer magnetic materials, exhibiting of the torque in such a way that ~l becomes parallel with the ~ paramagnetism, ferromagnetism, and diamagnetism, it magnetic field B, as shown in Fig. 1(b). Indeed, in the config- became essential to develop techniques to measure magneti- uration of Fig. 1(b), we have that h ¼ 0 for which Eq. (2) zation in these. While many experimental methods have shows that s ¼ 0 and all movement stops. Thus, the mag- been proposed, along with the existence of high tech equip- netic moment is defined so that the torque exerted on a cur- rent carrying loop, when placed in an external magnetic ment for research, a basic design requirement for educational 8 purposes has eluded us so far.3,4 In this paper, we present an field, is proportional to l. experimental technique to determine the magnetization of a permanent magnet using readily available lab instruments B. Magnetization namely as a pair of Helmholtz coils, an ammeter, a power For our experimental discussion, we consider a cylindrical supply, and a stopwatch. While this technique is applicable permanent magnet (composed of iron atoms) of radius R and for both introductory physics (calculus and algebra based) and height l, as shown in Fig. 2. In each atom, its electrons move undergraduate Physics/Engineering majors, the extent of around the nucleus on orbits similar to that shown in Fig. 1. exploration rests with the instructor. As a quick in-class Thus, each iron atom behaves like a microscopic loop with demo, an introductory class can observe the effect of an exter- magnetic moment ~lFe. In iron and other ferromagnetic mate- nal magnetic field on the oscillation period of a permanent rials, all the magnetic moment vectors are aligned and the magnet without determining any value of magnetization. In net magnetic moment l of the magnet is equal to NlFe, an advanced lab setting, students can explore numerical val- where N is the number of iron atoms in the magnet. The ues of magnetization of the four types of permanent magnets, magnetization M of the magnet, with a volume V is defined viz., neodymium iron boron (NdFeB), samarium cobalt as5,9 (SmCo), alnico, and ceramic or ferrite magnets. II. THEORY A. Magnetic moment It is well known that in a wire loop with area A, in which a current i is flowing, we define a vector known as the Fig. 1. (a) Magnetic moment of a current carrying loop at an angle h with “magnetic moment” (symbol ~l), perpendicular to the plane respect to B~ and (b) ~l k B~. 275 Am. J. Phys. 87 (4), April 2019 http://aapt.org/ajp VC 2019 American Association of Physics Teachers 275 Fig. 2. Cylindrical permanent magnet of radius R and height l. l M ¼ : (3) Fig. 3. Helmholtz coils of diameter D, consisting of N turns. Image copyright: V https://www.emworks.com/application/numerical-simulation-of-helmholtz-coil. For the cylindrical magnet of Fig. 2, the volume is given by the following equation: V ¼ pR2l: (4) C. Magnetic field at the center of a pair of Helmholtz coils The schematic of a pair of Helmholtz coils is shown in Fig. 3. Each has a diameter D and consists of N turns of wire. They are positioned so that they have a common axis, and the distance between the coil centers is adjusted so that it is equal to half the diameter D. Under these conditions, the Helmholtz coils generate a magnetic field along the common Fig. 4. Cylindrical permanent magnet suspended from strings. axis of the coils, which is uniform in the vicinity of the mid- point between the coil centers. The magnetic field B is given by the following equation:10,11 16l Ni B ¼ pffiffiffiffiffiffiffiffi0 : (5) D 125 Here, the current i is measured in Amps, B is measured in tesla, D ¼ 0:21 m, N ¼ 119, and l0 is a constant equal to 1:256 Â 10À6 T:m=A. III. DISCUSSION: OSCILLATION OF A MAGNET IN A MAGNETIC FIELD A permanent magnet is suspended, in a stirrup like Fig. 5. Top view of the cylindrical magnet oscillating between the arrangement (Fig. 4), so that the magnet points in the Helmholtz coils. Fig. 6. Pictorial representation of the experimental set-up. (a) Schematic and (b) actual working rendition. 276 Am. J. Phys., Vol. 87, No. 4, April 2019 B. Barman and A. Petrou 276 north-south direction. The magnetic field B~ generated by the Table I. Summary of the experimental parameters as well as the calculated Helmholtz coils is in the same direction such that the Earth’s value of M. magnetic field does not influence our measurement of mag- 2 2 À1 netization. This is an important criterion which has to be m (kg) R (m) l (m) l (kg m ) lðAm Þ MðAm Þ kept in consideration while performing this experiment. If 0.0511 0.0046 0.125 6:681 Â 10À5 3.047 ð3761:2Þ104 we now consider Fig. 5, which is a top view of Fig. 4, dis- place the magnet by a small angle /0 from the direction of ~ B, and release it from rest, the magnet will oscillate around between the centers of the coils. Hence, the choice of length the equilibrium position as indicated by the curved arrows for the permanent magnet is paramount to determining M shown in Fig. 5. The time it takes for the magnet to swing precisely. More significantly, the distance between the coils from left to right and back to the starting position is called should be equal to the radius in order to achieve this unifor- the period of oscillation (symbol T). By applying Newton’s mity.13,14 The period of 10 oscillations was measured as a second law of rotational motion, we can write (using the function of the current in the Helmholtz coils; the resulting small angle approximation) data were linearized using Eq. (9) and plotted. A least- squares fit was used to plot the data as shown in Fig. 7.In d2/ s ¼ I ¼ C/ þ lB sin / ½C þ lB /; (6) order to improve the accuracy of these measurements, it is dt2 important that the permanent magnet oscillates in a horizon- tal plane, keeping the small-angle approximation in mind. ~ where / is the angle that the magnet axis forms with B at Any vertical mode of oscillation would introduce a consider- any instant of time t. Here, C is the torsional constant that able amount of error in the measurement of the magnetiza- describes the mechanical properties of the supporting strings tion. Vernier calipers were used to measure the radius R and and l is the magnetic moment of the magnet. I is the moment the length ‘ of the magnet, while a balance was used to mea- of inertia (in our configuration) of the cylindrical magnet of sure the mass m of the magnet in kilograms. mass m and is given by the following equation:12 1 1 V. EXPERIMENTAL RESULTS AND DISCUSSION I ¼ ml2 þ mR2: (7) 12 4 The experimental data were plotted with B on the x-axis andðÞ 2p=T 2 on the y-axis as shown in Fig. 7 followed by The solution to the 2nd order differential equation in / is modeling using Eq. (9) to calculate the magnetic moment l. given by Having determined the slope (and the corresponding mag- netic moment l), together with volume V, we were able to 2p / ¼ / cos t : (8) determine the magnetization M of the magnet.