The Magnetic Torque Oscillator and the Magnetic Piston

Transcription

1 The Magnetic Torque Oscillator and the Magnetic Piston Martin Connors and Farook Al-Shaali, Athabasca University, Athabasca AB, Canada A agnet suspended in a unifor agnetic field like that of the Earth can be ade to oscillate about the field. The frequency of oscillation depends on the strength (agnetic oent) of the agnet, that of the external field, and the oent of inertia of the agnet. It is easily shown and verified by experient that a siple but nontrivial expression represents this otion well, but that only the product of the agnetic oent and agnetic field can be deterined. The repulsion of two agnets can be used to deterine the agnetic oent, and in turn the external field can be deterined. The results are reasonably good considering the very siple equipent required, and the experient allows quantitative investigation of agnetis. A agnet s strength is well-indicated by its agnetic oent. When placed in an external agnetic field B, a torque t arises on the agnet to align it with the external field. This effect is well-known since it is what akes a copass needle line up with agnetic north. More precisely, the torque is equal to t = B sin q, where q is the angle between the external field and the agnetic oent vector (directed fro the south pole to north pole in a regular agnet). Fro the angular for of Newton s second law t = Ia = I(d q/dt ), and assuing sall angles (q < 0 o or less), we can write the equation of otion for the agnet as d q B = q dt I, where I is its oent of inertia about the suspension axis. This is the equation for a siple haronic Fig.. Two agnets attracting each other with a thread between the do not hang vertically due to their attept to align with the Earth s agnetic field, which is not, in general, horiontal. The level is ade of aluinu (a protective cover over the bubble was reoved as the screws holding it in place were agnetic). oscillator with angular frequency ω = B I. So, uch like a siple pendulu in a gravitational field, a agnet will oscillate in the agnetic field if properly suspended. This oscillation ay be easily seen in a copass, but in that application it is not desired. Therefore, good copasses are liquid-filled to dap out any oveent and leave the needle aligned with the horiontal coponent of the Earth s agnetic field. A useful, if often hidden, aspect of odern technology is that strong agnets ay be produced cheaply. Most earphones, and all hard drives, use 440 DOI: 0.9/ The Physics Teacher Vol. 45, October 007

2 strong agnets, so it is very difficult not to be surrounded by strong agnets in odern life. In their raw for, sall strong agnets ay be purchased in hardware stores. 3 If a pair of such strong cylindrical agnets is carefully brought together with a thread placed between the, as shown in Fig., the thread will be held in place between the, and the agnets ay be suspended in space and react to Earth s agnetic field. The torque referred to above will ake the north-seeking, uch like a copass needle. Since there is very little daping, oscillation (usually present or easily started) about the (nearly) vertical axis fored by the thread is quite striking and will continue for a very long tie. A ore subtle aspect is that the agnet attepts to align with Earth s field in three diensions. Alignent parallel to the vertical part of the field, present everywhere except the agnetic equator, is not possible due to the weight of the agnet. 4 Nevertheless, as Fig. illustrates, the agnets do not hang vertically. Since it is the intent of this experient to deterine the horiontal coponent of Earth s agnetic field, the thread ust be weighted to hang ore vertically. It is also iportant that the thread not have any twist in it that would produce a torsional torque. In this experient, a tape easure weighing about 500 g was tied to the botto of the thread about one eter below the agnets and the thread suspended fro a tack near ceiling level. The sall agnets used here are disks (i.e., short cylinders) that rotate about a central diaeter. The appropriate forula for the oent of inertia, which depends on the ass M, radius R, and height H of the cylinder, can be found in soe introductory physics texts 5 as I = MH + MR 4. In our case, we vary I by adding ore agnets of the sae radius and thickness, so the oent of inertia should increase with the square of the nuber of agnets stacked. We ust ake an assuption in order to deterine how the agnetic oent varies as agnets are stacked. When a agnetic field is applied to any agnetically susceptible aterials, randoly oriented doains in the aterial align and increase the agnetic oent. 6 A ferroagnetic aterial as found in a strong agnet ay be regarded as an extree case of a paraagnetic aterial in that it is saturated; we assue that further application of a agnetic field will not increase the alignent of doains as they are already very highly aligned. Thus we can assue that the overall agnetic oent siply increases directly as the nuber of agnets. Using N agnets, each of thickness h, ass, and agnetic oent, the forula for the oscillation frequency becoes B ω( N ) =. h N + 4 R () This forula allows ready testing. By adding agnets to each side of the string, N can be varied in units of. In our case, N was varied fro to 4. Since larger sources of error were expected, the other required quantities of agnet ass and diensions were deterined soewhat roughly with a postal scale and ruler. Twelve agnets had a ass of 80 g, thus was kg. Their length when together was 3.8 c, so h was The diaeter was 8.5 by direct easureent, so radius R was Multiple oscillations of the stacked agnets were observed and tied using a wristwatch with a stopwatch setting. When two agnets were used, 00 oscillations were tied and 74 s elapsed; with 4 agnets, 0 oscillations took slightly over 40 s. The period P is siply the tie elapsed divided by the nuber of oscillations. Using Vernier s Graphical Analysis 7 the period P data values were entered, the frequency was deterined as a new calculated colun (π/p), and a curve fit was done based on Eq. (). The results are shown in Fig., where the soewhat coplex curve passes very near all data points. The good fit confirs that the basic theory behind Eq. () is correct: agnetic oents add when agnets stick together, and the oent of inertia behaves as it should. The only paraeter in the fit is referred to as A in the fit function, and its value of corresponds to the product B in SI units. 8 In order to disentangle these values, one of the ust be deterined. A siple way to do this is to balance agnetic force against gravitational force. For two agnets approxiated as interacting dipoles (agnetic oents), both forces ay be easily calculated, and this is shown below. First, since this paper tries to show how to do things siply and inexpensively, we show how to re- The Physics Teacher Vol. 45, October

3 Frequency (H) [ Auto Fit For: Data Set Frequency y = sqrt(a/(/* ^*x^+/4*0.0095^)) A: /-.45E-00.5 RMSE: 0.00 P Linear Fit For: Data Set P^ y = x+b (Slope): 0.08 b(y-intercept): 0.43 Correlation: ] vise Eq. () so that a straight line can be plotted to allow paraeters to be deterined. The dependence on N is ade ore explicit if it is taken to the nuerator by using the period P as the independent variable: P( N ) = π h N + 4 R. () B This equation can be ade linear by squaring both sides to get P π ( N ) = ( 3 h N + R ). B Fig.. Graphical Analysis fit. Data points show the frequency of oscillation as a function of the nuber of agnets. The fit corresponds to Eq. () and has only one paraeter. The good fit using a coplex curve indicates that the theory is basically correct. (3) We have plotted our data in this for of P as a function of N (for convenience using Graphical Analyis). The results and a straight-line fit are created in Fig. 3. A correlation coefficient of indicates a straight line, so the correlation coefficient of deterined in linear fitting with Graphical Analysis supports the linear relation shown by Eq. (3). In turn, as already shown with the nonlinear for, this iplies that the theory is very accurate and analysis should be reliable. For siplicity we will base further analysis on the paraeter already derived using Eq. (). Those who do not have Graphical Analysis should be able to readily figure out how to do a linear fit on graph paper and fro it derive the needed values using Eq. (3). N ] [ N Fig. 3. Data plotted in the linearied for corresponding to Eq. (3) in the text. The representation by a straight line is very good and analysis could be done using graph paper. Magnetic Dipole Moent fro Magnetic Piston By aking a agnetic piston, the agnetic oent can be found with very little instruentation. We saw above that a unifor agnetic field exerts a torque on a dipole (and in fact it exerts no force; a copass needle does not try to pull the user toward the agnetic pole, only to orient itself). Obviously, agnets also exert forces, and this is characteristic of nonunifor agnetic fields. Sall agnets have coplex (and nonunifor) fields close to the. If the agnets are far enough apart, their interaction can be regarded as that of dipoles. Modern strong agnets allow big enough spacing that the dipole approxiation is good while significant force is present. 9 This allows us to arrange agnets carefully in order to balance agnetic force against gravity. In this way, with an expression for the force between dipoles, we can deterine the dipole oent. A narrow clear plastic tube allows placing one stack of agnets (one agnet alone would easily rotate) above another without rotation fro agnetic torque taking place. In this way, the upper agnet stack will float in the air above the lower one. This setup can be referred to as a agnetic piston and is illustrated in Fig. 4. Along the axis of a dipole at the origin pointing along the direction, the field is given 0 by 44 The Physics Teacher Vol. 45, October 007

4 B = 0 (4) 4π 3, 7 where 0 = 4π 0 [ T / A ]. This field is nonunifor even along the axis since it falls off rapidly. The force is siply equal to the gradient of the agnetic field ties the coponent of the dipole oent parallel to and is along that axis. For two identical agnets of oent, we find that the agnitude of the force is F = 3 0 4π 4. The agnetic piston was ade by taping four stacked agnets to a tabletop and placing another four inside a tight plastic tube where they could freely slide up and down, with the tube taped in place above the other four agnets so that the agnets in the tube were suspended by agnetic force. In this configuration, 5.7 c separated the faces of the floating agnets fro those below. The value appropriate for the centers of the agnets, which are the effective dipole positions, is 6.3 c. The repulsive agnetic force upward ust have been equal to the gravitational force on the floating agnets. Thus 4 4g = 3 ( ) 4 π 4, and we can solve for the agnetic oent of a agnet as g = = A. 3 This is coparable to, but slightly saller than, the value found for a siilar agnet using a coil. 3 The fit in the previous section gave a value for the product B of , so B = nt. Field calculators online 3 show an expected horiontal coponent of 4,480 nt, so our value is likely about 30% too large. Soe of the error in this approach coes fro the sensitivity of the force equation for interacting dipoles so that sall errors in easuring and approxiation of positions of agnets, as those of dipoles, result in errors in in an equation that is quite sensitive to. Assuing this is the only error and is roughly 3, the error equation = Fig. 4. The agnetic piston was ade using two agnet stacks and a clear plastic coin tube. The lower stack of four agnets is taped to the tabletop to prevent rotation due to agnetic torque fro the floating upper stack. The walls of the plastic tube (also taped securely) prevent significant rotation of the floating stack. A finger ay be inserted to push down on the upper stack and force it to ove until it coes to a force balance position. would lead us to expect only a 0% error. It is thus likely that there was soe influence of the vertical coponent of the Earth s field, which at the latitude this experient was perfored is uch larger than the horiontal coponent. If this is the case, the experient should work better at lower latitudes than those of central Canada, where our run was done, since the Earth s agnetic field would be ore horiontal. It would also be possible to devise a better for of ounting if a achine shop is available, perhaps taking inspiration fro historical instruents. 4 The intent of this paper was to show how to get eaningful results with the siplest possible equipent, so the inaccuracy is not of uch concern here. Conclusions Using only very siple equipent, one can show that the for of the equation for a agnetic oscillator The Physics Teacher Vol. 45, October

5 is correct. The period of such an oscillator deterines only the product B, but if another ethod can be found to deterine the agnetic dipole oent, then the horiontal coponent of Earth s agnetic field can be deterined. With a siple agnetic piston, the dipole oent ay be deterined but is subject to nonnegligible error. Using it to subsequently deterine Earth s field resulted in a value coparable to that expected. This experient allows easureent techniques and otherwise nonintuitive physical quantities to be investigated at very low cost. References. The torque on a coil is derived in Chap. 0 of D. Giancoli, Physics, 6th ed. (Pearson Prentice Hall, Upper Saddle River, NJ, 005), and the relation to peranent agnets is explained.. This result follows iediately by solving the equation of otion with calculus; a noncalculus discussion of various types of haronic oscillators is found in Chap. 3 of J. Touger, Introductory Physics: Building Understanding (Wiley, New York, 006). 3. See the recent TPT article by Mike Moloney, Strong little agnets, Phys. Teach. 45, (Sept. 007) and an earlier, related article by G. Hageseth, A. J. Phys. 37, (May 969). The agnets used in this experient were purchased at Lee Valley Tools local store, but siilar agnets are now widely available. 4. If allowed to pivot freely about the center of ass, a dip indicator would result. These were historically used as scientific instruents, including on expeditions to find the north agnetic pole (the dip pole ). Usually the for was like a vertical copass or dip needle. See Dip_Needle/Dip_Needle.htl. 5. See, for exaple, D. Halliday, R. Resnick, and J. Walker, Fundaentals of Physics, 8th ed. (Wiley, New York, 007), p This is known as paraagnetis; the opposite behavior, diaagnetis, is also possible. Deeper understanding of agnetis is based on advanced concepts. Reading accessible to those slightly beyond the first year level can be found in the Feynan Lectures on Physics, Vol. II (Addison Wesley, Reading, MA, 964) gives inforation about this inexpensive and useful progra. 8. If SI units are consistently used, the details of the coplex agnetic units can be avoided. 9. The agnetic field very near the agnet is in general quite coplex, although it ay have soe siplifying characteristics such as syetry 0. This is a odification of the spherical coordinate for given in P. Lorrain and D. Corson, Electroagnetic Fields and Waves, nd ed. (Freean, New York, 970) in Chap. 7. A ore exact for based on the agnetiation of the agnet and its diensions is given by M. Connors, Measureent and analysis of the field of disk agnets, Phys. Teach. 40, (May 00) and could be reworked to allow to be deterined by easuring the agnetic field along the axis. However, the intent here is to use inial instruentation.. J.D. Jackson, Classical Electrodynaics, nd ed. (Wiley, New York, 975), p Earth field easureents are usually given in units of nanotesla (nt) leads to the International Geoagnetic Reference Field and also the version specific to Canada. PACS codes: 0.50.Pa, Martin Connors is professor and Canada Research chair at Athabasca University. Since 996 he has worked there on basic research and on techniques supporting the rapid growth in distance education in physics and astronoy. He has an interest in instruentation that can be used by hoe study students. Centre for Science, Athabasca University, University Drive, Athabasca AB, Canada T9S 3A3; artinc@ athabascau.ca Farook Al-Shaali is the course coordinator for physics and astronoy at Athabasca University. His background is in geophysics and Earth agnetis, and he has worked extensively on distance education physics courses, including labs to be done at hoe. Centre for Science, Athabasca University, University Drive, Athabasca AB, Canada T9S 3A3; farooka@ athabascau.ca 444 The Physics Teacher Vol. 45, October 007