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Magnetostatics Analysis, Design, and Construction of a Loudspeaker Calin Galeriu, Becker College, Worcester, MA M aking a loudspeaker is a very rewarding hands-on activity that can be used to teach about electromagnetism and sound waves. Several loudspeaker designs have been described in this magazine.1-4 The simplest loudspeaker4 has only a magnet, a coil, and three plastic cups. The simpler devices3,4 require a powerful amplified output, e.g., from a boom box. The more complex devices1,2 can operate using the smaller electric current from a CD player earphone output. Unfortunately, the procedure to make a more efficient loudspeaker is lengthy and less recommended to some high school students, involving a hot glue gun, a safety razor, five-minute epoxy, etc. Our loudspeaker, a variation of Heller’s,2 is both simple in construction and efficient in operation. An analysis of the magnetic field distribution helped us in the design of this loudspeaker. In its simplest form a loudspeaker is just a plastic (or paper) cup with a coil of wire glued (or taped) to its bottom and with a permanent magnet inside or near the coil. The sound is produced by the vibrations of the bottom of the cup, which functions as a loudspeaker diaphragm. These vibrations are the result of the electromagnetic interaction between the permanent magnet and the electric current flowing through the coil. What matters in the operation of the loudspeaker is the force perpendicular to the bottom of the cup, causing the vibrations. As a cross product, the electromagnetic force is perpendicular to both the magnetic field and the electric current. Maximizing the electromagnetic force perpendicular to the bottom of the cup is key to improving the performance of the loudspeaker. This happens when, at the position of the coil, we have a strong magnetic field with a radial direction (in a cylindrical reference system), as shown in Fig. 1. Fig. 1. In the plane of the loop of wire the electric current element has a tangential direction. A magnetic field with a radial direction will produce an electromagnetic force perpendicular to the plane of the loop. Sometimes a second cup, of the same size, is used to hold inside the permanent magnet and the cup with the coil. This second cup will also work as a loudspeaker enclosure and help dampen the out-of-phase sound waves produced at the rear of the speaker. When the loudspeaker diaphragm moves forward to create a layer of compressed air in front of the speaker, it also creates a layer of rarefied air in the back. In the absence of a loudspeaker enclosure, the 180oout-of-phase sound waves will interfere in the room with the in-phase sound waves, resulting in reduced loudspeaker efficiency, especially at low frequencies. Magnetostatics analysis To calculate, visualize, and analyze the magnetic field lines for different magnet configurations, we have used the Maxwell SV 9.0 computer program. The software and the getting started guides can be downloaded from Ansoft’s website.5 Two tutorials are provided, one for an electrostatic problem and one for a magnetostatic problem. The Maxwell SV program can solve two-dimensional problems, which means that the analyzed device must have translational or rotational symmetry. The electromagnetic field calculated is a cross section of the structure. The loudspeakers that we built use disk magnets and have cylindrical symmetry. Five steps are required to calculate the magnetic field distribution for any of the loudspeaker designs investigated. In the first step a drawing is made in which the dimensions of the solid parts (magnet, steel core, etc.) are specified. We have used a grid of 0.03125 in and a 2-in by 2-in spatial domain. In the second step the solid parts are assigned material proper- Fig. 2. (a) A magnet. (b) Two magnets. (c) A magnet and a steel core. (d) A magnet and a steel core inside a steel cup. (e) A magnet on a metal bottle cap. Magnets are shown in blue and steel parts are shown in gray. The dotted line is the axis of rotational symmetry. DOI: 10.1119/1.3502508 Fig. 3. Magnetic field lines for a cylindrical magnet. The figure shows the right half of a vertical cross section. The rotation symmetry axis is at the left side of the figure. The Physics Teacher ◆ Vol. 48, November 2010 537 ties (NdFeB, steel). In the third step, free boundary conditions are defined. In the fourth step, the computer solves the Maxwell equations using the finite element method. In the fifth step, the calculated magnetic field lines are plotted. What we get is the right half of a vertical cross section, with the rotation symmetry axis at the left side of the figure. Five magnet configurations have been investigated, all shown in Fig. 2. The simplest loudspeaker, with just a magnet inside a coil, is not the most efficient because some of the magnetic field lines travel far from the coil and don’t contribute to the electromagnetic force, as shown in Fig. 3. If two magnets are used, then the magnetic field is stronger, and the sound produced is louder. A disadvantage of this setup is the cost. But there is also an added bonus. The two magnets can be placed on each side of the bottom of a second cup, where they stick together. The cup with the coil is then placed inside the cup with the magnets, as shown in Fig. 4. This makes aligning the coil over the inner magnet a lot easier. Furthermore, in this setup all the magnetic field lines reaching the coil have the same radial direction (outward or inward). Of course, on the other side of the bottom of the second cup the radial direction of the field lines is reversed, as seen in Fig. 5. In the absence of the second cup, if the coil extends over both regions then the total electromagnetic force on the Fig. 4. A loudspeaker with two cylindrical magnets and two plastic cups. Magnets are shown in blue and the coil is shown in red. Fig. 5. Magnetic field lines for two magnets. In the upper half the radial component of the magnetic field is pointing one way, while in the lower half it is pointing the other way. Fig. 7. Magnetic field lines for a magnet and a steel core inside a steel cup. The coil will be placed in the gap, between the two steel parts, where the magnetic field is strong. 538 coil will be reduced, since forces from some turns will oppose forces from other turns. The same discussion applies to the case of just one magnet. In this situation the sound will be loudest when the lowest turn of the coil is at the center of the magnet. Heller2 makes a similar observation. If a steel core is used inside the coil, then again all the magnetic field lines reaching the coil have the same radial direction, as seen in Fig. 6. A great improvement in the performance of the loudspeaker is obtained when the magnet and the steel core are placed inside a steel cup. This configuration confines almost all of the magnetic field inside the metal and through the coil, as seen in Fig. 7, resulting in a large electromagnetic force. The magnetic field lines stay inside the metal because in this way the magnetic field energy is minimized. Although this loudspeaker produced a really loud sound with just one magnet, we do not recommend it for classroom use for several reasons. Manufacturing the steel parts is not trivial. The magnet placed inside the steel cup has a tendency to stick to the wall, and one needs great skill and a couple of toothpicks to move it to the center. Once the steel core is placed over the magnet, the three parts become inseparable, hiding the details of the construction from future students, and thus reducing the educational value of the experiment. Fig. 6. Magnetic field lines for a magnet (on the bottom) and a steel core (on top). If the coil is placed around the steel core, then all the magnetic field lines intersecting the coil have the same radial direction. Fig. 8. Magnetic field lines for a magnet on a metal bottle cap. This is an effective way to keep the magnetic field concentrated near the coil. The Physics Teacher ◆ Vol. 48, November 2010 We have found that a compromise between simplicity and efficiency can be achieved if a metal bottle cap is used instead of the two steel parts. The magnetic field, while not as confined as in the previous case, is still concentrated near the coil, as seen in Fig. 8. Because this simple design uses only one rare-earth magnet but still produces a nice loud sound, this loudspeaker is our Fig. 9. An intermediate step during the “winner.” Next we give construction of the loudspeaker. a detailed step-by-step procedure on how to make such a loudspeaker. Materials needed For each student: • Two plastic cups (7 oz, flexible) • One cylindrical rare-earth magnet6 (1/2-in diameter, 3/16-in height) • Approximately 2 m of thin enamel-coated magnet wire (30 gauge) • One metal bottle cap For students to share: • 2-in wide Scotch packaging tape • Scissors • Sandpaper • AA batteries • Toothpicks • An audio cable from a sacrificed set of earphones, with exposed ends Construction of the loudspeaker Cut a square piece of Scotch tape (2 in x 2 in) and roll it on a AA battery with the sticky side out in such a way that one end gets attached to the other end, making a cylindrical tube 2 in long. Hold the battery from the “+” end, to prevent damaging the adhesive at the “–” end. Slide the battery through the Scotch tape tube until about half an inch of tape extends over the “–” end. Sand out about half an inch of the enamel insulation and leave about 10 in of free wire at both ends. Wind the coil, about 20 turns and about a quarter-inch long, half an inch from the end of the Scotch tape, as shown in Fig. 9. Twist the Fig 10. A finished loudspeaker. A second cup will be used to hold the metal bottle cap with the magnet and the first cup with the coil, as shown in Fig. 4 for a related design. wire ends a couple of times to make sure that the coil will not unwind, but not too tight because that could prevent the battery from sliding out. Make four evenly spaced cuts in the exposed section of the Scotch tape, as shown in Fig. 9 by the dotted lines. With the help of two toothpicks bend the Scotch tape squares inside, one by one, starting with two that are facing each other. Make all four Scotch tape squares lie one on top of the other on the flat bottom of the battery. The Scotch tape square on top will have the sticky side exposed, and this side will be used to tape the coil to the center of the bottom of a plastic cup, as shown in Fig. 10. Press on the battery and from the inside on the bottom of the cup to make sure that the bond is strong. Remove the battery. Very carefully cut the excess Scotch tape, slowly approaching the coil in a spiral move. Make sure you don’t cut the wire too! Place the magnet in the center of the metal bottle cap. Place the bottle cap on the bottom of the second plastic cup. Slowly slide the first cup inside the second cup, until the magnet is inside the coil. Connect the coil ends to the audio cable and turn the CD player on. Gently adjust the position of the coil relative to the magnet until the loudest sound is heard. Conclusion We have used Maxwell SV 9.0, a free and powerful computer program, to analyze the magnetic field distribution for some very simple loudspeakers. Through conceptual and experimental investigations we have designed a simple, safe-tomake, low-cost, but nonetheless powerful loudspeaker. Every student we asked was able to successfully build one. Acknowledgments The author wishes to thank the anonymous referees for The Physics Teacher ◆ Vol. 48, November 2010 539 helpful comments, and Mr. Tate Ostiguy for manufacturing the two steel parts in the Machine Shop at Bay Path High School. References Frames of REFERENCE 1. Allen Keeney and Brant Hershey, “Making your own dynamic loudspeaker,” Phys. Teach. 35, 297–299 (May 1997). 2. Peter Heller, “Drinking-cup loudspeaker – A surprise demo,” Phys. Teach. 35, 334 (Sept. 1997). 3. Molly Johnson and Virginia Stonick, “Sound science – A simple and robust hands-on loudspeaker activity,” Phys. Teach. 37, 350–351 (Sept. 1999). 4. Rhett Herman, “As simple as possible,” Phys. Teach. 40, 182–183 (March 2002). 5. The Maxwell SV software and the getting started guides can be downloaded from www.ansoft.com/maxwellsv. 6. Neodymium magnets can be ordered from K&J Magnetics, www.kjmagnetics.com. This site also has information about the safe handling of the magnets. original, unedited film now available on DVD Clearly Demonstrates and Explains Inertial and Accelerated Frames of Reference AAPT and Ztek Co. have transferred the complete 30-minute unedited film to DVD to preserve it for physics educators. The DVD also includes a Barcode & Chapter Guide. The film may be played uninterrupted from the beginning or it may be started, with a remote control or barcode reader, at any one of 21 points. Product by Ztek Demo Segment Available at www.ztek.com AAPT Member Price: $159 Calin Galeriu is teaching physics and math at Becker College and math at Bay Path Regional Vocational Technical High School. He has received a BS degree in physics from the University of Bucharest, an MA degree from Clark University, and a PhD degree from Worcester Polytechnic Institute. Becker College, 61 Sever St., Worcester, MA, 01609; Calin.Galeriu@ becker.edu American Association of Physics Teachers To order visit thephysicsstore.com or call 301-209-3333 New from AAPT! Preconceptions in Mechanics Members: $28 Non-Members: $35 Order yours now at 540 This second edition of Charles Camp and John J. Clement’s book contains a set of 24 innovative lessons and laboratories in mechanics for high school physics classrooms that was developed by a team of teachers and science education researchers. Research has shown that certain student preconceptions conflict with current physical theories and seem to resist change when using traditional instructional techniques. This book provides a set of lessons that are aimed specifically at these particularly troublesome areas: Normal Forces, Friction, Newton’s Third Law, Relative Motion, Gravity, Inertia, and Tension. The lessons can be used to supplement any course that includes mechanics. Each unit contains detailed step-by-step lesson plans, homework and test problems, as well as background information on common student misconceptions, an overall integrated teaching strategy, and key aspects of the targeted core concepts. A CD of all duplication materials is included. www.aapt.org/store The Physics Teacher ◆ Vol. 48, November 2010