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Teaching Electrical Concepts via Gravitational Analogies Page 1 Using Gravitational Analogies To Introduce Elementary Electrical Field Theory Concepts Printed 3:36 PM 5/2/17 Sue Saeli, Dept of Physics, SUNY-Buffalo State College, 1300 Elmwood Ave, Buffalo, NY 14222 Dan MacIsaac, Dept of Physics, SUNY-Buffalo State College, 1300 Elmwood Ave, Buffalo, NY 14222 <[email protected]>. Please direct correspondence regarding this manuscript to MacIsaac. Keywords: analogies, electric, gravity, field, potential, force PACS codes: 01.40Gb, 01.55, 41.90 Abstract: Familiar gravitational phenomena and conceptual analogies are useful in explaining those introductory electrical concepts associated with field theory since the electrical ideas are unfamiliar, abstract and difficult to visualize. These analogies emphasize the underlying continuity of fields in physics and support the spiral development of student understanding. We find the following four tables to be particularly useful in reviewing and summarizing these comparisons after students have conducted through appropriate touchstone activities and discourse as part of the process of making sense of electric fields. Acknowledgements: This manuscript addressed requirements for PHY690: Masters' Project at SUNY-Buffalo State College, and was informed by comments of students participating in PHY622 at SUNY-BSC during summer, 2003. Portions were supported by NSF DUE 0302097. All figures by Mr. Matt Coia. Some terms were coined in discussion with Dr. David Cole of Northern Arizona University; Mr. John Burgholzer of Amherst HS also provided significant comment. Other ideas were informed by the comments and curricula of The ASU Modeling Physics (REF1) project and personnel, particularly Mr. Gregg Swackhamer and Mr. Larry Dukerich, and the Workshop Physics (REF2) curriculum by Professor Priscilla Laws of Dickinson College. Errors and omissions are the responsibility of the authors. Introduction: Conceptual analogies from more familiar gravitational phenomena are useful in explaining introductory electrical concepts based upon field theory since the electrical ideas are unfamiliar, abstract and difficult to visualize. These analogies emphasize the underlying continuity of field concepts in physics and they support the spiral development of student understanding. We find the following four tables to be particularly useful in summarizing and reviewing these comparisons after students have worked through appropriate activities analyzed via extended student discourse(REF3). Teaching Electrical Concepts via Gravitational Analogies Page 2 Table 1: Introductory Analogies Between Gravitational And Electrical Forces Gravitational Electrical Comments Forces: Newton's Universal Law of Gravitation and the Coulomb Law for electric forces. Matter has a fundamental property called mass, measured in kg, which has just one sign: positive. Students may ask and need to be told that so-called "anti-matter" has positive mass (and reversed electric charge). Matter has another fundamental property called charge, measured in Coulombs which can have two signs: positive or describes the gravitational force negative. Hence Electric and direction, where rˆ forces can be repulsive or describes the direction and attractive. negative means attractive. Gravitational force is therefore qq always attractive. F k 1 2 rˆ Note we are talking about point masses and charges or perfect spherical distributions of mass and charge. This is a non-calculus treatment (easily extended). mm Fg G 1 2 2 rˆ r e The magnitude of this force is written: mm Fg G 1 2 2 r 11 G 6.67x10 Nm/kg 2 2 INSERT FIG1G.JPG HERE r2 or in magnitude only: Fe k q1q2 rˆ r2 where in SI units: where in SI units: Printed 3:36 PM 5/2/17 k 9x10 9 Nm 2 /C 2 INSERT FIG1E.JPG HERE Some have even coined the phrase "inertial charge" for mass to exploit this analogy. k , the gravitational force, Fg , is usually much smaller than the electrical force, F e (have students work both forces Since G is much smaller than for 2 protons and 2 electrons and compare / discourse). Students may not be familiar with rˆ (read as r-hat) notation (REF3), but will need it if they go on in physics. As well, this notation is needed to make conceptual sense of centripetal acceleration, so if it is new, now's the right time to discuss it. Note the tiny stick man in the figures defines rˆ as a unit vector pointing to the other point mass or charge. rhat really contains direction information only. Your students might also not recognize the "absolute value" notation used to strip direction from a vector, or the triple bar definition sign. These notations require explicit explanation and repeated student use. If your state exam requires a particular notation, use that as well from the start of the year. Teaching Electrical Concepts via Gravitational Analogies Page 3 Table 2: Introductory Analogies Between Gravitational And Electrical Fields Gravitational Electrical Comments Vector Fields For a small mass (compared to that of the Earth) on or very near the surface of the Earth we can group together known terms and solve: Fg G mearth m2 m mG earth2 2 rearth rearth further define Fg m g where g Similarly, with the electrical force there is a field around a given point charge Q (or spherically symmetrical distribution of charge Q) and and it is useful to talk about the field strength around that charge. Gmearth 2 rearth Fe k and is readily calculable, producing the F qE famous g = 9.8 N/kg pointing towards defining e Q the center of the Earth on the surface of E k 2 the Earth. r , and where therefore, is readily calculable Now we can talk about the local field for uniform electric fields strength of the Earth's gravitation field say those very near a charged at the Earth's surface, g , being the smooth spherical shell with ratio of the gravitational force a "test charge Q or between two mass" (a mass much smaller than that parallel plates with opposite charges as: of the Earth very near the Earth's surface). E F g m F q INSERT FIG2E.JPG HERE INSERT FIG2G.JPG HERE q1q2 Q 2 qo k 2 r r gives The gravitational field strength units of force per unit mass or N/kg, which should be shown (by students) to be the same as the more commonly used m/s2. Henceforth, the field units are to be pedagogically preferred. Printed 3:36 PM 5/2/17 Although the universal law of gravitation formula will work with any two point or spherically symmetric masses, we most commonly experience the downwards force of gravity at the Earth's surface, g . In that case one of the masses becomes the mass of the earth and the distance is the radius of the earth. Students should perform this calculation of the gravitational field strength g to practice using exponentials in their calculator under teacher supervision. The idea is to make g slightly less mysterious as a local value for gravitational field strength and to permanently move to N/kg units, which we have students show are equivalent to m/s2 and conceptually preferred. We actually walk around the room with a plumb bob mo ," and determine the strength and direction of g -- note analogy to "a vanishingly small test charge qo ." We – we call this "a vanishingly small test mass stand on tables, and hold the bob in corners, change in determining that g doesn't measurably direction and size in the room. We start fields off by having students sketch a figure (usually on a whiteboard) to explain the relationship between g in the classroom, g on the surface of the Earth at the equator and N and S poles, and g in space near the Earth's surface. This develops a better understanding of g and makes explicit the The corresponding units for E both a point in space and near field analogy near the electric field strength are, that of a Van de the surface of a charged shell like therefore, force per unit Graaf generator. charge or N/C, again with more common units of V/m We want to establish and reinforce the analogies (more on V later). between E and g . Showing the units of g as N/kg helps to solidify the analogy when comparing E An important value of to E (and can help clarify issues regarding to N/C for 6 know is E 3x10 N/C or gravitational fields). Have students demonstrate these units are equivalent. V/m – the dielectric breakdown strength of the Also establish the equivalence of E between Earth's atmosphere at STP. When this field strength is parallel plates (REF6) and g in a room on the exceeded, air will be torn Earth's surface. apart (ionized) and will conduct; we will see static Parallel plates can be attached to a Van deGraff electricity sparks drawn generator and ground to demonstrate E with a through the air. Any spark packing peanut on a thread. This is also useful for present means we know an the E near a Van deGraff sphere. instant minimum value for E (REF5). Note traditionally important numeric values of g and E . Students should memorize these. Teaching Electrical Concepts via Gravitational Analogies Page 4 Printed 3:36 PM 5/2/17 Tables 3 & 4: Introductory Analogies Between Gravitational And Electrical Potential Energy and Potential Potential Gravitational Electrical Comments Energy Gravitational potential energy is the Electric potential energy is defined analogously to stored energy a mass can have due to the gravitational potential energy discussed its gravitational attraction to another Electric potential energy can previously in the course. A topographic contour mass and depending on the be found similarly, with more map should also be examined (REF7) and the separation difference. At the Earth's variations possible due to thought experiment of walking a wheelbarrow about surface we find this by assuming a different possible signs of a contour lines or perpendicular to contour lines locally uniform field strength and charge. should be whiteboarded & discussed. Also the path direction: taken by a loose ski or bowling ball free to fall down from a mountain peak on a topo map. PE q E h qV e PEg m g h when E and h are in the same direction when g and h are in the same direction INSERT FIG3G.JPG HERE Potential Gravitational potential (a rarely used term) is the gravitational potential energy per unit mass or for uniform gravitational fields: Vg PEg m mgh m gh where the units are J/kg and assuming a mass displacement parallel to g . We actually coin a term for this we call "liftage," somewhat analogous to "plumbing head" – where a scalar figure expresses where water can flow due to the use of a water tower in a water distribution system: INSERT FIG4E.JPG HERE INSERT FIG3E.JPG HERE Electric potential is defined as electric potential per unit charge, which turns out to be a very practical measure for electric phenomena : Ve PEe q qEh q Eh Electric potential is more commonly known as voltage and for uniform fields: Ve Eh the units of electrical potential are J/C or volts. Again, note the analogy comparing E between two parallel plates and g within a room on the Earth's surface. This relationship and the role of path dependence should be explored and exploited in activities like Arons' homework questions or the Modeling Physics work sheets. Students should try sketching lines of E and V in experimental situations and discuss . When we talk about potential the analogy becomes less useful since we rarely discuss gravitational potential. The idea can be explained simply in terms of “liftage”. Use the example of a water tower that holds the water for a municipality above the level of all the users’ bathrooms. Therefore, the “liftage” is dependent on the height of the water not the mass of the water. In other words, as long as there is water in the tower above the level of the bathroom, there can be water flowing in the bathroom. It doesn’t matter how much you need or how much is in the tower. Teaching Electrical Concepts via Gravitational Analogies Page 5 Using these tabulated analogies: Printed 3:36 PM 5/2/17 We have used this tabulated comparative approach in several instructional contexts, and have found the least successful way for students to learn these ideas is by presenting these complete tables in an early formal lecture, although students are often happiest with such lectures. Rather, we suggest that the ideas be formally presented in tabular form AFTER students have struggled with the individual ideas in discussions, after worksheets and concrete hands-on activities both reviewing gravitational ideas and examining electrical phenomena. While the ideas in these tables could be presented as final formal review notes for the section on introductory electric concepts, though students seem unhappy with delayed presentation of these formalisms. Our preferred balance is to reconstruct the information one table at a time while moving through subject as teacher-led "mini-lecture" summaries from student language and ideas. The idea is to warrant student negotiated meaning and ideas while moving towards standardized language and formalisms when enough experience and discourse has been completed when students are ready for the formalism and require reassurance. Hence each of these four tables usually appear in our student notes as summary interludes amongst other activities, although we have tried constructing these kind of tables in the back pages of student notebooks as well. Formalizing pieces only at intervals when students are ready for such (and are requesting such) works well for us. First, the instructor should review and enlarge notation for the universal gravitation relationship. Particularly review calculation skills and examples using scientific notation in calculators. As Hewitt points out in Conceptual Physics, this is a good time to have students find and compare the gravitational attraction between an electron and a proton with the gravitational attraction. Since this would be just for review, students should be able to complete this task quickly and with very little guidance. Next, students can calculate the electrical attraction between these same two particles. Although this calculation is new they should have no difficulty because of the similarities to the gravitation equation. Take some time to discuss the results. These two exercises should be done together within one presentation in order to highlight the similarities. As with any analogies, it is important to point out any differences to help to mitigate confusion and in some cases to enhance the discussion. For example, the comparison between the G and k allows deeper understanding of the relationship between Fg and Fe. In the next table, is the relationship between gravitational field strength and electric field strength. Students have difficulty seeing that “little g” is a measurement of field strength. At this point, a useful exercise would be to show how m/sec2 represents the same units as N/kg. Until students have the understanding that g , the acceleration due to gravity, is a way of representing the field strength, the analogy is too weak. Besides, time spent with units is time well spent. The third table deals with potential energy. This is the time to show that all energy is the same. This allows the instructor to review the concept of energy. Finally, the last table introduces electric potential. Without some development of the concept of gravitational potential the analogy with electric potential will be useless. In order to be successful students will need to visualize how water gets into their upstairs bathroom without a pump. A simple diagram of the local water tower next to “their” house should help them to see that the water coming from the tower has the potential to return to the same height and that this idea does not depend on the mass of the water. Although it is the analogies that have been the focus of this paper, this visual aid is crucial since the idea of “liftage” is not a concept most teachers spend time teaching. As they say “a picture is worth a thousand words.” When this idea is accepted, the concept of electric potential can be introduced. This approach works well not only to give students a more concrete way of learning about these electrical concepts, but also to reinforce the gravitational concepts. Our time is so limited with each individual topic there is very little opportunity for repetition. There are numerous benefits when teachers use opportunities, like this, to overlap units or refer to previously discussed material. Teaching Electrical Concepts via Gravitational Analogies Page 6 References: Printed 3:36 PM 5/2/17 REF1. D. Hestenes, G. Swackhamer & L. Dukerich, Modeling Physics Curriculum (ASU, Tempe AZ, 2004). <http://modeling.asu.edu/Curriculum.html>. See second semester (electricity & magnetism) curricular materials (worksheets and activities), particularly Units E1: Charge and Field and E2: Potential. A freely reproducible HS curricular resource based on Physics Education Research (PER) touchstone curricular activities. REF2. P. Laws, The Workshop Physics Activity Guide Module 4: Electricity & Magnetism (Wiley, Hoboken NJ, 2004). For a comparative development see especially Unit 21: Electrical and Gravitational Energy, p. 573-593. REF3. For an outstanding vignette demonstrating advance student discourse in the introductory physics class, see D. Desbien, M. Joshua & K. Falconer, RTOP4 Video 4. (SUNY-BSC, Buffalo NY, 2003). Streamed QuickTime video available from <rtsp://media1.buffalostate.edu/physics/rtop4.mov>. REF4. R. Knight., Physics For Scientists and Engineers with Modern Physics: A Strategic Approach. (Pearson Addison Wesley, San Francisco, 2004). P. 808 develops the r-hat notation. The workbook is an excellent source of PER-inspired activities. REF5. R. Chabay & B Sherwood. Matter & Interactions VolII: Electric and Magnetic Interactions. (Wiley, Hoboken NJ, 2002). Section 14.8 is a case study of sparks in air, and this is an outstanding PER-influenced curriculum for studying electricity and magnetism. REF6. A. Arons, Teaching Introductory Physics (Wiley, Hoboken NJ, 1996). See problem 5.17, p76 of Volume II. REF7. B.S. Andreck, "Using Contour Maps to Teach Electric Field and Potential," Phys. Teach. 28, 499 (Oct, 1989). Arons, A.B. (1990). Teaching Introductory Physics, Part II, problem 5.17, p 76. Reading NY: John Wiley & Sons. Hestenes, D. Swackhamer, G, & Dukerich, L. (2004) Modeling Physics Curriculum. http://modeling.asu.edu/Curriculum.html. See second semester (electricity & magnetism) curricular materials (worksheets and activities), particularly Units E1: Charge and Field, and E2: Potential. Outstanding freely-reproducible HS curricular resource. LARRY CAN THESE UNITS BE UNLOCKED ON THE MODELING SITE??? Laws, P.W. (2004). The Workshop Physics Activity Guide, Module 4. Reading NY: John Wiley & Sons. See particularly Unit 21: Electrical and Gravitational Energy, p. 573-593. Chabay, R. & Sherwood, B. (2002). Matter & Interactions Vol II: Electric and Magnetic Interactions. Reading NY: John Wiley & Sons. See p484-495 for a case study of sparks in air. An outstanding resource for activities teaching introductory electric concepts including an intelligible introduction to the use of calculus in electrostatics. Hewitt?????? MacIsaac, Dan, Assistant Professor of Physics, SUNY- Buffalo State College. Hewitt, Paul G., Conceptual Physics, Prentice-Hall Inc., 2002, p. 507. Burgholzer, John Physics Teacher Amherst High School. Teaching Electrical Concepts via Gravitational Analogies Page 7 Fig1G: QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Printed 3:36 PM 5/2/17 Teaching Electrical Concepts via Gravitational Analogies Page 8 Fig1E: QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Printed 3:36 PM 5/2/17 Teaching Electrical Concepts via Gravitational Analogies Page 9 Fig2G: QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Printed 3:36 PM 5/2/17 Teaching Electrical Concepts via Gravitational Analogies Page 10 Fig2E: QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Printed 3:36 PM 5/2/17 Teaching Electrical Concepts via Gravitational Analogies Page 11 Fig3G: Printed 3:36 PM 5/2/17 QuickTime™ and a TIFF (U ncompressed) decompressor are needed to see this picture. Teaching Electrical Concepts via Gravitational Analogies Page 12 Fig3E: Printed 3:36 PM 5/2/17 QuickTime™ and a TIFF (U ncompressed) decompressor are needed to see this picture. Fig4G: Teaching Electrical Concepts via Gravitational Analogies Page 13 QuickTime™ and a TIFF (U ncompressed) decompressor are needed to see this picture. Printed 3:36 PM 5/2/17