Position Vectors in 3-d: Student Difficulty With Spherical Unit Vectors in Intermediate E&M Results Brant Hinrichs Drury University, 900 N. Benton Ave., Springfield, MO Introduction • Student difficulties in introductory mechanics have been studied extensively (e.g. see any paper by McDermott, Schaffer, et. al. of the last 15 years) [1] • Even student difficulties in introductory E&M have begun to be studied more extensively • But there are few studies of student difficulties in intermediate E&M • This poster presents results from a study of student difficulties with position vectors and spherical unit vectors in Griffith’s level E&M [2]] • Only one person got the correct answer! • Graduate students did no better than undergraduates • The results don’t seem to depend on text, teacher, year in school, type of school, class size, etc. • Nearly half of all students put answer A 1 • About twenty percent just listed r, θ, φ • Seventy percent (A+correct+other) explicitly included unit vectors in their answers • Half of all student had difficulty determining correct values for the spherical coordinates themselves • Students don’t understand what it means to express a vector as a linear superposition of unit vectors (A 2 , A 3 ) • Students misapply methods from Cartesian coordinates (A 1 ), (i.e. pattern matching) Observations r = ( x, y, z ) = x ˆ x + y ˆ y + z ˆ z r = ( r ,, " ) # r ˆ r + ! ˆ ! + " ˆ " Acknowledgements Miki & Eads Hinrichs for encouragement, Drury for a sabbatical to collect data, and the four faculty who let me into their classes always points radially outward from the origin - changes direction depending on θ and φ - doesn’t point in a constant direction like For example, for point 1, θ=π/2 and φ=0, so While for point 2, θ=π (φ doesn’t matter), so r ˆ z y x ˆ , ˆ , ˆ x z y x r ˆ ˆ cos ˆ 0 sin sin ˆ 0 cos sin ˆ 2 2 2 = + + = ! ! ! z z y x r ˆ ˆ cos ˆ sin sin ˆ cos sin ˆ ! = + + = " # " # " z y x r z y x r r r r ˆ cos ˆ sin sin ˆ cos sin ˆ ˆ 2 2 2 ! " ! " ! + + = + + = = x-axis y-axis z-axis x y z r r ˆ ! ! x z y 1 2 3 4 5 6 Expected Answer: r r ˆ 5 1 = r r ˆ 5 2 = r r ˆ 5 3 = r r ˆ 5 4 = r r ˆ 5 5 = r r ˆ 6 = Explanation Concept Test Please write in terms of and for the following six different points. Show all work. ˆ ! r ˆ r , ˆ ! , • My students –were very competent with spherical coordinates –but still didn’t seem to “get” spherical unit vectors • Goals of intermediate E&M: –understand the math and physics of Maxwell’s Integral equations (Electric & Magnetic) –functionally understand (i.e. be able to set up) 3-d vector integrations to calculate fields & potentials: • Position vectors are ubiquitous in Maxwell’s Integral equations –students should be able to write down for any given non-Cartesian geometry –start with spherically symmetric geometries Motivation E ( r ) = 1 4 !" o # ( $ r ) d $ a r % $ r 2 & r % $ r r % $ r S ’’ V ( r ) = 1 4 !" o # ( $ r ) d $ l r % $ r 2 C & #1) Integrate over source charge distribution (λ, σ, ρ)  modeled as point: - particles (λ) - patches (σ) - chunks (ρ)  pointed to by #2) Find field or potential at the field point pointed to by r ! r r , ! , " ( ) ˆ r , ˆ ! , ˆ " r and ! r r and ! r A hollow cone of radius a, and height h, is centered on the z-axis with its tip at the origin and it’s base in the + z direction. It has uniform surface charge density, σ, on its curved sides, but no charge on its base. Find the electric potential at a point on the z-axis above the cone. Example z y-axis x-axis h a S r r ! r r ! " a d ! z z t poin field r ge char source of patch generic a d ˆ = = = ! ! r = points to patch = ! r ˆ ! r 0 < ! r < a 2 + h 2 0 < ! " < 2# ! $ = tan %1 a h ( ) & ’ ( ( ) ( ( * + ( ( , ( ( ˆ ! r = sin ! " cos ! # ˆ x + sin ! " sin ! # ˆ y + cos ! " ˆ z Typical Incorrect Answers Type A 3 Type A 2 Type A 1 Explicitly include ˆ r , ˆ ! , ˆ " Type C Type B 2 Do not include ˆ r , ˆ ! , ˆ " Type B 1 Data Collection TABLE 1. Schools from which data was collected and details of how the concept test was given at each school. Institution Textbook N a How Given When Given Small private liberal arts college in the upper midwest, PLA Pollack & Stump [3] 12 of 12 As homework for credit After completing both Chp2 (Vector Calculus) in class and relevant homework from Chp2 Small public university in the upper midwest, SP Griffiths 6 of 6 Quiz After completing both lecture on Section 1.4 (curvilinear coordi- nates) and relevant homework Large public university in the southwest, undergraduates LP-ug Griffiths 14 of 26 Volunteers who stayed after class During the last week of a full year of intermediate E&M Large public university in the southwest, graduate students LP-g n/a 14 of 21 Volunteers who stayed after class During the last week of the first year of graduate school, in their quantum course a Indicates how many of the students officially registered for the course actually took the concept test. TABLE 2. Results for each school and composite totals for the concept test shown previously Answer b r, all School N Correct A1 A2 A3 B1 B2 C Other correct c PLA 12 0 6 2 1 2 1 0 0 6 SP 6 0 3 0 1 1 0 1 0 3 LP-ug 14 1 6 3 0 1 2 0 1 6 LP-g 14 0 6 1 1 4 0 2 0 7 46 100% 1 2% 21 46% 6 13% 3 7% 8 17% 3 7% 3 7% 1 2% 22 48% b Number of students at each school who made this kind of error. c Number of students at each school who got all r, θ, and φ values correct for all six points. References [1] L. C. McDermott and E. F. Redish, “Resource Letter: PER-1: Physics Education Research,” Am. J. Phys. 67, 755-767 (1999). [2] D. J. Griffiths, Introduction to Electrodynamics, 3 rd Edition, Upper Saddle River, New Jersey: Prentice Hall, 1999. [3] G. L. Pollack and D. R. Stump, Electromagnetism, 1 st Edition, San Francisco, California: Addison-Wesley, 2002.