Student Use of Geometric Reasoning in Upper-Division E&M Problems Context A Sequence of 5 Activities In a Junior-Level E&M course Students are asked to find: 1 The electric potential V on an axis due to 2 point charges 2 The electric potential V in all space due to a charged ring 3 The electric field in all space due to a charged ring 4 The magnetic vector potential, in all space, due to a spinning charged ring Radius = R Charge = Q Period = T What's Right with : Student Understanding of Vectors and Scalars Students have correctly remembered a relationship for finding the unit vector in a given direction. However, some students try to apply this to, not realizing that that the absence of vector makes this relationship inapplicable to. Students Who Check Dimensions Don’t Settle on Wrong Answers Do Your Students Know a Variable from a Constant? It’s Harder Than You Think. Derek, points to 2πR/T and says, "This is angular velocity, right?" Ryan, "Wait, this is a velocity, because there's distance per time." Bob, "Right, but it's angular velocity..." Ryan, "The units still don't work out though." Derek, points to 2πR/T and says, "This is tangential velocity. It would have to be divided by 2π to be ω, right?" Ryan, "Divided by R, because you want, like, radians per second." Derek, "Yes." (nods) Ryan, "The 2π/T would get us ω." 1-Minute Discussion of Dimensions Allen, referring to current, "So,...will it just be λ over T?" Tom, "λ over T? No, I don't think so." Allen, pointing to λ/T, "Ya' know, 'cause there's our length, there's our time....Yeah and there's where circumference would come in, so that's got to be right." Laura, "Wait, wouldn't Q pass through in time T?” Tom, "It should be all of Q." Allen, "So'd be all the Q's coming around." Laura, "Yeah, Q/T" 28-Second Discussion of Physical Meaning Student, Not Checking, Propagates Error Throughout Problem In Middle School and High School 3x + 2 = 17 A “variable” is often a specific unknown number to be “solved for.” Constants are numbers like π. In Intro Physics Current can be a volume or surface densityCurrent is a changing vector quantity Current in the Upper-Division: It’s Not Just a Scalar Any More In Intro Physics Courses:  Current in a circuit is treated as a scalar  Students use the “right hand rule” to find directions of magnetic forces and fields  With solenoids, the formula B = μ o IN/L, allows current to be treated as a scalar In Upper-Division Physics: Students often try to “pull out” current from the integral, unaware that the direction of the current is essential during integration. It takes students time to understand why a volume density has dimensions of 1/area instead of 1/volume. Current densities are a “different beast” than the mass densities with which students are most familiar. Confusing with Choosing the right drawing for Students who have not made a proper drawing, recognize that the vector forms an angle with the axis and confuse with. 5 The magnetic field, in all space, due to a spinning charged ring In Upper-Division Physics In this equation, what is a variable and what is a constant?  Students know universal constants such as G and µ o.  Students start to see equations as proportional relationships.  In a problem with a rocket launching to the moon using F = GmM e /r 2, students may not recognize that M e is constant while m and r are not.  For the equation above, students may see a sea of variables and not recognize that only terms with are “variable” during integration. This often makes it difficult for them know when they are “done” when trying to make an elliptic integral. Note that after integration, r,, and z become the relevant “variables.”  Students must deal with problems where vectors are changing while retaining a constant magnitude. For a ring of constant radius, students may assume that the position vector from the origin to the ring is also constant. This material is based upon work supported by the National Science Foundation under DUE Grant Nos , , , , Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF) Len Cerny and Corinne Manogue This work is part of the Paradigms in Physics project at Oregon State University. More information, including instructor’s guides for many of our activities, are available on our website: Acknowledgements We would like to thank members of the Paradigms team, especially Elizabeth Gire, Emily van Zee, and Janet Tate When drawing the picture above, students who draw the picture with angle equal to 0 o, 45 o, or 90 o, often make errors. Students using 0 o or 90 o, often make sign errors. Students using 45 o, often confuse sine and cosine. Drawing a small angle often works best.