Student Use of Geometric Reasoning in Upper-Division E&M Problems
Len Cerny and Corinne Manogue
Context
A Sequence of 5 Activities
In a Junior-Level
E&M course
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 = μoIN/L,
allows current to be treated as a scalar
 
 
o
I (r ' )dl '
A(r ) 
 

4 ring | r  r ' |
Current can be a volume or surface density
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.
Students often try to “pull out”
current from the integral, unaware
that the direction of the current is
essential during integration.

What's Right with ˆ 
:
 
1-Minute Discussion of Dimensions
Students are asked to find:
1
The electric potential V on an axis
due to 2 point charges
R

T
2
The electric potential V in all space
due to a charged ring
In Upper-Division Physics:
Current is a changing vector quantity
Students Who Check Dimensions
Don’t Settle on Wrong Answers
3 
The electric field E in all space
due to a charged ring
I
J

r

Confusing r̂ with ˆ
r̂

Students who have not
made a proper drawing,
recognize
that the vector

r forms an angle 
with the axis and confuse
r̂ with ˆ .

QT
J

In Middle School
and High School
Students have correctly remembered
a relationship for finding the unit
vector in a given direction.
However, some students try to apply
this to , notˆrealizing that that the
absence of vector makes this
relationship inapplicable to . ˆ
Choosing the right
drawing for ˆ
5

The magnetic field B ,
in all space,
due to a spinning charged ring
3x + 2 = 17
A “variable” is often a
specific unknown number to
be “solved for.” Constants are
numbers like π.
In Intro Physics
 Students know universal
constants such as G and µo.
 Students start to see equations
as proportional relationships.
When drawing the picture above, students
who draw the picture with angle  equal to
0o, 45o, or 90o, often make errors. Students
using 0o or 90o, often make sign errors.
Students using 45o, often confuse sine and
cosine. Drawing a small angle often works
best.
We would like to thank members of the
Paradigms team, especially
Elizabeth Gire, Emily van Zee, and
Janet Tate
Q
 o
A
4
2
RTd 

| r  r' |

Do Your Students Know a
Variable from a Constant?
It’s Harder Than You Think.

r'

v
v̂  
v 
Acknowledgements
T
Q
I
T
Radius = R
Charge = Q
Period = T
Student Understanding of Vectors and Scalars
ˆ

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"
Student, Not Checking, Propagates Error Throughout Problem
The magnetic
vector potential,

A in all space,
due to a spinning charged ring
 
r  r'


T
28-Second Discussion of Physical Meaning
4

r
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."
 In a problem with a rocket
launching to the moon using
F = GmMe/r2, students may not
recognize that Me is constant
while m and r are not.
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:
http://physics.oregonstate.edu/portfolioswiki
In Upper-Division Physics
In this equation, what is a variable
and what is a constant?

o  QR 2  2

 
A(r ,   z) 
4  T  
 sin  ' i  cos  ' jd '
r 2  2rR cos(   ' )  R 2  z 2
 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.
9653250, 0088901, 0231032, 0231194, 0618877.
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)