  We investigate the difficulties that students in calculus-based introductory
physics courses have with the concepts of symmetry, electric field, and
electric flux which are important for applying Gauss's law. The determination
of the electric field using Gauss's law requires determining the symmetry of a
particular charge distribution and predicting the direction of the electric
field everywhere if a high symmetry exists. Effective application of Gauss's
law implicitly requires understanding the principle of superposition for
electric fields. Helping students learn when Gauss's law can be readily applied
to determine the strength of the electric field, and then helping them learn to
determine the appropriate shape of Gaussian surfaces if sufficient symmetry
exists, can help develop their reasoning and problem-solving skills. We
administered free-response and multiple-choice questions and conducted
interviews with individual students using a think-aloud protocol to elucidate
the difficulties that students have with the concepts of symmetry, electric
field, and electric flux. We also developed a multiple-choice test that targets
these conceptual issues to obtain quantitative information about their
difficulties and administered it to 541 students in the introductory
calculus-based physics courses and to upper-level undergraduates in an
electricity and magnetism course and to graduate students enrolled in a TA
seminar course. We find that undergraduate students have many common
difficulties with these concepts.
