Students’ use of vector calculus in an intermediate electrodynamics course Laurens Bollen Supervisor: Mieke De Cock (KU Leuven) Co-supervisor: Paul van Kampen (DCU)

Introduction Goal: Investigate students’ use of vector calculus in an intermediate electrodynamics course Mathematical techniques (vector calculus) from previous courses Physical concepts are known from introductory E&M “ The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” E.P. Wigner, Nobel prize in physics in 1963 → Physics needs maths

Introduction

Outline Introduction Physical and mathematical background Research design Results Conclusions

Physical and mathematical background Introduction Physical and mathematical background Research design Results Conclusions

Physical and mathematical background

Important role in electrodynamics: Maxwell’s equations in differential form (Gauss’s law) (no name) (Faraday’s law) (Maxwell-Ampère’s law)

Research design Introduction Physical and mathematical background Research design Results Conclusions

Research design Electrodynamics course At three universities: KU Leuven, DCU, St Andrews 2 nd bachelor students majoring in physics/mathematics or similar Prior knowledge o A maths course including a chapter on vector calculus o Introductory course on E&M

Research design Evaluation of learning in traditional instruction Student interviews Development, implementation and assessment of tutorials Design of a pre- and post-test that evaluates students’ understanding Document student difficulties Improve insight in student thinking process Keep track of the elements that trigger a correct way of thinking Develop inquiry based worksheets to help students overcome difficulties Implement tutorials in electrodynamics KU Leuven, DCU and St Andrews(?) Evaluate and improve tutorials using results from pre- and post-test

Research design Four skills/competencies we want our students to acquire Structural understanding of vector operators Perform calculations with divergence and curl Interpret graphical representations of vector fields in terms of divergence and curl Conceptually understand Maxwell’s equations in differential form

Results Introduction Physical and mathematical background Research design Results Conclusions

Results Structural understanding of vector operators (first year): pretest KU Leuven and DCU Comparable results for KU Leuven and DCU Only 10-20% gives an conceptual explanation A lot of students’ descriptions were incorrect or inaccurate Focus on evaluation (>50% gave a formula) At least 10% makes an error against the vector/scalar character Interpret (i.e. write down everything you think of when you see) the following operations. “The divergence is a measure for how the field is changing.”

Results Structural understanding of vector operators (second year): Interviews KU Leuven Confirmation of findings in pretests Many students tend to think about divergence as a measure for the spreading of the field, and curl as the bending of the field (dictionary definitions) “The divergence measures how strongly field lines spread apart. When you have a charge that generates an electric field, there will be a strong divergence since the field lines spread apart as you go further from the charge."

Results Structural understanding of vector operators (second & third year): new pretest + post-test

Results Structural understanding of vector operators (second & third year): new pretest + post-test Earlier findings are generalizable Only 0-20% shows good understanding of divergence and 10-40% of curl in all three universities Results get better after instruction, but there is still a lot of room for improvement → Clarify conceptual meaning of divergence and curl

Results Performing calculations Generally, students are fairly well trained in calculating divergence and curl after their first year Results differ from university to university Most errors are linked to setting up the equation (symmetry of physics problems), rather than of algebraic nature → No further intervention needed

Results Interpretation of graphical representations of vector fields Pretest Purely mathematical KU Leuven45%25%20%5% DCU10% 5%0% St Andrews40% 20% Indicate where the divergence/curl is nonzero for the following vector fields in the xy-plane. The z-component is zero everywhere. Show your work.

Results Interpretation of graphical representations of vector fields Pretest Purely mathematical Post-test Mathematical and physical The magnetic field of an infinite current carrying wire along the z-axis The electric field of a charged infinitely long cylinder with radius R.

Results Interpretation of graphical representations of vector fields When students interpret field vector plots They are often inconsistent/unclear about their reasoning Many students use approaches based on descriptions, e.g. “spreading” or “curly” fields Calculations often work, but only in Cartesian coordinates Conceptual approaches (paddlewheel, box-mechanism, integrals) are more efficient, but often misunderstood When a physical context is added, a lot of students prefer their naive thinking above (Maxwell’s) equations

Results Interpretation of graphical representations of vector fields When students interpret field vector plots They are often inconsistent/unclear about their reasoning Many students use approaches based on descriptions, e.g. “spreading” or “curly” fields Calculations often work, but only in Cartesian coordinates Conceptual approaches (paddlewheel, box-mechanism, integrals) are more efficient, but often misunderstood When a physical context is added, a lot of students prefer their naive thinking above (Maxwell’s) equations “If you look at the figure, it seems evident that there is a divergence, so I rather think that there is something wrong with the use of the formula than with my interpretation of the sketched figure.” → More focus on teaching how to interpret field vector plots

Results Interpretation of graphical representations of vector fields When students sketch vector fields During interviews, most participants initially made a field line diagram, but failed to interpret it Students describe the divergence as a measure for spreading field lines and curl as bending field lines Students confuse field line diagrams with field vector plots → Clarify difference field line and field vector diagrams → Teach students to jump between representations

Results Conceptual understanding of Maxwell’s equations

Results Conceptual understanding of Maxwell’s equations All four laws are poorly understood Both on post-test and during interviews, number of correct answers was disappointingly low The local character of the equations is often misinterpreted Students prefer the integral form of Maxwell’s equations, but fail to link it to the differential form About the magnetic field of a current carrying plate: “The curl will be nonzero everywhere here, since the current density J is constant and not zero here [at the location of the plate], and there is no changing electric field. So the curl of the magnetic field must be nonzero." → Focus on conceptual understanding and local character of Maxwell’s equations in differential form

Conclusions Introduction Physical and mathematical background Research design Results Conclusions

Identified difficulties A structural understanding of divergence and curl o Students focus on evaluation o They lack conceptual understanding Doing calculations that involve vector operators o Students are fairly trained in doing calculations The interpretation of div & curl in graphical representations o Students are inconsistent in their problem solving strategies o Divergence as spreading field lines, curl as bending field lines Conceptual understanding of Maxwell’s equations o All 4 laws are poorly understood o Local character is misinterpreted

Conclusions Focus in intervention (tutorials) Conceptual understanding of divergence and curl Link between different representations / solution strategies o Graphical interpretation (using box-mechanism, paddlewheel, integral) o Calculation + interpretation o Direct interpretation of Maxwell’s equations Difference between field line and field vector diagrams Fields with 1/r² (spherical) and 1/r (cylindrical) symmetry Conceptual understanding and local character of Maxwell’s equations in differential form

Thanks for your attention!

Back-up slide: number of students Pretest numbers (typically drop during the year) * Only pretest, no post-test † Implementation of tutorials Interviews (1h) 8 students at KU Leuven that finished the course # studentsKU LeuvenDCUSt Andrews * † 49* † 85 †