Conservative Fields

Prerequisites

Students should be able to:

  • Find $d\vec{r}$ for a straight or curved path.
  • Evaluate the dot product between two vectors qualitatively.
  • Read vector field maps.
  • Use chop-multiply-add ideas to construct an integral.
  • Estimate integrals by approximation.

Reading: GVC § Independence of PathFinding potential Functions

In-class Content

  • Counting Paths (SGA - 30 min)
  • Lecture - 20 min
    • What is the difference between the line integral and the flux integral (given that both use the dot product)?
    • What does it mean for a field to be conservative?
      • Path independent
      • Corresponds to a surface (scalar field) that it is the gradient of
      • Is curl free Note: Curl is needed for 3 different ideas in this class: Conservative Work, Going from $\vec A$ to $\vec B$, and getting from $\vec B$ to $\vec J$.
  • QUIZ
  • Work (SGA - 30 min)
  • Lecture - 10 min

Optional In-class Content

  • Murder Mystery Method (SGA - 50 min) (alternatively a homework problem - see below)
  • Conservative Fields (lecture) (Math 3.5: Independence of Path, Math 3.6: Conservative Vector Fields, Math 3.7: Finding Potential Functions)
  • Equivalent Statements (lecture)

Homework for Static Fields

  1. (PathIndependence) Students explicitly compute the work required to bring a charge from infinity using two different paths. Include part (e) as an additional path that cannot be solved by integration. (You must invoke path-independence of conservative fields.)

  2. (MurderMystery) This question is adapted from the in-class activity


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