# 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.
• 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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