## Activity: Work By An Electric Field (Contour Map)

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.
What students learn
• Students practice doing a vector line integral of an electric field.
• Students match the electric field to a surface representing the potential and see that a positive line integral of the electric field corresponds to a drop in the potential (a negative change) $\int \vec{E} \cdot \vec{dr} = -\Delta V$
• Media

Estimate Work on A Path: The attached vector field represents an electric field. Draw a path starting at the red star and ending at the purple triangle. Estimate the work done on a test charge by the field along the path you drew. The longest vector shown on the vector field has a magnitude of $15$ electric field units.

Prep: Change the beginning and end points to vary the level of difficulty of the integration as needed.

1. Chop, Multiply, & Add (what we most want): Select a path tangent to the electric field, chop up the path, estimate the distance between points and magnitudes of electric fields, multiply, and add.

2. Estimation with Average Value: Estimate an average electric field along the path, estimate the length of the path, and multiply.

3. Potential Pitfall: Students may incorrectly find the change in potential by saying that the lengths of the arrows are the values of the potential.

Potential Pitfall: Students may not consider units. This is a good place to discuss how distance on the vector map can mean physical distance or electric field magnitude.

Discussion: Units We have intentionally left off SI units for electric field. For electric field, students can either use N/C, which helps students to think about the electric field as force per charge, or V/m, which helps students to think about the electric field as the gradient of the potential. Discussing both is useful.

Path Independence: Some students have trouble distinguishing the path independence of the work done by a conservative field and the path independence of the total displacement. These students will argue that the work done on two different paths is the same because the total displacement is the same. It is nice to have a non-conservative field on hand to give to these students to demonstrate that, although the displacement is path independent, the work is not always path independent.

Relate Representations: How is the work done by the electric field related to the surface?

Discussion: Gradient and Slope This is a good place to talk about how the electric field is the gradient(slope) of the surface. You're multiplying the slope by a “run” to get a “rise”.

Discussion: Most students know that electric field vectors point toward negative charges and that the negative charge lies at the bottom of a potential well. However, most students are really surprised by the negative sign in the relationship:$\int \vec{E} \cdot \vec{dr} = -\Delta V$. This is the big bang for this activity.

SUMMARY PAGE

What Students Learn:

• Students practice doing a vector line integral of an electric field.
• Students match the electric field to a surface representing the potential and see that a positive line integral of the electric field corresponds to a drop in the potential (a negative change) $\int \vec{E} \cdot \vec{dr} = -\Delta V$

Time Estimate: 20 min

Introduction

• Students should have practice with doing vector line integrals. We suggest the Vector Integrals activity as a warmup.
• Students should know that the electric field in the negative gradient of the electric potential.

Whole Class Discussion / Wrap Up:

• Emphasize the Chop, Multiply, Add approach to doing the vector line integral. Allow several groups to share how they performed the integral.
• Talk about how since the electric field is the gradient (slope) of the potential, the line integral corresponds to multiplying the slope by the “run” to get the “rise”.
• But there is an overall negative sign, so a positive slope corresponds to a “drop”.

• group Number of Paths

group Small Group Activity

30 min.

##### Number of Paths

Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.
• group Vector Integrals (Contour Map)

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

• group Electric Field of Two Charged Plates

group Small Group Activity

30 min.

##### Electric Field of Two Charged Plates
• Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
• Students should know that
1. objects with like charge repel and opposite charge attract,
2. object tend to move toward lower energy configurations
3. The potential energy of a charged particle is related to its charge: $U=qV$
4. The force on a charged particle is related to its charge: $\vec{F}=q\vec{E}$
• group Electric Potential of Two Charged Plates

group Small Group Activity

30 min.

##### Electric Potential of Two Charged Plates
Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.
• assignment Line Sources Using Coulomb's Law

assignment Homework

##### Line Sources Using Coulomb's Law
Static Fields 2022 (5 years)
1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance $s$ straight out from the midpoint, starting from Coulomb's Law.
2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.
• assignment Electric Field from a Rod

assignment Homework

##### Electric Field from a Rod
Static Fields 2022 (4 years) Consider a thin charged rod of length $L$ standing along the $z$-axis with the bottom end on the $xy$-plane. The charge density $\lambda$ is constant. Find the electric field at the point $(0,0,2L)$.
• group Equipotential Surfaces

group Small Group Activity

120 min.

##### Equipotential Surfaces

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
• group Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 2022 (8 years)

Integration Sequence

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

• group Charged Sphere

group Small Group Activity

30 min.

##### Charged Sphere

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
• assignment_ind Vector Differential--Rectangular

assignment_ind Small White Board Question

10 min.

##### Vector Differential--Rectangular
Static Fields 2022 (7 years)

Integration Sequence

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

Keywords
E&M Path integrals
Learning Outcomes