## Activity: 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.
• group Small Group Activity schedule 30 min. build Yellow plastic surface, Big Whiteboard (1 per group), Contour map for a parallel plate capacitor in a plastic sleeve (1 per group), Ruler (1 per group), Dry-erase pens and erasers (1 each per student), Student handout (1 per student) description Student handout (PDF)
What students learn
• How to interpret a surface plot (plastic surface model) of the potential due to a parallel plate capacitor
• The potential near the center of two charged plates is a plane (constant slope)
• Equipotential curves represent a single value of potential
• Equipotential curves for the center of two charged plates are equally spaced horizontally
• The height of zero potential is arbitrary/chosen for convenience
• Media

Before you is a plastic surface representing the electric potential between two charged plates. A $1~cm$ height difference corresponds to an electric potential difference of $1~V$.

Interpret the Surface
Rank the three points by the value of the electric potential from greatest to least.

This question is very straightforward and intuitive for students.

Mark three points on the surface that are separated by equal (non-zero!) changes in potential.

Points should represent equal changes in height. They need not be along a straight line!

Identify (and mark) all the points with the same value of potential as your three points. What patterns do you notice?

Answer: Students should notice that the points with the same potential are straight lines that are equally spaced horizontally.
Extension: What shape of surface would yield lines not equally spaced horizontally? Curved?

Connect Representations
Align your surface with your contour map. How are you making this alignment? Where is the surface a good approximation of the potential? Where is the approximation less good?

Answer: The surface should be in the middle of the capacitor with the lower end of the surface at the negative plate.

Discussion: How did you align the surface to the contour map? Ask students to share their alignment strategies:

• Matching Contours: Each contour represents a single value of potential. The surface should go in the region where the contours are parallel and equally spaced on the contour map.
• Negative Plate: Students might know or have intuition that the lowest potential will be aligned with the negative plate.
Extension: Is the value of the potential negative at the location of the negatively charged plate?

Generate a Graph
Sketch a graph of the potential ($V$) vs. distance from the negative plate ($x$).

• Describe the relationship between potential and distance from the negative plate.
• Propose an equation to describe the electric potential as a function of the distance, $x$, from the negative plate. Where did choose for the location of $V=0$?

Extension: Write a symbolic equation for the potential.

Extension: What do you think the potential is not negative near the negative plate?

Extension: Why is it reasonable that the curves of constant potential are straight lines?

SUMMARY PAGE

What Students Learn:

• How to interpret a surface plot (plastic surface model) of the potential due to a parallel plate capacitor
• The potential near the center of two charged plates is a plane (constant slope)
• Equipotential curves represent a single value of potential
• Equipotential curves for the center of two charged plates are equally spaced horizontally
• The height of zero potential is arbitrary/chosen for convenience

Time estimate: 15 minutes

Equipment:

• Yellow plastic surface
• Big Whiteboard (1 per group)
• Contour map for a parallel plate capacitor in a plastic sleeve (1 per group)
• Ruler (1 per group)
• Dry-erase pens and erasers (1 each per student)
• Student handout (1 per student)

Introduction

• Try doing this activity BEFORE you talk about the electric field of a parallel plate capacitor and follow up with the “Electric Potential of a Parallel Plate Capacitor” surface activity.
• Students should know that a parallel plate capacitor is two charged planes separated by a distance.
• Students should be familiar with the term “potential” and maybe know the potential due to a point charge (not required but useful for one of the challenge questions).
• NOTE: We chose to have a uniform charge density on the plates rather than make the plates conductors. Therefore, the equipotential curves intersect the plates far from the center.

Whole Class Discussion / Wrap Up:

• Introduce the term “equipotential”
• Have the students propose an equation to describe the electric potential as a function of the distance, x, from the negative plate: $V_{-\rightarrow +} = m\;x$ It's linear! Students will learn later that the slope, $m$, is the magnitude of the electric field.

• 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: $\vec{F}=q\vec{E}$
• assignment Flux through a Paraboloid

assignment Homework

##### Flux through a Paraboloid
Static Fields 2021 (5 years)

Find the upward pointing flux of the electric field $\vec E =E_0\, z\, \hat z$ through the part of the surface $z=-3 s^2 +12$ (cylindrical coordinates) that sits above the $(x, y)$--plane.

• 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.
• group Work By An Electric Field (Contour Map)

group Small Group Activity

30 min.

##### 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.
• group Gravitational Potential Energy

group Small Group Activity

60 min.

##### Gravitational Potential Energy

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.
• group Gravitational Force

group Small Group Activity

30 min.

##### Gravitational Force

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.
• 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 Ideal Gas Model

group Small Group Activity

30 min.

##### Ideal Gas Model

Students consider whether the thermo surfaces reflect the properties of an ideal gas.
• keyboard Electrostatic potential of four point charges

keyboard Computational Activity

120 min.

##### Electrostatic potential of four point charges
Computational Physics Lab II 2023 (2 years)

Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.
• group Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

##### Changes in Internal Energy (Remote)

Warm-Up

Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.

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Learning Outcomes