## Activity: 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 a Parallel Plate Capacitor” 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 Small Group Activity schedule 30 min. build Yellow plastic surface (1 per group), Contour map for parallel plate capacitor with plastic sleeve (1 per group), Big whiteboard (1 per group), Dry-erase markers and erasers (1 each per student), Student handout (1 per student) description Student handout (PDF)
What students learn
• Potential and potential energy are different. The value of potential is independent of the sign of charge of the test particle.
• Force and energy are both ways to understand how charged objects interact.
• Review that electric field and electric potential are related to force and potential energy.
• Electric field vectors are perpendicular to equipotential surfaces and are short if the curves are closely spaced.
• 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 changes plates (near the center of the plates) and explore the properties of the electric potential.
• assignment Electric Field from a Rod

assignment Homework

##### Electric Field from a Rod
AIMS Maxwell AIMS 21 AIMS Maxwell AIMS 21 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 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 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 Line Sources Using Coulomb's Law

assignment Homework

##### Line Sources Using Coulomb's Law
AIMS Maxwell AIMS 21 Static Fields Winter 2021
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.
• group Electric Field Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electric Field Due to a Ring of Charge
AIMS Maxwell AIMS 21 AIMS Maxwell AIMS 21 Static Fields Winter 2021

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use Coulomb's Law $\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the electric field, $\vec{E}(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $\vec{E}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• 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.
• assignment Linear Quadrupole (w/ series)

assignment Homework

Power Series Sequence (E&M)

AIMS Maxwell AIMS 21 Static Fields Winter 2021

Consider a collection of three charges arranged in a line along the $z$-axis: charges $+Q$ at $z=\pm D$ and charge $-2Q$ at $z=0$.

1. Find the electrostatic potential at a point $P$ in the $xy$-plane at a distance $s$ from the center of the quadrupole.
2. Assume $s\gg D$. Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.
3. A series of charges arranged in this way is called a linear quadrupole. Why?

• group Electrostatic Potential Due to a Pair of Charges (with Series)

group Small Group Activity

60 min.

##### Electrostatic Potential Due to a Pair of Charges (with Series)
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.
• assignment Electric Field of a Finite Line

assignment Homework

##### Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
2. Perform the integral to find the $z$-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the $s$-component as well!)

What Students Learn:

• Potential and potential energy are different. The value of potential is independent of the sign of charge of the test particle.
• Force and energy are both ways to understand how charged objects interact.
• Review that electric field and electric potential are related to force and potential energy.
• Electric field vectors are perpendicular to equipotential surfaces and are short if the curves are closely spaced.

Time Estimate: 20 minutes

Equipment:

• Yellow plastic surface (1 per group)
• Contour map for parallel plate capacitor with plastic sleeve (1 per group)
• Big whiteboard (1 per group)
• Dry-erase markers and erasers (1 each per student)
• Student handout (1 per student)

Introduction:

• 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 a Parallel Plate Capacitor” 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: $\vec{F}=q\vec{E}$

Whole Class Discussion / Wrap Up:

• Have students articulate their reasoning how the negative charge will move. Make sure force and energy reasoning are articulated by students.
• Have students propose relations between the electric field vectors they've draw and the equipotential surfaces. The correct orientation (perpendicular) may be difficult to perceive if the students haven't draw the arrows carefully. The relation between the strength/spacing may be easier for students to see.

## Electric Field of Two Charged Plates

Before you is a plastic surface and a contour map each representing the electric potential . A 1 cm height difference corresponds to an electric potential difference of 1 V.

Consider the Motion of a Positive Charge: If you were to place a positively charged particle at rest at the blue square, which way do you expect the particle to move?

1. What direction is the force on the charged particle?
2. Does the charged particle move toward higher or lower electric potential?
3. Does the electric potential energy increase, decrease, or stay the same?

Discussion: This question is trying to get students to articulate a resolution to force and energy descriptions of this situation.

Answer: Toward the negative plate, moves toward lower potential, potential energy decreases, yes they agree.

Consider the Motion of a Negative Charge: If you were to place a negatively charged particle at rest at the blue square, which way do you expect the negative charged particle to move?

1. What direction is the force on the charged particle?
2. Does the charged particle move toward higher or lower electric potential?
3. Does the electric potential energy of the system increase, decrease, or stay the same?

Answer: Toward the positive plate, moves toward higher potential, potential energy decreases, yes they agree because potential is independent of the sign of the charge by potential energy is not.

Student Ideas: Students might conflate potential with potential energy and either think that both will increase or both will decrease. If they think it will increase, ask them to think about energy conservation (the potential energy decreases if the kinetic energy increases and energy is conserved). If they think they will both decrease, remind students that potential doesn't depend on the sign of the test charge, so the surface should not rotate between this case and the previous case.

Consider the Electric Field at the Blue Square: Draw a vector on the contour map to indicate $\vec{E}$ at the blue square.

• How is the vector oriented with respect to the contour lines?

Student Ideas: Students might know that the electric field should be perpendicular to the equipotential lines. If they use this reasoning, try to get them to reconcile it with a superposition approach.

Consider the Electric Field at Several Points: Draw vectors at several additional points to represent $\vec{E}$, making sure the lengths of the vectors are qualitatively accurate. Choose points near the middle and edges of the map.

• How do the electric field vectors near the middle compare with the vectors near the edge of the map?

Answer: the electric field near the edge will be pointing is different directions (not parallel to each other) and will be shorter.

• How are the electric field vectors related to the equipotential lines?

Answer: should be perpendicular. This may not be easy to see if students have not been careful about how they've drawn the vectors.
Student Ideas: Students might know that the electric field should be perpendicular to the equipotential lines. If they use this reasoning, try to get them to reconcile it with a superposition approach.

Keywords
Learning Outcomes