## Student handout: 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 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.

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?

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

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?

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

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

• Explain your reasoning.
• Does your answer depend on the sign of the charge?

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

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?

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

• 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 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 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.
• keyboard Electrostatic potential of spherical shell

keyboard Computational Activity

120 min.

##### Electrostatic potential of spherical shell
Computational Physics Lab II 2022

Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.
• 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.
• 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 Electric Field Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electric Field Due to a Ring of Charge
Static Fields 2022 (7 years)

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

##### Linear Quadrupole (w/ series)

Power Series Sequence (E&M)

Static Fields 2022 (5 years)

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 $\vec{r}$ in the $xy$-plane at a distance $s$ from the center of the quadrupole. The formula for the electrostatic potential $V$ at a point $\vec{r}$ due to a charge $Q$ at the point $\vec{r'}$ is given by: $V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert}$ Electrostatic potentials satisfy the superposition principle.
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.

• 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!)

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Learning Outcomes