## Activity: 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.
• Media
• 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 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 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.
• assignment Electric Field and Charge

assignment Homework

##### Electric Field and Charge
divergence charge density Maxwell's equations electric field AIMS Maxwell AIMS 21 Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
2. Find a formula for the charge density that creates this electric field.
3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
• 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 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 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 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 Gravitational Field and Mass

assignment Homework

##### Gravitational Field and Mass
AIMS Maxwell AIMS 21

The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}

This problem explores the consequences of the divergence theorem for this shell.

1. Using the given value of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: $r<a$, $a<r<b$, and $r>b$.
2. Discuss the physical meaning of the divergence in this particular example.
3. For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius $Q$, where $a<Q<b$. ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
4. Discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

• 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.
• 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.

## Charged Sphere

Each prompt will have the goal of the prompt and additional questions you may want to ask each group (or the whole class) as they work.

The horizontal direction of the surface represents distance, r in this case. The surface of the sphere is marked by an indent in the surface near the lowest corner. The height of the surface corresponds to the value of electric potential. Read the introduction below to the class before beginning. In the last 5 minutes of class, pass out the Post-Activity Evaluation.

Important things to note (you may choose to put this on the board): The surface and the contour map represent the SAME system and the SAME variables (distance and electric potental). Electric potential goes to 0 when r goes to infinity.

Your group has a plastic surface and a contour map that represent the electric potential due to a charged sphere as a function of position. The electric potential is zero infinitely far away from the sphere. Solve the following problems together and discuss the results.

You are employed by an electronics company called Crabapple Technologies. Crabapple wants to put a small charged nanoprobe at the blue circle.

1. If the nanoprobe moves further away from the sphere, how will the electric potential change? What if the nanoprobe moves closer to the sphere?

Goal: General trend of electric potential. How does electric potential change with respect to distance?

Additional Questions: Prompt students to look at the continuous change of the potential (i.e. not just picking one point and comparing the two values) or have them choose several points to compare.

2. Optional if first surface activity Identify other points on the surface where the electric potential is the same as the potential at the blue circle and draw a line to connect them. Do the same for the orange star and the green square.

1. Align your surface with the contour map. How are you making your alignment?
2. How could the nanoprobe move so that the electric potential remains constant?

Goal: Introducing contour lines - places where the value of potential is the same. What do contour lines represent? Explicit combination of representations.

Guide: How can you tell the value of potential? What feature of the surface represents potential? Find one spot where the potential is the same as at the blue circle and draw a dot. Do this again for several other points and connect them with a smooth line. What do these lines mean? Why are they drawn on the map? What information do the contour lines give you? How could you relate this to the lines you just drew on your surface? What main features could you use to align the surface to the map?

3. Sketch a graph of the potential $V$ vs. distance from the center of the sphere $r$. Remember to label your axes.
2. Why is it reasonable to represent the information from the surface in a graph with only 2 axes?

Goal: Visualize the function V(r)

Guide: What are you graphing? What axes should you plot? What will the axes look like? Where is potential zero? How does potential change with respect to distance? How could you represent that on your graph? Remember that height represents the value of electric potential.

Additional Questions/Comments: Note where students are putting V=0. Ask them to label it on their graph. If students have trouble with the graph inside of the sphere, ask them "do you need to think about inside the sphere if we are talking about a probe outside the sphere?" If they want to graph within the sphere, they can use the surface to determine what the graph should look like.

4. Indicate the direction of the field at the blue circle on the contour map. Explain your reasoning.

Goal: Start the students with something intuitive. They should know Coulomb force points along the line between charged objects, and can use that to determine electric field direction. The students will have to infer that the sphere is negatively charged.

Possible instructor question: Is the sphere positively or negatively charge? How can you tell?

Guide: Locate the point on the contour map. What direction is the Coulomb force in? What is the qualitative magnitude (i.e. is the force very very small? very, very big? somewhere in between?)? Remember that this is a sketch of the vector, so the magnitude does not need to be exact.

5. Locate a point where you would expect the electric field to be larger. How do you know it's larger?

Goal: Compare field at two different points. Begin to connect field to change in electric potential.

Guide: In what direction is the field at the blue circle? Does the field change depending on the position of the nanoprobe? Does the electric field get smaller or larger as the nanoprobe moves closer to the sphere?

Additional questions: In what way is the change in potential different at this point?

Note: Students may locate a point on one representation (graph, contour map, surface) - have them locate the point on all representations.

6. Is the rate of change of electric potential with respect to r positive, negative, or zero?
7. Compare $\frac{dV}{dr}$ at the two points from (4) and (5). Which one has a larger magnitude?

Goal: Introduction to relating potential to field. We already talked about the field, now we will examine the change in potential energy. Compare $\frac{dV}{dr}$ and begin to make a connection between magnitude of field and magnitude of $\frac{dV}{dr}$.

Guide: Rate of change is represented by the slope.

8. Sketch a graph of the electric field vs. distance from the center of the charged sphere. Remember to label your axes.

Goal: Visualize the function E(r)

Guide: What are you graphing? What axes should you plot? How does field change with respect to distance? How could you represent that on your graph? Is field positive or negative? Note that if students make the field positive, they are plotting the magnitude of the field. The component of the field is negative since it points opposite to increasing r.

9. The Crabapple employee handbook states:
There is a relationship between electric potential and electric field; field is the negative gradient of potential. $\vec{E}(\vec{r}) = -\vec{\nabla} V(\vec{r})$ Do you agree? Support your answer with evidence from this activity.

Goal: Develop a relationship between potential and field based on knowledge gained in this activity.

Guide: Direct students to their answers from (4), (5), (6). Does the magnitude of the field seem to change w.r.t. a change in potential? Which direction is force in?

Keywords
E&M Introductory Physics Electric Potential Electric Field
Learning Outcomes