## Student handout: 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.
What students learn
• Add the potential due to each charge to calculate the potential due to a collection of charges.
• Equipotential surfaces are 3D surfaces where the potential is a constant value.
• The spacing between equipotential surfaces, by convention, is such that the change in potential is the same for adjacent equipotential surfaces.
• Therefore, close spacing means the potential is changing quickly with distance; wide spacing means the potential is changing slowly.
• Considering equipotential surfaces is only one of many ways to visualize the electric potential in space.
• Inverse square force law means that the potential changes faster closer to the source---far away, the potential changes slowly.

Start with a Simpler Case: The electrostatic potential due to a particle with charge $q$ is: $V(r)=\frac{kq}{r}$

where $k$ is the electrostatic constant and $r$ is the distance from the particle.

On your whiteboard, identify all the points with the same value of potential around a single point charge. Repeat for several different values of potential.

• What shapes have you drawn?
• If you wanted the difference in potential represented by the shapes to be equal, how are they spaced?

Add Complexity: Draw equipotential surfaces for the potential due to 4 particles with equal, positive charge arranged in a square.

Examine a New Case: Repeat for a quadrupole: 2 positively charged particles and 2 negatively charged particles arranged in a square, with “like” charged particles on opposite corners.

Extend to New Surfaces: The red surface represents the potential of a quadrupole in the plane of the charges (at $z=0$ cm). What would the potential look like in the $z=1$ cm plane? What would be different? What about the $z=-1$ cm?

• 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.
• 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 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.
• 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.
• keyboard Electrostatic potential and Electric Field of a square of charge

keyboard Computational Activity

120 min.

##### Electrostatic potential and Electric Field of a square of charge
Computational Physics Lab II 2023 (2 years)

Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.
• 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.
• computer Using Technology to Visualize Potentials

computer Mathematica Activity

30 min.

##### Using Technology to Visualize Potentials
Static Fields 2023 (6 years)

Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrate several different ways of plotting the potential.
• 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 Electric Field Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electric Field Due to a Ring of Charge
Static Fields 2023 (8 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups 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.

• assignment Linear Quadrupole (w/ series)

assignment Homework

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

Power Series Sequence (E&M)

Static Fields 2023 (6 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.

3. A series of charges arranged in this way is called a linear quadrupole. Why?

Keywords
E&M Quadrupole Scalar Fields
Learning Outcomes