Activity: Electrostatic Potential Due to a Pair of Charges (without Series)

Static Fields 2023 (4 years)
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). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.
• group Small Group Activity schedule 30 min. build Medium whiteboards/markers/erasers, balls to represent the charges, voltmeter, coordinate axes on the ceiling description Student handout (PDF)
What students learn
• The superposition principle for the electrostatic potential;
• How to calculate the distance formula $\frac{1}{|\vec{r} - \vec{r}'|}$ for a simple specific geometric situation;
Electrostatic Potential from Two Charges
• Find a formula for the electrostatic potential $V(\vec{r})$ that is valid everywhere in space for the following two physical situations:
• Two charges $+Q$ and $+Q$ placed on a line at $z'=D$ and $z''=-D$.
• Two charges $+Q$ and $-Q$ placed on a line at $z'=D$ and $z''=-D$, respectively.
• Simplify your formulas in the limiting cases of:
• the $x$-axis
• the $z$-axis
• Discuss the relationship between the symmetries of the physical situations and the symmetries of the functions in these limiting cases.
• computer Visualizing Flux through a Cube

computer Computer Simulation

30 min.

Visualizing Flux through a Cube
Static Fields 2023 (6 years) Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.
• assignment Differential Form of Gauss's Law

assignment Homework

Differential Form of Gauss's Law
Static Fields 2023 (6 years)

For an infinitesimally thin cylindrical shell of radius $b$ with uniform surface charge density $\sigma$, the electric field is zero for $s<b$ and $\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s$ for $s > b$. Use the differential form of Gauss' Law to find the charge density everywhere in space.

• assignment Linear Quadrupole (w/ series)

assignment Homework

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?

• assignment Electric Field from a Rod

assignment Homework

Electric Field from a Rod
Static Fields 2023 (5 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 $(0,0,2L)$.
• assignment Linear Quadrupole (w/o series)

assignment Homework

Static Fields 2023 (4 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 $\vec{r}$ on the $x$-axis at a distance $x$ from the center of the quadrupole.

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

• assignment Gauss's Law for a Rod inside a Cube

assignment Homework

Gauss's Law for a Rod inside a Cube
Static Fields 2023 (4 years) Consider a thin charged rod of length $L$ standing along the $z$-axis with the bottom end on the $x,y$-plane. The charge density $\lambda_0$ is constant. Find the total flux of the electric field through a closed cubical surface with sides of length $3L$ centered at the origin.
• group Electrostatic Potential Due to a Ring of Charge

group Small Group Activity

30 min.

Electrostatic Potential Due to a Ring of Charge
Static Fields 2023 (8 years)

Power Series Sequence (E&M)

Warm-Up

Ring Cycle Sequence

Students work in small groups to use the superposition principle $V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}$ to find an integral expression for the electrostatic potential, $V(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $V(\vec{r})$ 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 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)
Static Fields 2023 (6 years)

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 and Charge

assignment Homework

Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2023 (4 years) Consider the electric field $$\vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases}$$
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.
• 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!)

Learning Outcomes