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.
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.
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 Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
60 min.
Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics
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 Computational Activity
120 min.
electrostatic potential python
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 usingnumpy and matplotlib.
keyboard Computational Activity
120 min.
group Small Group Activity
30 min.
computer Mathematica Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
coulomb's law electric field charge ring symmetry integral power series superposition
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 Homework
Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
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.
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.