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

Static Fields 2024 (7 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.
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.

Keywords
Electric potential Point Charges Distance Formula
Learning Outcomes