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). Students then evaluate the limiting cases of the potential on the axes of symmetry.
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 \(z''=-D\).
- 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.
It may help to do the The Distance Formula (Star Trek) activity before this one. There is an alternative version of this activity Electrostatic Potential Due to a Pair of Charges (with Series) in which students find series expansions of the potential along the axes of symmetry.
Students typically know the iconic formula for the electrostatic potential of a point charge \(V=\frac{kq}{r}\). We begin this activity with a short lecture/discussion that generalizes this formula in a coordinate independent way to the situation where the source is moved away from the origin to the point \(\vec{r}^{\prime}\), \(V(\vec{r})=\frac{kq}{|\vec{r} - \vec{r}^{\prime}|}\). (A nice warm-up (SWBQ) to lead off the discussion:
Introductory SWBQ Prompt: “Write down the electrostatic potential everywhere in space due to a point charge that is not at the origin.” The lecture should also review the superposition principle. \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0}\sum_{i}\frac{q_i}{|\vec{r} - \vec{r_i}|} \]
This general, coordinate-independent formula should be left on the board for students to consult as they do this activity.