Two Point Charges

Students should be able to:

• Write the electric potential due to a point charge not at the origin.
• Sketch equipotential surfaces due to a point charge.
• Use superposition for scalar fields.
• Use power series to approximate a function.

1. (DeltaPractice)

   You have a charge distribution composed of two point charges: one with charge $+3q$ located at $x=-d$ and the other with charge $-q$ located at $x=+d$.

   1. Sketch the charge distribution.

   2. Write an expression for the \emph{volume} charge density $\rho (\vec{r})$ everywhere in space.

2. (Quadrupole)

   Consider a series 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 $P$ in the $xy$-plane at a distance $s$ from the center of the quadrupole.

   2. Assume $s»D$. 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. Is it possible to find the electric field at $P$ from your answer to the first part of the problem? If you answered that it is possible, find the electric field. If you answered that it is not possible, explain your answer.

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

   5. Briefly describe in words something you learned from doing this problem that you would like to remember for the future. Make your statement using good scientific writing, as you would in a research paper.

3. (WritingI)

   Using the handout “Guiding Questions for Science Writing” to suggest topics that you should address, write up your analysis of the activity entitled Electrostatic Potential From Two Charges. You do not need to do the calculations from every case, but your analysis should include some comparison of different cases, as we discussed in class after the activity. To help us with the grading process, please turn in this writing assignment stapled separately from your other homework.

4. (DeltaPractice2)

   Sketch the volume charge density: $\rho (x,y,z)=c\,\delta (x-3)$