Two Point Charges

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Section 4.5 Two Point Charges

Start by writing down a formula for the electrostatic potential $V(\rr)=V(x,y,z)$ everywhere in space due to a single point charge that is not located at the origin. Then:

Activity 4.5.1.

Find the electrostatic potential everywhere in space for the following two cases:

Two charges $+Q$ and $+Q$ are placed on a line at $z=+D$ and $z=-D\text{,}$ respectively.
Two charges $+Q$ and $-Q$ are placed on a line at $z=+D$ and $z=-D\text{,}$ respectively.

Hint.

You may be tempted to use the iconic equation for the electrostatic potential to give an answer along the lines of

\begin{equation*} V=k\left(\frac{q_1}{r_1}+\frac{q_2}{r_2}\right) \end{equation*}

But this expression doesn't provide enough information to describe $r_1$ and $r_2\text{.}$ This is a good example of the need to “unpack” the iconic equation. Your final answers should explicitly involve $x,y,z$ and $D\text{.}$

Activity 4.5.2.

Simplify your expressions, assuming that you are evaluating the potential along the $z$-axis. Then do the same, this time assuming you are on the $x$-axis.

Hint.

In some cases, the contributions from the two terms cancel. How does this cancellation relate to the symmetries of the problem?