Question

The two halves of the rod in FIGURE EX25.35 are uniformly charged to {Q. What is the electric potential at the point indicated by the dot?

Short answer

The electric potential at the dot point is zero

Step-by-step solution

Theory of electric  potential at  the charge point

The left side has indicted as positive side +Q whose length is $\frac{1}{2}$and the right side is negative -Q whose length is $\frac{1}{2}$.

A the potential energy of a charge is

$V=\frac{1}{\left(4\pi {\epsilon }_0\right)}\left(-\partial q/\sqrt{{\left(d\right)}^2+{\left(x\right)}^2}\right)$due to the negative charge at point P.

$\partial q$has divided in both positive and negative charge based on the direction from the central point of the rod.

$V=\frac{1}{\left(4\pi {\epsilon }_0\right)}\left(\partial q/\sqrt{{\left(d\right)}^2+{\left(x\right)}^2}\right)$due to the positive charge at point P.

Step 2:  Theory of net charge equation 

Total summation of both charge is

$\begin{array}{rcl}V& =& V_{+}+V_{-}=\left[\frac{1}{\left(4\pi {\epsilon }_0\right)}\left(-\partial q/\sqrt{{\left(d\right)}^2+{\left(x\right)}^2}\right)\right]+\left[V=\frac{1}{\left(4\pi {\epsilon }_0\right)}\left(\partial q/\sqrt{{\left(d\right)}^2+{\left(x\right)}^2}\right)\right]\\ & =& 0\end{array}$

The electric potential at the P point is zero volt