Answer
We can rank the magnitude of the electric potential the rod produces at those three points:
$(a) \gt (b) \gt (c)$
Work Step by Step
We can write the general expression for the electric potential at point P due to a system of charges:
$V = \frac{1}{4\pi~\epsilon_0}~\sum \frac{q_i}{r_i}$
Note that $\sum q_i$ is equal for all three points a, b, and c, since $\sum q_i$ is the total charge on the uniform rod.
We can find the range of distances $r_i$ from each of the three points to the rod:
point a:
The shortest distance from point a to the middle of the rod is $d$
The longest distance from point a to each end of the rod is $\sqrt{(L/2)^2+d^2}$
point b:
The shortest distance from point b to the rod is $d$
The longest distance from point b to the far end of the rod is $\sqrt{L^2+d^2}$
point c:
The shortest distance from point c to the rod is $d$
The longest distance from point c to the far end of the rod is $L+d$
The average distance from a point to the length of the rod is greatest for point c, next greatest for point b, and least for point a.
We can rank the magnitude of the electric potential the rod produces at those three points:
$(a) \gt (b) \gt (c)$
