(a) Define a coordinate system along the perpendicular bisector of the rod with the +x axis through points A and B as shown below and the origin at the center of the rod.
Potential difference along a path from point A to point B.
The magnitude of the electric field due to the rod, for points near the rod ( ), along the perpendicular bisector of the rod is
which decreases from point A to point B, as shown above. Since we've defined the axis to be perpendicular to the rod with the origin at the rod, then
The potential difference along a path from point A to point B is
Because the electric field decreases from point A to point B, it is not constant, thus an integral is required. For a straight-line path from point A to point B, . The dot product simplifies to since and are both in the +x direction. Thus,
Note that to simplify the expression, we used the fact that the difference in natural logs can be written as the log of the ratio of the quantities.
(b) The potential difference from A to C can be calculated along any path. Since we already know the potential difference between A and B, then let's choose a path from A to B and B to C, along the legs of the right triangle shown below.
Potential difference along a path from point A to point C.
Point C is close enough to the perpendicular bisector that the electric field at point C is the same as the electric field at point B. That is, the electric field due to a long rod is independent of the vertical distance from the perpendicular bisector, as long as the point is near to the perpendicular bisector and near to the rod. Thus, from point B to point C, the electric field is constant and to the right.
The potential difference along the path A to B and B to C. We already know that
From B to C, is perpendicular to along the path, thus and . In other words, point C is at the same potential as point B. As a result, the potential difference from A to C is