A straight rod of length L has a positive charge Qdistributed along its length. Find the electric potential due to the rod at a point located a distance d from one end of the rod along the line extending from the rod.

MODEL:
Model the charges as a simple shape, such as a line or a disk. Assume the charge is uniformly distributed.

VISUALIZE:
For the pictorial representation:
Draw a picture and establish a coordinate system.
Identify the point P at which you want to calculate the electric potential.
Divide the total charge Q into small pieces of charge ΔQ using shapes for which you already know how to determine V. This division is often, but not always, into point charges.
Identify distances that need to be calculated.

SOLVE:
The mathematical representation is V=ΣVi.
Use superposition to form an algebraic expression for the potential at P.
Let the (x, y,z) coordinates remain as variables.
Replace the small charge ΔQ with an equivalent expression involving a charge density and a coordinate, such as dx, that describes the shape of charge ΔQ. This is the critical step in making the transition from a sum to an integral because you need a coordinate to serve as an integration variable.
Express all distances in terms of the coordinates.
Let the sum become an integral. The integration will be over the coordinate variable that is related to ΔQ. The integration limits for this variable will depend on the coordinate system you have chosen. Carry out the integration, and simplify the result.

ASSESS:
Check that your result is consistent with any limits for which you know what the potential should be.
No information is provided on the rod's cross-section. For simplicity, assume that its diameter is much smaller than the rod's length. You can, then, model the rod as a line of charge.

To most efficiently solve this problem, you should divide the rod into pieces of charge that consist of:

a) thin lines of charge of length L and a very small cross-section.
b) thin "slices" of the rod cut parallel to the axis of the rod.
c) thin "slices" of the rod cut perpendicular to the axis of the rod.