Electric Potential at a Point along the Axis of a Quadrupole
To determine the electric potential at a point along the axis of a quadrupole, we first need to understand the configuration of the quadrupole. A quadrupole consists of two dipoles lined up in opposite directions.

Electric Potential Due to Individual Dipoles
- Each dipole creates its own electric field, which contributes to the overall electric potential at a point along the axis of the quadrupole.
- The electric potential due to a single dipole decreases as 1/r^2, where r is the distance from the dipole.
- For a dipole aligned along the z-axis, the electric potential at a point z along the axis is given by V = k * p / (z^2 + a^2)^(3/2), where k is the electrostatic constant, p is the dipole moment, z is the distance along the axis, and a is the separation between the charges in the dipole.

Net Electric Potential of the Quadrupole
- Since the two dipoles in a quadrupole are aligned in opposite directions, their electric potentials at a point along the axis will have opposite signs.
- The net electric potential at a point z along the axis of the quadrupole is the sum of the electric potentials due to the two dipoles, taking into account their directions.

Resulting Electric Potential
- When the two dipoles are lined up in opposite directions, the electric potentials at a point along the axis of the quadrupole will partially cancel each other out.
- The resulting electric potential will be the difference between the magnitudes of the electric potentials due to the two dipoles.
- The precise calculation of the electric potential at a point z along the axis of the quadrupole will depend on the specific values of the dipole moments, distances, and other relevant parameters.