Unit potential point charges emit a spherically expanding potential. Spheres are characterized by their time and space origin from which we can calculate their radius. The velocity of the emitter is also used to calculate the gradient of the potential sphere.

To facilitate our geometry we can use r and it’s related symbol family as first class symbols with well defined meanings.

- r is used for the general geometrical form of r as a variable radius from an emitting unit potential to a receiving unit potential which may be itself.
- Note 1 : there are conditions where a point charge could encounter it’s own potential one or more times at a given instant t. However, each of those would have a different emission point along the path history and therefore a different r.
- Note 2 : r is only defined between path histories. No r exists unless both endpoints are a point charge, including the self.

- r-vector means a specific r with an origin and a point of action where the Dirac sphere intersects a unit potential.
- r-hat means the unit radii with a length of 1 unit of distance which perhaps we can relate to the Planck length, the field speed @, and t-hat, the unit time. We have a lot of flexbility here as it is only convention. It may make sense to bridge to Planck’s constants. We also know that r-hat = @ t-hat, where @ is the speed of the potential.

I’m not quite clear on the level of mathematical sophistication necessary to define the expanding emission from a unit potential. We can examine the slope and rate of change of the slope. If absolute velocity is zero we only have an electric field. It is only when the point charge is moving that we introduce what is called magnetism, but which is really a different shape to what is a scalar field.

- unit negative potential : -1/r with slope 1/r
^{2} - unit positive potential : 1/r with slope -1/r
^{2}

What information do we need to define a vector radius?

- origin : x, y, z along a point charge path
- elapsed time : converts to radius by r = @t
- theta and phi polar coordinates
- Alternately instead of specifying time or radius length, one can specify the spatial coordinate x’, y’, z’ where the expanding potential intersects a unit potential point charge.
- Each specific r is defined by six numbers
- x, y, z, t, theta, phi or,
- x, y, z, r, theta, phi or,
- x, y, z, x’, y’, z’

I am hopeful that this first-class treatment of radius r will be useful in the geometry of unit potential point charge paths and action.

*J Mark Morris : Boston : Massachusetts*