Field Geometry of Spheres

Electrostatics and Gravity, and Shells and Spheres

The anatomy of the sphere must include density. Density, wether of mass or charge, is innately involved with the "physical" geometry.
Also radius is no longer a constant. Its treatment needs to be a variable for the integrations to proceed.

First, We would like to consider a hollow sphere - or shell...
Of constant density.

Here is the mass of this shell covering the intire sphere. One intuitive,
The other derived.
We choose to leave shell mass in the deferential form because of a convenient integration to obtain total sphere mass made up of successive shells.
Treat this as an identity for now. (I will not integrate it while working in shells.)
The ring shell, as well as the intire shell, has this thickness
(This is a trivial thing. I don't think I will use it.)

 Diagram
We will use this diagram:

An observer at point P.
The observer will be measuring acceleration or force from composit masses in the form of a sphere.

A distance Z from the center of the sphere to the observer at point P.
The center point of the sphere was chosen on a "hunch".
It is a secret for now; later to be revealed and later to be convenient.
Z is a constant.

A distance D to either infinitesimal masses or charges.
The distance D is the operating line for the actual field laws.
Distance D is a variable.

The radius r of the sphere.
At first, r may not seem a variable. But my treatment throughout is one of a variable. Radius r is a variable.

Acceleration g equation:
Follows the inverse square law. Purely a geometrical law, where distribution is the only cause of attenuation. But there I go on a different subject...

The mass in this equation can be considered a single body "point-mass" (of which there is no such thing, but that is another story again.)

The lower case d will become the upper case D when we start to integrate.
This is the accepted generic gravitational equation that we will use as a starting point.

Before we begin, I would like to emphasize one more thing...
Shown is my crude but dramatic aid in vector visualization...
The observer has two large masses abreast. And one mass straight ahead to observe. The two masses to either side are extremely dense and huge in diameter. Yet they do not influence the observers measurement of force or acceleration in any way.
It is like they do not exist!
Force and Acceleration are vectors. And as such, in this example, have their lateral components cancelled.
Superposition explains this with beautiful mathematics; But - psychologically - not to feel the presents of a couple of monstrous bodies can be quite unnerving.

Therefore - and to not use vector algebra - we can, thanks to symmetry, use only cosine acceleration - which is the straight ahead component. The deliberate construction of rings has allowed this symmetry.

Rings

Each incremental circular ring shell has a volume and a mass.

Substituting incremental mass into the acceleration g equation...

We have too many variables...
We should gravitate toward Z. (Sorry for the inappropriate pun.)

...And retaining ANYTHING of Z. It has something to do with that secret I mentioned.

Use the cosine rule to eliminate cos(alpha).
Good trade: we gained a Z, and eliminated cos(alpha).
Don't worry about dr. I am only going to use it later in an identity; and not in any integration.
We can eliminate more...
The derivative of the cosine rule will allow a sin substitution, and eliminating the sin(theta) variable.

A third objective is a better integration variable...
Change integration variable from theta (zero to pi)
to

Here is why I kept the derivative form. It is so easy to integrate the shells into a solid.

Just stack the shells like an onion.

I forgot to mention the secret (if you havent noticed), and why Z was so carefully maintained:
In electrostatics and gravity, and with shells and solid spheres - all mass can be represented by a POINT: the center of mass!! Something interesting happens when we make Z less than r. In other words, take the observer inside the sphere. We have to change the limits, but that is all that is necessary to look inside the sphere shell.

Weightless!

Just because of geometry! Their is NOTHING else at work here. Even inverse-square force and accelaration are due totally to geometry. I believe that Newton himself proved this with just a ruler making a couple of lines in a circle. Newton was obsesed with geometry.

It makes one wonder how many more physical laws are due solely to geometry...
Use the other equation (outside) for all remaining shells (if any) where the observer is outside. Both forms are continuous and compatable.