﻿ The Correspondence Between Statically and Dynamically Determined Force Fields with Multiple Particles
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
U.S.A.

The Correspondence Between
Statically and Dynamically
Determined Force Fields
with Multiple Particles

## Background

Consider the following two situations in mathematical physics.

## Static and Dynamic Situations

• A generic charge is distributed over some geometric region R and its density is known. The generic charge could be gravitational mass, electrical charge, magnetic charge or nucleonic charge (the charge associated with the nuclear strong force). The generic charge density is given by a function ρ(ζ), where ζ is a position vector. The force vector at any location Z is determined. This is the static situation.
• The centers of n charges of magnitude qi traverse the same geometric region R as above. A probability density function Pi(ζ) for the center of the each charge is determined based on the proportion of the time the center spends at various locations. The expected value of the force at any location can be computed based upon those probability density functions. This is the dynamic situation.

Suppose the function giving the force dF on a unit charge at position Z due to a charge of dq located at position ζ is given by

#### dF = F(s)dq

where s is the distance between point ζZ and point Z and F(s) is a vector function . This distance is given by |Z−ζ|. The increment of charge is given by ρ(Z)dR, where dR is a volume element.

The intensity vector G of the field at Z is then given by

#### G(Z) = ∫R F(|Z−ζ|)ρ(ζ)dR

The intensity at Z is constant so the expected value of G, E{ }, is just G; i.e.,

## The Dynamic Situation

On the other hand in the dynamic situation the field intensity at Z fluctuates. When the center of the charge qi is at point ζi the force Hi on a unit charge at point Z due to that charge is given by

#### Hi = F(|Z−ζi|)qi

The total force H is then given by the summation over the charges. Thus the force at point Z is

#### H(Z) = Σi F(|Z−ζi|)qi

H(Z) is a function of time because each ζi is a function of time.

Let Pi(ζ) be the probability density function representing the proportion of the time the distributed charge center spends at a point ζ. For a point traveling along a path at a velocity of v the probability density function is given by

#### Pi(ζ) = 1/(Ti|vi(ζ|)

where Ti is the total time required for particle i to traverse its path.

The expected value of the vector of field intensity H is then given by

## The Correspondence

Note that if the charge density ρ(ζ) of the static case is given by ∫VσiqiPi](ζ) then the dynamic case problem is mathematically identical in expected value to the static case problem. This correspondence points up the significance of the time-spent probability density function to mechanics. It is as significant as the spatial distribution of any charge.

## Periodicity of Atomic and Subatomic Motion

The rate of rotation of an electron in a hydrogen atom, according to the Bohr model, is easily determined. The electrostatic force between an electron and a proton has to be balanced by the centrifugal force on the electron; i.e.,

#### Ge²/r² = mrω²

where G is a constant, e is the electrical charge of the electron and the proton, m is the mass of the electron and r is the orbit radius for the electron. Therefore

#### ω² = Ge²(m/r³)

The frequency ν is equal to ω/(2π). This works out to be 6.6×1015 times per second; i.e., 6.6 quadrillion times per second. Any observation will involve a time-average and at the above frequency the observation will be equal to the expected value. Thus the observed world is the world of the dynamic appearances of physical systems. In the case of electrons this would be elliptical rings.

## Conclusions

Consider a static charge distribution with a density function ρ(ζ) where the total charge is n. Let Pi(ζ) be the time-spent probability density function for the i-th particle. traversing its periodic path. This dynamic system has the same expected value for force at a point Z as a static charge distribution density equal to ΣiPi(ζ)qi.