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An Equivalence of Statically and Dynamically Determined Fields

Background

Consider the following two situations in mathematical physics.

Suppose the function giving the effect 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 Z and point ζ. This distance is given by |Z−ζ|. The increment of charge is given by ρ(ζ)dR.

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

G(Z) = ∫R F(|Z−ζ|)ρ(ζ)dR

The intensity G may be a scalar, a vector or a tensor. The intensity at Z is constant so the expected value of G, E{ }, is just G; i.e.,

E{G(Z)} = G(Z) = ∫R F(|Z−ζ|)ρ(ζ)dR

On the other hand in the dynamic situation the field intensity at Z fluctuates over time. When the charge Q is at point ζ the effect on a unit charge at point Z is given by

H(t) = F(|Z−ζ|)Q

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

P(ζ) = 1/(T|v(ζ|)

where T is the total time required to traverse the path.

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

E{H(Z)} = ∫R F(|Z−ζ|)QP(ζ)dR

The Correspondence

Note that if the charge density ρ(ζ) is given by P(ζ)Q then the dynamic case problem is mathematically identical 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.

Thus for any dynamic situation of a particle there exists a static distribution of charge having the same expected value of its effect throughout space as the dynamic particle. The observed world is necessarily a world of spatial and temporal averages. At the atomic and subatomic levels the frequencies at which particles execute periodic paths is so high that all that can hoped to be observed are time averages. (The electron in a hydrogen atom executes its orbit at a rate of several quadrillion times per second,)

The preceding states the existence of a static equivalent for a particle executing a periodic path. It is easily generalized to multiple particles. However it is for point particles and point particles possessing a field may have infinite energies. It is essential for the result to be extended to particles with distributed charge, particularly ones with spherically symmetric distributions.

If the effect of the particles being considered is a force that is inversely proportional to distance squared then the force due to a spherically distributed charge on a unit charge at a point in space outside of the distribution is the same as if the charge were concentrated at the center of the distribution. Then the distribution of charge in the equivalent static distribution would be based upon the probability density distribution of the centers of the particles in the dynamic situation. However the validity of the existence of a static equivalent to a dynamic pattern of particles is not just limited to force laws that are inversely proportional to distance squared.

For the general case the equivalent static charge distribution is the sum of the separate effects of the particles in the dynamic situation. For a single particle the distribution of charge is derived from the probability distribution of the center of the charge. Given the location of the center of the charge distribution the distribution of the charge is determined. Spherical symmetry is important because otherwise the orientation of the particle in space would have to be considered. Although it might be tedious to mathematically construct the distribution of charge for the static equivalent there is no doubt that it exists. For example, for a spherical electron rotating around a proton in a hydrogen atom the equivalent static distribution of electrostatic charge would be that of a solid tubular ring.

Conclusions

A particle with a generic charge of Q which traverses a periodic path with a spent-time probability density function of P(ζ) has the same expected value effect as a static charge distribution density equal to P(ζ)Q. This also applies to multiple particle systems. When the charge is spherically distributed a modified version of this also applies.

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