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An Extension of the Correspondence Between Statically and Dynamically Determined Force Fields |
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Consider the following two situations in mathematical physics.
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
where s is the distance between point ζ and point Z and F(s) is a vector function. F(s) is general but the most important cases are the ones in which F(s) dependent on the inverse separation distance squared. This distance between Z and ζ is given by |Z−ζ|. The increment of charge is given by ρ(ζ)dR, where dR is a volume element.
The intensity vector G of the field at Z is then given by
The intensity at Z is constant so the expected value of G, E{ }, is just G; i.e.,
On the other hand in the dynamic situation the field intensity at Z fluctuates. When the center of the distributed charge Q is at point ζ the force dH on a unit charge at point Z due to the charge of dq at ξ is given by
The charge density dq at ξ is a function of the distance between ξ and ζ, say σ(|ξ-ζ|). The total force H is then given by the integration over the volume V of the distributed charge. Thus the force at point Z is
H(Z) is a function of time because ζ is a function of time.
Let P(ζ) be the probability density function representing the proportion of the time the distributed charge center spends at point ζ. For a point traveling along a path at a velocity of v the probability density function is given by
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
Note that if the charge density ρ(ζ) is given by [∫_{V}σ(|ξ-ζ|)dV]P(ζ) 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.
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.,
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
The frequency ν is equal to ω/(2π). This works out to be 6.6×10^{15} 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.
Consider a generic charge distribution with a density function σ(s) where s is the distance from the center ζ of the distribution and ζ traverses a periodic path with a spent-time probability density function of P(ζ). This dynamic system has the same expected value for force at a point Z as a static charge distribution density equal to P(ζ)Q where Q is equal to ∫_{V}σ(|ξ-ζ|)dV.
With subatomic particles executing cycles at rates on the order of quadrillions of times per second the observed quantities are equal to their expected values.
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