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The Correspondence Between Statically and Dynamically Determined Force Fields with Multiple Particles |
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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 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
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 charge q_{i} is at point ζ_{i} the force H_{i} on a unit charge at point Z due to that charge is given by
The total force H is then given by the summation over the charges. Thus the force at point Z is
H(Z) is a function of time because each ζ_{i} is a function of time.
Let P_{i}(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
where T_{i} 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
Note that if the charge density ρ(ζ) of the static case is given by ∫_{V}σ_{i}q_{i}P_{i}](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.
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 static charge distribution with a density function ρ(ζ) where the total charge is n. Let P_{i}(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 Σ_{i}P_{i}(ζ)q_{i}.
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