Free space equation

On the other hand, Equation (49) can be used in the derivation of the classical Clausius-Mossotti equation (Equation (9)) so that, for an isolated gas molecule in free space, Equation (49) can be written as... 

The solution to (6.1) is of direct importance for modelling planetary motion on a spherical orbit. It is noted that the coefficient of dr becomes infinite on the spherical shell r = 2m and, the solution therefore goes singular at both r = 0 and = 2m, known as the Schwarzschild radius. For any known macroscopic body the radius = 2/cM/c falls well inside the body where free-space equations are not valid and (6.1) does not yield an appropriate solution. 

In these expressions N is the number of dipoles per unit volume, e the electron charge, m the electron mass, T the damping constant, coq the resonance radian frequency of the harmonically bound electron, co the radian frequency of the field, and the permittivity of free space. Equations (12) and (13) are sketched in Fig. 1. The range of frequencies where increases with frequency is referred... 

From the fact that f/conmuites with the operators Pj) h is possible to show that the linear momentum of a molecule in free space must be conserved. First we note that the time-dependent wavefiinction V(t) of a molecule fulfills the time-dependent Schrodinger equation... 

The central equations of electromagnetic theory are elegantly written in the fonn of four coupled equations for the electric and magnetic fields. These are known as Maxwell s equations. In free space, these equations take the fonn ... 

In free space (p = 0, J= 0, <[) = constant), the equations for the potentials decouple and take the following simple fonn ... 

Equation (A 1,6.8). along with the definitions (Al.h.S) and (Al.6.6) constitute the central equation for the propagation of electromagnetic waves in free space. The fonn of section A 1,6.4 admits hamionic solutions of the fonn... 

In order to rmderstand how light can be controlled, we must first review some of tire basic properties of tire electromagnetic field [8], The electromagnetic tlieory of light is governed by tire equations of James Clerk Maxwell. The field phenomena in free space with no sources are described by tire basic set of relationships below ... 

Above we described tire nature of Maxwell s equations in free space in a medium, two more vector fields need to be... 

These new quantities allow us to directly relate properties of tire media to E and H. In essence tliey afford us tire opportunity to quantify tire field-matter interaction. The media response to tire fields is described generally in tenns of tire polarization, P and tire magnetization, M. (We note tliat in free space P and M=Q and we recover equation (C2.15.1 ), equation (C2.15.2 ), equation (C2.15.3 ) and equation (C2.15.4 ) above.)... 

Maxwell s equations can be combined (61) to describe the propagation of light ia free space, yielding the following scalar wave equation ... 

As shown in Chapter 6, these functions arise in all central-force problems, (hat is, systems composed of two interacting spherical objects in free space. The fundamental differential equation involved is... 

In this equation, cfo and q2 are the charges, r is the distance separating them, and e is the dielectric constant, which has a value of 1 for a vacuum or free space. This force law has no directional component 211... 

The polarization vectors vanish in free space, so that in the absence of charge and matter D = e0E, H = —B and the Maxwell equations are ... 

The electromagnetic field in free space is described by the pair of coupled equations... 

The electromagnetic field in free space is described by the electric field vector E and the magnetic field vector H, which in the absence of charges satisfy Maxwell s equations... 

A fluidized bed reactor has a substantial free space above the main level of the catalyst for purpose of disengaging entrainment. In this region plug flow may be assumed to prevail. An overall appropriate model accordingly will consist of well mixed and bypass zones in parallel followed by a plug flow zone. The fraction of flow in bypass is 1-a and the fraction of vessel volume in plug flow is 2. Find the transfer function and equations for the responses to step and impulse inputs of tracer. 

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