Lorentz radius

The most important new feature of the Lorentz transformation, absent from the Galilean scheme, is this interdependence of space and time dimensions. At velocities approaching c it is no longer possible to consider the cartesian coordinates of three-dimensional space as being independent of time and the three-dimensional line element da = Jx2 + y2 + z2 is no longer invariant within the new relativity. Suppose a point source located at the origin emits a light wave at time t = 0. The equation of the wave front is that of a sphere, radius r, such that... 

As we know from the discussion of magnetic sectors, an ion of velocity v entering a uniform magnetic field B perpendicular to its direction will move on a circular path by action of the Lorentz force (Chap. 4.3.2), the radius of which is determined by Eq. 4.13 ... 

On physical grounds, relaxation of permanent dipoles is expected to be highly dependent on temperature this is in contrast with Lorentz oscillators, the dielectric behavior of which is relatively insensitive to changes in temperature. Debye (1929, Chap. 5) derived a simple classical expression for the relaxation time of a sphere of radius a in a fluid of viscosity tj ... 

To identify the invariant quantity under Lorentz transformation it is noted that a light wave emitted from a point source at time t = 0 spreads to the surface of a sphere, radius r, such that... 

The reason is that the measuring rod that moves with the rotating circumference suffers Lorentz contraction, but not for measurement along the radius. The only interpretation is that euclidean geometry does not apply to K. ... 

Lorentz calculated EL in the following way. A spherical region within the dielectric, centred on the point X at which EL is required, is selected. The radius is chosen so that, as viewed from X, the region external to the spherical boundary can be regarded as a continuum, whereas within the boundary the discontinuous atomic nature of the dielectric must be taken into consideration. EL can then be written... 

A minor variant of the SI system, called MKS by experimentalists in nonlinear optics, is called SI here and defined below. Alas, there are also other systems "rationalized" or "unrationalized", Heaviside-Lorentz, "atomic" (where me = c = h = 1), and so on. The "rationalized" and "unrationalized" versions differ in how they apportion the pesky factor An (surface area of a sphere with unit radius that is involved in surface integrals) between the various electrical and the magnetic variables. 

PROBLEM 2.7.2. Another application of the Lorentz force is the selection of particles by a mass spectrometer. A dilute plasma of ions, each bearing a charge q and mass m, is accelerated by an electric field to a uniform velocity v, then drifts into a uniform magnetic field B, in which ions of different mass describe circular orbits of different radius r. The mass / charge ratio is given by m/q = rB/v. 

Now let the radius R of the ball Or tend to infinity. The surface integral over a sphere Cr vanishes due to radiation conditions (8.85) and (8.86), and we arrive at a mathematical formulation of the Lorentz lemma ... 

We now can use the Lorentz lemma to derive the reciprocity relations for the Green s electromagnetic tensor. Let us assume that the electric dipoles with moments a and b are located at points with the radius-vectors r and r",... 

The refractive index of the ethane/propane mixture, needed for calculation of the scattering vector, was determined from experimental density measurements and an empirical Lorentz-Lorenz relationship (12). Incorporating the errors from the viscosity and refractive index calculations into the Stokes-Einstein relation results in a maximum error in the hydrodynamic radius of approximately 5%. 

Rl has been deduced for the case of the rotational axis being normal to the primary beam. The common Lorentz formulation is valid if a crystallite is rotated within the beam by co = 27i. For a certain number of crystallites with a rotational radius less than the beam radius this is true. Crystallites outside this radius experience a rotation coeff that is dependent upon their rotational radius and the width of the primary beam. It can be given as ... 

The force and torque exerted on a solid particle were obtained in the form of a power series with respect to RJl, where is the particle radius and I is the distance from the center of the particle to the wall. Lorentz derived an asymptotic expression for the motion of a sphere along the normal... 

The results obtained by Wang and Dhir are consistent with earlier observations by Lorentz et al. [22], who calculated site density from measured superheat values and plotted the site densities as a function of active site radius for organic fluids and for water, respectively. Their results are illustrated in Fig. 15.13. 

FIGURE 15.13 Number density as a function of calculated cavity radius (adapted by Carey [4] from the results of Lorentz et al. [22], with permission of Taylor Francis, Washington, DC. All rights reserved). 

Most properties of synchrotron radiation may be derived starting from classical electrodynamics, where an oscillating dipole is subjected to a Lorentz transformation. Assuming that relativistic electrons move on curved trajectories in a bending magnet of the radius R, the radiated power... 

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