By Leibniz Gottfried Wilhelm
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Extra resources for A Unitary Principle of Optics, Catoptrics, and Dioptrics
Let us start by considering a point radiator that is small compared to the radiating wavelength, and could be an oscillating free or bound electron. If this radiating particle is sufficiently small, we can represent its density n(r) as a Dirac delta function‡ such that the current density, J (r, t) [Eq. 20) In rectangular coordinates δ(r) is shorthand notation for the product δ(x)δ(y)δ(z), where the delta function has the properties δ(x) = ‡ See Appendix D. 0 ∞ for x = 0 for x = 0 31 32 S O F T X-R A Y S A N D E X T R E M E U L T R A V I O L E T R A D I A T I O N The normalization condition of the delta function is ∞ −∞ δ(x) d x = 1 The delta function has the additional property that 0 for x = a ∞ for x = a δ(x − a) = with the normalization condition ∞ −∞ δ(x − a) d x = 1 This leads to the so-called sifting property ∞ −∞ f (x)δ(x − a) d x = f (a) The transformed current density Jkω for a point source radiator can now be determined for use in Eq.
6), and form the difference of H · [Eq. 2)] − E · [Eq. 1)] to obtain H · (∇ × E) − E · (∇ × H) = −H · ∂D ∂B −E· −E·J ∂t ∂t Recalling the vector identity ∇ · (A × B) = B · (∇ × A) − A · (∇ × B), and using Eqs. 27) is the differential form of Poynting’s theorem. The time derivative terms on the right side of the equation represent the rate of change of energy per unit volume (energy density) stored in the magnetic and electric fields, respectively. The rightmost term represents the rate of energy dissipation per unit volume associated with the current density J.
325. The “corpuscles” (see his p. 197 as well) are what we now call electrons. P. Thomson) the classical theory of scattering is presented in the manner followed by modern texts. 7) can be obtained by the same procedures. Using Eq.