Continuum Radiation

Abstract

The experimentally determined Coulomb force, F, between two static point charges, q ₁ and q ₂ is (Coulomb, 1785)

$$F = \frac{{{q_1}{q_2}}}{{{R^2}}}{n_r}$$

(1-1)

where R is the distance between the charges, and n _r is a unit vector directed from one charge to the other. The static electric field, E, of a point charge q ₁ is defined so that

$$F = {q_2}E,\,where\,E = \frac{{{q_1}}}{{{R^2}}}{n_r}$$

(1-2)

and R is the distance from q ₁. Integrating equation (1-2) over a closed spherical surface we, obtain Gauss’s law

$$\oint\limits_S {E \cdot nds} = 4\pi q = 4\pi \int\limits_V {\rho dv} $$

(1-3)

where ρ is the charge density,$\oint\limits_S {}$ denotes the closed surface integral, E·n is the component of E which is normal to the surface element d s, and $\int\limits_V {\rho dv}$ is the amount of charge within the closed surface. Using the equations of vector analysis, Eq. (1-3) may be expressed as Poisson’s equation (Poisson, 1813)

$$\nabla \cdot E = - {\nabla ^2}\varphi = 4\pi \rho $$

(1-4)

where

$$E = - \nabla \varphi $$

(1-5)

and φ is called the scalar electric potential. For one static point charge, q, we have

$$\varphi = {q \mathord{\left/ {\vphantom {q R}} \right. \kern-\nulldelimiterspace} R}$$

(1-6)

where R is the radial distance from the charge. For an electrostatic dipole consisting of two charges, +q and −q separated by a distance a along the z axis

$$\varphi = a\,q\,\cos {\theta \mathord{\left/ {\vphantom {\theta {{R^2}}}} \right. \kern-\nulldelimiterspace} {{R^2}}}$$

(1-7)

so that

$$\begin{gathered} {E_r} = \frac{{2d}}{{{R^3}}}\cos \theta \,{n_r} \hfill \\ {E_\theta } = \frac{{d\,\sin \theta }}{{{R^3}}}\,{n_\theta } \hfill \\ \end{gathered} $$

(1-8)

and

$${E_\varphi } = 0$$

where the dipole moment d = a q, the angle θ is the angle between the z axis and the radial direction, R is the radial distance from the dipole, and n _θ and n _r are unit vectors in the θ and the radial direction.

“What is light? Since the time of Young and Fresnel we know that it is wave motion. We know the velocity of the waves, we know their lengths, and we know that they are transverse; in short, our knowledge of the geometrical conditions of the motion is complete. A doubt about these things is no longer possible; a refutation of these views is inconceivable to the physicist. The wave theory of light is, from the point of view of human beings, certainty.”

Hertz, 1889

“It is now, I believe, generally admitted that the light which we receive from the clear sky is due in one way or another to small suspended particles which divert the light from its regular course... There seems to be no reason why the color of the compound light thus scattered should not agree with that of the sky... Suppose for distinctness of statement, that the primary ray is vertical, and that the plane of vibration is that of the meridian. The intensity of the light scattered by a small particle is constant, and a maximum for rays which lie in the vertical plane running east and west, while there is no scattered ray along the north and south line. If the primary ray is unpolarized, the light scattered north and south is entirely due to that component which vibrates east and west, and is therefore perfectly polarized.”

Lord Rayleigh, 1871