GENERATION, CONTROL, AND MEASUREMENT 139 



For actual sources of finite size, the inverse-square law can be applied, 

 without integration, with an error of less than 1 per cent when the largest 

 dimension of the source or receiver is not more than one-tenth the dis- 

 tance between the two. This is a useful relation for predicting the irradi- 

 ance obtainable at known distances from finite sources. 



The opposite extreme from a point source is a uniformly distributed 

 source, such as an overcast sky or a large bank of fluorescent lamps with 

 closely spaced tubes. For infinite, uniformly distributed sources, the 

 irradiance is independent of the distance. A parallel beam of radiant 

 energy has similar properties and is approximated by a small source, such 

 as a projection-lamp filament at the focus of a large paraboUc mirror or 

 plano-convex lens. Lamps mounted in reflectors used for general irradi- 

 ation have properties that are intermediate between those of a point 

 source and those of a distributed source. 



LAMBERT COSINE LAW 



The irradiance produced by a parallel beam of radiant flux is usually 

 measured by determining the incident flux per unit of area normal (per- 

 pendicular) to the beam. However, for surfaces that are inclined from 

 the normal by an angle of incidence 6, the irradiance is decreased because 

 the beam is distributed over a larger area, and H = (J/d-) cos d. In 

 regard to sources, the cosine law states that the radiance or brightness 

 of a black body is independent of the direction from which it is observed. 

 For a finite plane source the intensity is independent of the angle at which 

 it is observed, but the projected apparent area and total flux are propor- 

 tional to the cosine of the angle of emittance. The angle of emittance 6 

 is included in the quantities radiance and brightness. 



FRESNEL's law of REFLECTION 



When a beam of radiant energy is incident to a smooth surface, specu- 

 lar reflection occurs such that the angle of incidence is equal to the angle 

 of reflection, both measured from an axis normal to the surface. The 

 proportion reflected from transparent surfaces is determined by the refrac- 

 tive indexes of the substances and the angle of incidence 6, as given by 

 Fresnel's law. For normal incidence {d = 0) the proportion of the energy 

 reflected R from a beam of intensity / is 



/o Kn-i + ni/ 



where h = intensity of incident beam, 

 / = intensity of reflected beam, 

 R = proportion reflected, 



712 = refractive index of transparent substance, and 

 Ui = refractive index of medium from which energy enters. 



