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5.1. PHOTOMETRY AND GEOMETRICAL OPTICS
- Spectral response of an eye V 00 is shown in Fig. 5.1.
V 1.
0.
0.
a
Fig. 5.1.
A=1.5melm far ii..-0.555,um
PART FIVE
OPTICS
- Luminous intensity I and illuminance E: dcto. (^) 71C dC2 ' dS •
- Illuminance produced by a point isotropic source:
I cos a E — (^) r 2
where a is the angle between the normal to the surface and the direction to the source.
- Luminosity M and luminance L:
= del)emit^ L— clil M (^) dS , dt1 AS cose • (5.1c)
- For a Lambert source L = const and luminosity
M = aL. (5.1d)
(5.1a)
(5.1b)
- Relation between refractive angle 0 of a prism and least deviation angle a: sin a+0 0 2 — n sin 2 '
where n is the refractive index of the prism.
- Equation of spherical mirror:
s -t--— s =^ R '
where R is the curvature radius of the mirror.
- Equations for aligned optical system (Fig. 5.2):
S,n'^ n^ f'^ xe=fr.
s s s
Fig. 5.2.
- Relations between focal lengths and optical power:
f ' = n'^ f^ f'^71 re— (1) ' W ' f •
- Optical power of a spherical refractive surface:
n' R
n
(5.1i)
- Optical power of a thin lens in a medium with refractive index no:
(n— no) ( -14- 37 , (^) (5.1j)
where n is the refractive index of the lens.
- Optical power of a thick lens:
0 =01+02 — (^) n 0 102, (5.1k)
(5.1e)
(5.1f)
(5.1g)
(5.1h)
where d is the thickness of the lens. This equation is also valid for a system of two thin lenses separated by a medium with refractive index n.
5.4. Determine the luminosity of a surface whose luminance
depends on direction as L = Lo cos 0, where 0 is the angle between
the radiation direction and the normal to the surface. 5.5. A certain luminous surface obeys Lambert's law. Its lumi-
nance is equal to L. Find:
(a) the luminous flux emitted by an element AS of this surface into a cone whose axis is normal to the given element and whose aperture angle is equal to 0; (b) the luminosity of such a source.
5.6. An illuminant shaped as a plane horizontal disc S = 100 cm
in area is suspended over the centre of a round table of radius R
= 1.0 m. Its luminance does not depend on direction and is equal
to L = 1.6.104cd/m2. At what height over the table should the
illuminant be suspended to provide maximum illuminance at the circumference of the table? How great will that illuminance be? The illuminant is assumed to be a point source.
5.7. A point source is suspended at a height h = 1.0 m over the
centre of a round table of radius R = 1.0 m. The luminous intensity I
of the source depends on direction so that illuminance at all points
of the table is the same. Find the function I (0), where 0 is the angle
between the radiation direction and the vertical, as well as the lu-
minous flux reaching the table if I (0) = I, = 100 cd.
5.8. A vertical shaft of light from a projector forms a light spot
S = 100 cm2in area on the ceiling of a round room of radius R =
= 2.0 m. The illuminance of the spot is equal to E = 1000 lx.
The reflection coefficient of the ceiling is equal to p = 0.80. Find the maximum illuminance of the wall produced by the light reflected from the ceiling. The reflection is assumed to obey Lambert's law. 5.9. A luminous dome shaped as a hemisphere rests on a horizon- tal plane. Its luminosity is uniform. Determine the illuminance at
the centre of that plane if its luminance equals L and is independent
of direction. 5.10. A Lambert source has the form of an infinite plane. Its
luminance is equal to L. Find the illuminance of an area element
oriented parallel to the given source. 5.11. An illuminant shaped as a plane horizontal disc of radius
R = 25 cm is suspended over a table at a height h = 75 cm. The
illuminance of the table below the centre of the illuminant is equal to E0 = 70 lx. Assuming the source to obey Lambert's law, find its luminosity. 5.12. A small lamp having the form of a uniformly luminous sphere
of radius R = 6.0 cm is suspended at a height h = 3.0 m above the
floor. The luminance of the lamp is equal to L = 2.0.104cd/m
and is independent of direction. Find the illuminance of the floor directly below the lamp. 5.13. Write the law of reflection of a light beam from a mirror in vector form, using the directing unit vectors e and e' of the inci-
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dent and reflected beams and the unit vector n of the outside normal to the mirror surface. 5.14. Demonstrate that a light beam reflected from three mutually perpendicular plane mirrors in succession reverses its direc- tion. 5.15. At what value of the angle of incident 01is a shaft of light reflected from the surface of water perpendicular to the refracted shaft? 5.16. Two optical media have a plane boundary between them. Suppose Oi„ is the critical angle of incidence of a beam and 01 is the angle of incidence at which the refracted beam is perpendicular to the reflected one (the beam is assumed to come from an optically denser medium). Find the relative refractive index of these media if sin Oler/sin 01= 1 = 1.28. 5.17. A light beam falls upon a plane-parallel glass plate d=6.0 cm in thickness. The angle of incidence is 0 = 60°. Find the value of deflection of the beam which passed through that plate. 5.18. A man standing on the edge of a swimming pool looks at a stone lying on the bottom. The depth of the swimming pool is equal to h. At what distance from the surface of water is the image of the stone formed if the line of vision makes an angle 0 with the normal to the surface? 5.19. Demonstrate that in a prism with small refracting angle 0 the shaft of light deviates through the angle a (n — 1) 0 regard- less of the angle of incidence, provided that the latter is also small. 5.20. A shaft of light passes through a prism with refracting angle 0 and refractive index n. Let a be the diffraction angle of the shaft. Demonstrate that if the shaft of light passes through the prism symmetrically, (a) the angle a is the least; (b) the relationship between the angles a and 0 is defined by Eq. (5.1e). 5.21. The least deflection angle of a certain glass prism is equal to its refracting angle. Find the latter. 5.22. Find the minimum and maximum deflection angles for a light ray passing through a glass prism with refracting angle 0 = 60°. 5.23. A trihedral prism with refracting angle 60° provides the least deflection angle 37° in air. Find the least deflection angle of that prism in water. 5.24. A light ray composed of two monochromatic components passes through a trihedral prism with refracting angle 0 = 60°. Find the angle Da between the components of the ray after its pass- age through the prism if their respective indices of refraction are equal to 1.515 and 1.520. The prism is oriented to provide the least deflection angle. 5.25. Using Fermat's principle derive the laws of deflection and refraction of light on the plane interface between two media.
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