# Problems in General Physics → Optics → Photometry and Geometrical Optics

 5.1. Making use of the spectral response curve for an eye (see Fig. 5.1), find: (a) the energy flux corresponding to the luminous flux of 1.0 lm at the wavelengths 0.51 and 0.64 μm; (b) the luminous flux corresponding to the wavelength interval from 0.58 to 0.63 μm if the respective energy flux, equal to Φe = 4.5 mW, is uniformly distributed over all wavelengths of the interval. The function V(λ) is assumed to be linear in the given spectral interval. Free solution >> 5.2. A point isotropic source emits a luminous flux Φ = 10 lm with wavelength λ = 0.59 μm. Find the peak strength values of electric and magnetic fields in the luminous flux at a distance r = 1.0 m from the source. Make use of the curve illustrated in Fig. 5.1. Free solution >> 5.3. Find the mean illuminance of the irradiated part of an opaque sphere receiving (a) a parallel luminous flux resulting in illuminance E0 at the point of normal incidence; (b) light from a point isotropic source located at a distance l = 100 cm from the centre of the sphere; the radius of the sphere is R = 60 cm and the luminous intensity is I = 36 cd. Free solution >> 5.4. Determine the luminosity of a surface whose luminance depends on direction as L = L0 cos θ, where θ is the angle between the radiation direction and the normal to the surface. Free solution >> 5.5. A certain luminous surface obeys Lambert's law. Its luminance is equal to L. Find: (a) the luminous flux emitted by an element ΔS of this surface into a cone whose axis is normal to the given element and whose aperture angle is equal to θ; (b) the luminosity of such a source. Free solution >> 5.6. An illuminant shaped as a plane horizontal disc S = 100 cm2 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*104 cd/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. Free solution >> 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(θ), where θ is the angle between the radiation direction and the vertical, as well as the luminous flux reaching the table if I(0) = I0 = 100 cd. Free solution >> 5.9. A luminous dome shaped as a hemisphere rests on a horizontal plane. Its luminosity is uniform. Determine the illuminance at the centre of that plane if its luminance equals L and is independent of direction. Free solution >> 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. Free solution >> 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. Free solution >> 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*104 cd/m2 and is independent of direction. Find the illuminance of the floor directly below the lamp. Free solution >> 5.16. Two optical media have a plane boundary between them. Suppose θ1cr is the critical angle of incidence of a beam and θ1 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 θ1cr/sin θ1 = η = 1.28. Free solution >> 5.17. A light beam falls upon a plane-parallel glass plate d=6.0 cm in thickness. The angle of incidence is θ = 60°. Find the value of deflection of the beam which passed through that plate. Free solution >> 5.21. The least deflection angle of a certain glass prism is equal to its refracting angle. Find the latter. Free solution >> 5.24. A light ray composed of two monochromatic components passes through a trihedral prism with refracting angle θ = 60°. Find the angle Δα between the components of the ray after its passage 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. Free solution >> 5.33. Find the optical power and the focal lengths (a) of a thin glass lens in liquid with refractive index n0 = 1.7 if its optical power in air is Φ0 = -5.0 D; (b) of a thin symmetrical biconvex glass lens, with air on one side and water on the other side, if the optical power of that lens in air is Φ0 = +10 D. Free solution >> 5.34. By means of plotting find: (a) the path of a ray of light beyond thin converging and diverging lenses (Fig. 5.7, where OO' is the optical axis, F and F' are the front and rear focal points); (b) the position of a thin lens and its focal points if the position of the optical axis OO' and the positions of the cojugate points P, P' (see Fig. 5.5) are known; the media on both sides of the lenses are identical; (c) the path of ray 2 beyond the converging and diverging lenses (Fig. 5.8) if the path of ray 1 and the positions of the lens and of its optical axis OO' are all known; the media on both sides of the lenses are identical. Free solution >> 5.35. A thin converging lens with focal length f = 25 cm projects the image of an object on a screen removed from the lens by a distance l = 5.0 m. Then the screen was drawn closer to the lens by a distance Δl = 18 cm. By what distance should the object be shifted for its image to become sharp again? Free solution >> 5.37. A thin converging lens is placed between an object and a screen whose positions are fixed. There are two positions of the lens at which the sharp image of the object is formed on the screen. Find the transverse dimension of the object if at one position of the lens the image dimension equals h' = 2.0 mm and at the other, h" = 4.5 mm. Free solution >> 5.42. Figure 5.9 illustrates an aligned system consisting of three thin lenses. The system is located in air. Determine: (a) the position of the point of convergence of a parallel ray incoming from the left after passing through the system; (b) the distance between the first lens and a point lying on the axis to the left of the system, at which that point and its image are located symmetrically with respect to the lens system. Free solution >> 5.52. An optical system is located in air. Let OO' be its optical axis, F and F' are the front and rear focal points, H and H' are the front and rear principal planes, P and P' are the conjugate points. By means of plotting find: (a) the positions F' and H' (Fig. 5.11a); (b) the position of the point S' conjugate to the point S (Fig. 5.11b); (c) the positions F, F', and H' (Fig. 5.11c, where the path of the ray of light is shown before and after passing through the system). Free solution >>