methods for determining the rays that actually form these images. This will considerably facilitate the solution of such problems, for example, as the construction of an object positioned at some distance from a flat mirror, or of the image of a large object in a small lens or in a lens half covered by an opaque screen, and determining the position of an observer’s eye for simultaneous viewing several im- ages produced by optical systems. In Secs 32-34 pay attention to the rules used to End graphically the focus of the rays in a beam that has passed through an optical system (for example, Problems 555-557). In solving these problems, study the specific uses of the equations of spherical mirrors and lenses for calculating the position of the images produced by systems with con- verging beams. The problems in the sections that follow should be solved in the order in which they are presented because many of them are based on the results of previous problems. 506. How should a point source, a flat object and a screen be placed for the outline of the shadow on the screen to be similar to that of the object? 507. An electric lamp is placed into a frosted glass sphere of radius 20 cm and is suspended at a height of 5 m above the floor. A ball of radius 10 cm is held under the lamp at a height of 1 m. Determine the dimensions of the shadow and half-shadow cast by the ball. At what height should the ball be placed for the shadow on the floor to disappear? What will the dimensions of the half-shadow be in this case? What should the diameter of the ball be for the dimensions of its shadow to be the same irrespective of the distance from the ball to the floor? 508. The following simple method can be used to compare the luminous intensity of two sources: a thick rod D and sources S, and S2 some distance away are placed in front of a semi—opaque screen AB (Fig. 167). The sources are so ar- ranged that the half-shadows AO and OB are of the same luminance. 9-1218
130 _ PROBLEMS In what directions should the sources be moved so that the half—shadows cast by them are in contact all the time? What patterns will be observed when the sources are shifted in any other directions? 509. The image of an object is obtained using a box with a small aperture (Fig. 168). The depth of the box EC = 20 cm, the distance to the object CD = 20 cm and the diameter of the aperture C is d = 1 mm. B \\\\Q»*"·*"/,}.7.81 Q. //,/ "`°*•S, _ ETT;/r§'s&g.E;ZT;-E-p Fig. 167 Fig. 168 Can the parts of an object 2 mm in size be distinguished on the image in these conditions? 510. What will the shape of a light spot be if the di- mensions of the mirror are small and those of the source are large? 511. In one of his notes M.V. Lomonosov poses the fol- lowing question: "Any colour if moistened with water becomes deeper. Why?" The colour of the surfaces of bodies that can be impregnated with water does indeed grow darker and richer after moistening. Explain this phenomenon. 512. One of the expressions of the laws of propagation of light is Fermat’s principle assert-ing that light always propagates along the shortest paths. Consider the following case: light is emitted from a source A (Fig. 169), then reflected from a mirror and reaches a point B. Prove that the path ACB as determined by the law of reflection is the Shortest of all possible paths of the ray. 513. Two pins A and B arranged as shown in Fig. 170 are
CHAPTER rv. oprxcs 131 stuck in front of a mirror. What arrangement of the images of these pins will be seen by an observer in different view- ing positions? In what position of the eye will the image of the pins be superimposed on each other? 514. An object O’O and a mirror AC are placed as shown in Fig. 171. Construct the image of this object in the mir- ror. Where should the eye be placed to observe the image of the entire object? 515. A desk—lamp is placed in front of a mirror. What will be the change in the distance between the lamp and its image if the mirror is drawn 5 cm away from the lamp? A B é B \ 0\ / \ 0 " K" " s “ 0 / A \ $ A Fig. 169 Fig. 170 Fig. 171 516. A man stands in front of a mirror and looks at himself with one eye. What place should be covered in the mirror so as to keep the image of the other eye out of vision? 517. A ball is placed on a horizontal table. At what angle to the plane of the table should a mirror be placed to have the image of the ball moving vertically when the ball is brought towards the mirror? 518. A light ray is incident on a mirror. The mirror is turned through 1° about the axis lying in the plane of the mirror perpendicular to the ray. Through what angle on will the reflected ray be turned in this case? What distance .2: will the light spot move on a screen set perpendicularly to the reflected ray at a dis- tance l = 5 m from the mirror? 519. A mirror 1 m high hangs on a wall. A man stands a distance of 2 m away from the mirror. What is the height of the portion of the opposite wall in the room that can be 9*
132 pnonnnms seen by the man in the mirror without changing the posi- tion of his head? The wall is 4 m from the mirror. 520. Determine graphically the positions of the eye when an observer can simultaneously see in a flat mirror of finite 0 A dimensions the image of a ° point and a section of a stra- ight line placed with respect B to the mirror as shown in mm: Fig. 172. Fig. 172 521. When M.V. Lomonosov was attempting to increase the incendiary power of lenses he designed the device, shown in Fig. 173, and called it the catoptric-dioptric incendiary instrument. In this case A2, A2, A2,A5 are flat mirrors and B2, B2, B2, B2, B2 are convergent lenses. B4 Determine the angles at which A #M#»»· _, the mirrors should be- positioned .-~“"` V i and the minimum dimensions of 6; BJ these mirrors that will ensure the equality of the luminous fluxes E { incident on each lens. The diame- B, B, ter of the lenses is d and the opti- A A cal axes of the lenses B2, B2, B2, ' _ ’ B 2 form angles of ;|;45° with the Fig- 173 direction of the primary beam. 522. A point source of light and itis two images produced by two mirrors lie at the vertices o an equi ateral triangle. as az / U ·-_`.____·;__ M1-; 0- 4 Fig. 174 Determine the position of the mirrors with respect to the source and the angle between them. 523. Prove that a source and its two images in mirrors
CHAPTER IV. OPTICS 133