From a convex surface, parallel rays when reflected are made to diverge; convergent rays are reflected less convergent; and divergent rays are rendered more divergent. It is the nature of all convex mirrors and surfaces to scatter or disperse the rays of light, and in every instance to impede their convergence. The following figure shows the course of parallel rays as reflected from a convex mirror. AEB is the convex surface of the mirror; and KA, IE, LB, parallel rays falling upon it. These rays, when they strike the mirror, are made to diverge in the direction AG, BH, &c. and both the parallel and divergent rays are here represented as they appear in a dark chamber, when a convex mirror is presented to the solar rays. The dotted lines denote only the course or tendency of the reflected rays, towards the virtual focus F, were they not intercepted by the mirror. This virtual focus is just equal to half the radius CE.
figure 19.
The following are some of the properties of convex mirrors: 1. The image appears always erect, and behind the reflecting surface. 2. The image is always smaller than the object, and the diminution is greater in proportion as the object is further from the mirror, but if the object touch the mirror, the image at the point of contact is of the same size as the object. 3. The image does not appear so far behind the reflecting surface as in a plain mirror. 4. The image of a straight object, placed either parallel or oblique to the mirror is seen curved in the mirror; because the different points of the object are not all at an equal distance from the surface of the mirror. 5. Concave mirrors have a real focus where an image is actually formed; but convex specula have only a virtual focus, and this focus is behind the mirror; no image of any object being formed before it.
The following are some of the purposes to which convex mirrors are applied. They are frequently employed by painters for reducing the proportions of the objects they wish to represent, as the images of objects diminish in proportion to the smallness of the radius of convexity, and to the distances of objects from the surface of the mirror. They form a fashionable part of modern furniture, as they exhibit a large company assembled in a room, with all the furniture it contains, in a very small compass, so that a large hall with all its objects, and even an extensive landscape, being reduced in size, may be seen from one point of view. They are likewise used as the small specula of those reflecting telescopes which are fitted up on the Cassegrainian plan, and in the construction of Smith’s Reflecting Microscope. But on the whole, they are very little used in the construction of optical instruments.
Properties of Concave speculums.
Concave specula have properties very different from those which are convex; they are of more importance in the construction of reflecting telescopes and other optical instruments; and therefore require more minute description and illustration. Concave mirrors cause parallel rays to converge; they increase the convergence of rays that are already converging; they diminish the divergence of diverging rays; and, in some cases, render them parallel and even convergent; which effects are all in proportion to the concavity of the mirror. The following figures show the course of diverging and parallel rays as reflected from concave mirrors.
Fig. 20 represents the course of parallel rays, and AB, the concave mirror on which they fall. In this case, they are reflected so as to unite at F, which point is distant from its surface one fourth of the diameter of the sphere of the mirror. This point is called the focus of parallel rays, or the true focus of the mirror. And, since the sun beams are parallel among themselves, if they are received on a concave mirror, they will all be reflected to that point, and there burn in proportion to the quantity of rays collected by the mirror. Fig. 21. shows the direction of diverging rays, or those which proceed from a near object. These rays proceeding from an object further from the mirror than the true focal point, as from D to A and to B, are reflected converging and meet at a point F, further from the mirror than the focal point of parallel rays. If the distance of the radiant, or object D, be equal to the radius CE, then will the focal distance be likewise equal to the radius: That is, if an object be placed in the center of a concave speculum, the image will be reflected upon the object, or they will seem to meet and embrace each other in the centre. If the distance of the radiant be equal to half the radius, its image will be reflected to an infinite distance, for the rays will then be parallel. If, therefore, a luminous body be placed at half the radius from a concave speculum, it will enlighten places directly before it at great distances. Hence their use when placed behind a candle in a common lantern; hence their utility in throwing light upon objects in the Magic Lantern and Phantasmagoria, and hence the vast importance of very large mirrors of this description, as now used in most of our Light Houses, for throwing a brilliant light to great distances at sea to guide the mariner when directing his course under the cloud of night.
figure 20.
figure 21.
When converging rays fall upon a concave mirror, they are reflected more converging and unite at a point between the focus of parallel rays and the mirror; that is, nearer the mirror than one half the radius; and their precise degree of convergency will be greater than that wherein they converged before reflection.
Of the images formed by Concave Mirrors.
If rays proceeding from a distant object fall upon a concave speculum, they will paint an image or representation of the object on its focus before the mirror. This image will be inverted, because the rays cross at the points where the image is formed. We have already seen that a convex glass forms an image of an object behind it; the rays of light from objects pass through the glass, and the picture is formed on the side farthest from the object. But in concave mirrors the images of distant objects—and of all objects that are farther from its surface than its principal focus—are formed before the mirror, or on the same side as the object. In almost every other respect, however, the effect of a concave mirror is the same as that of a convex lens, in regard to the formation of images, and the course pursued by the rays of light, except that the effect is produced in the one case by refraction, and in the other by reflection. The following figure represents the manner in which images are formed by concave mirrors. GF represents the reflecting surface of the mirror; OAB, the object; and IAM, the image formed by the mirror. The rays proceeding from O, will be carried to the mirror, in the direction OG, and according to the law that the angle of incidence is equal to the angle of reflection, will be reflected to I, in the direction GI. In like manner the rays from B, will be reflected from F to M, the rays from A, will be reflected to a, and so of all the intermediate rays, so that an inverted image of the object OB, will be formed at IM. If the rays proceeded from objects at a very great distance the image would be formed in the real focus of the mirror, or at one-fourth the diameter of the sphere from its surface; but near objects, which send forth diverging rays, will have their images formed a little farther from the surface of the mirror.
figure 22.
If we suppose a real object placed at IM, then OB will represent its magnified image, which will be larger than the object, in proportion to its distance from the mirror. This may be experimentally illustrated by a concave mirror and a candle. Suppose a concave mirror whose focal distance is five inches, and that a candle is placed before it, at a little beyond its focus, (as at IM)—suppose at five and a half inches,—and that a wall or white screen receives the image, at the distance of five feet six inches from the mirror, an image of the candle will be formed on the wall which will be twelve times longer and broader than the candle itself. In this way concave mirrors may be made to magnify the images of objects to