Both convex and concave mirrors are formed of portions of a sphere. A convex speculum is ground and polished in a concave dish or tool which is a portion of a sphere, and a concave speculum is ground upon a convex tool. The inner surface of a sphere brings parallel rays to a focus at one fourth of its diameter, as represented in the following figure, where C is the centre of the sphere on which the concave speculum AB is formed, and F the focus where parallel rays from a distant object would be united, after reflection, that is, at one half the radius, or one fourth of the diameter from the surface of the speculum. Were a speculum of this kind presented to the sun, F would be the point where the reflected rays would be converged to a focus, and set fire to combustible substances if the speculum be of a large diameter, and of a short focal distance. Were a candle placed in that focus, its light would be reflected parallel as represented in the figure. These are properties of concave specula which require to be particularly attended to in the construction of reflecting telescopes. It follows, from what has been now stated, that if we intend to form a speculum of a certain focal distance,—for example, two feet, it is necessary that it should be ground upon a tool whose radius is double that distance, or four feet.
figure 18.
Properties of Convex Mirrors.
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.