To find the focal distance of a concave glass. Take a piece of paste-board or card paper, and cut a round hole in it, not larger than the diameter of the lens; and, on another piece of paste-board, describe a circle whose diameter is just double the diameter of the hole. Then apply the piece with the hole in it to the lens, and hold them in the sun-beams, with the other piece at such a distance behind, that the light proceeding from the hole may spread or diverge so as precisely to fill the circle; then the distance of the circle from the lens is equal to its virtual focus, or to its radius, if it be a double concave, and to its diameter, if a plano-concave. Let d, e, (fig. 12,) represent the diameter of the hole, and g, i, the diameter of the circle, then the distance C, I, is the virtual focus of the lens.9
The meniscus represented at E, fig. 5, is like the crystal of a common watch, and as the convexity is the same as the concavity, it neither magnifies nor diminishes. Sometimes, however, it is made in the form of a crescent, as at F, fig. 5, and is called a concavo-convex lens; and, when the convexity is greater than the concavity, or, when it is thickest in the middle, it acts nearly in the same way as a double or plano-convex lens of the same focal distance.
Of the IMAGES formed by convex lenses.
It is a remarkable circumstance, and which would naturally excite admiration, were it not so common and well known, that when the rays of light from any object are refracted through a convex lens, they paint a distinct and accurate picture of the object before it, in all its colours, shades, and proportions. Previous to experience, we could have had no conception that light, when passing through such substances, and converging to a point, could have produced so admirable an effect,—an effect on which the construction and utility of all our optical instruments depend. The following figure will illustrate this position. Let L, N, represent a double convex lens, A, C, a, its axis, and OB, an object perpendicular to it. A ray passing from the extremity of the object at O, after being refracted by the lens at F, will pass on in the direction FI, and form an image of that part of the object at I. This ray will be the axis of all the rays which fall on the lens from the point O, and I will be the focus where they will all be collected. In like manner BCM, is the axis of that parcel of rays which proceed from the extremity of the object B, and their focus will be at M; and since all the points in the object between O, and B, must necessarily have their foci between I and M, a complete picture of the points from which they come will be depicted, and consequently an image of the whole object OB.
figure 13.
It is obvious, from the figure, that the image of the object is formed in the focus of the lens, in an inverted position. It must necessarily be in this position, as the rays cross at C, the centre of the lens; and as it is impossible that the rays from the upper part of the object O, can be carried by refraction to the upper end of the image at M. This is a universal principle in relation to convex lenses of every description, and requires to be attended to in the construction and use of all kinds of telescopes and microscopes. It is easily illustrated by experiment. Take a convex lens of eight, twelve, or fifteen inches focal distance, such as a reading glass, or the glass belonging to a pair of spectacles, and holding it, at its focal distance from a white wall, in a line with a burning candle, the flame of the candle will be seen depicted on the wall in an inverted position, or turned upside down. The same experiment may be performed with a window-sash, or any other bright object. But, the most beautiful exhibition of the images of objects formed by convex lenses, is made by darkening a room, and placing a convex lens of a long focal distance in a hole cut out of the window-shutter; when a beautiful inverted landscape, or picture of all the objects before the window, will be painted on a white paper or screen placed in the focus of the glass. The image thus formed exhibits not only the proportions and colours, but also the motions of all the objects opposite the lens, forming as it were a living landscape. This property of lenses lays the foundation of the camera obscura, an instrument to be afterwards described.
The following principles in relation to images formed by convex lenses may be stated. 1. That the image subtends the same angle at the centre of the glass as the object itself does. Were an eye placed at C, the centre of the lens LN, fig. 13, it would see the object OB, and the image IM under the same optical angle, or, in other words, they would appear equally large. For, whenever right lines intersect each other, as OI and BM, the opposite angles are always equal, that is, the angle MCI is equal to the angle OCB. 2. The length of the image formed by a convex lens, is to the length of the object, as the distance of the image is to the distance of the object from the lens: that is, MI is to OB :: as Ca to CA. Suppose the distance of the object CA from the lens, to be forty-eight inches, the length of the object OB = sixteen inches, and the distance of the image from the lens, six inches, then the length of the image will be found by the following proportion, 48 : 16 :: 6 : 2, that is, the length of the image, in such a case, is two inches. 3. If the object be at an infinite distance, the image will be formed exactly in the focus. 4. If the object be at the same distance from the lens as its focus, the image is removed to an infinite distance on the opposite side; in other words, the rays will proceed in a parallel direction. On this principle, lamps on the streets are sometimes directed to throw a bright light along a foot-path where it is wanted, when a large convex glass is placed at its focal distance from the burner; and on the same principle, light is thrown to a great distance from lighthouses, either by a very large convex lens of a short focal distance, or by a concave reflector. 5. If the object be at double the distance of the focus from the glass, the image will also be at double the distance of the focus from the glass. Thus, if a lens of six inches focal distance be held at twelve inches distance from a candle, the image of the candle will be formed at twelve inches from the glass on the other side. 6. If the object be a little further from the lens than its focal distance, an image will be formed, at a distance from the object, which will be greater or smaller in proportion to the distance. For example, if a lens five inches focus, be held at a little more than five inches from a candle, and a wall or screen at five feet six inches distant, receive the image, a large and inverted image of the candle will be depicted, which will be magnified in proportion as the distance of the wall from the candle exceeds the distance of the lens from the candle. Suppose the distance of the lens to be five and a half inches, then the distance of the wall where the image is formed, being twelve times greater, the image of the candle will be magnified twelve times. If MI (fig. 13.) be considered as the object, then OB will represent the magnified image on the wall. On this principle the image of the object is formed by the small object glass of a compound microscope. On the same principle the large pictures are formed by the Magic Lantern and the Phantasmagoria; and in the same way small objects are represented in a magnified form, on a sheet or wall by the Solar microscope. 7. All convex lenses magnify the objects seen through them, in a greater or less degree. The shorter the focal distance of the lens, the greater is the magnifying power. A lens four inches focal distance, will magnify objects placed in the focus, two times in length and breadth; a lens two inches focus will magnify four times, a lens one inch focus eight times; a lens half an inch focus sixteen times, &c. supposing eight inches to be the least distance at which we see near objects distinctly. In viewing objects with small lenses, the object to be magnified should be placed exactly at the focal distance of the lens, and the eye at about the same distance on the other side of the lens. When we speak of magnifying power, as, for example, that a lens one inch focal distance magnifies objects eight times, it is to be understood of the lineal dimensions of the object. But as every object at which we look has breadth as well as length, the surface of the object is in reality magnified sixty-four times, or the square of its lineal dimensions; and for the same reason a lens half an inch focal distance magnifies the surfaces of objects 256 times.
Reflections deduced from the preceding subject.
Such are some of the leading principles which require to be recognised in the construction of refracting telescopes, microscopes, and other dioptric