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The above is the direct experimental proof that the intensity of illumination varies inversely as the square of the distance from the source.
The theoretical proof of the law is furnished at once by Fig. 401, for since all the light which falls from the candle L on A is spread over four times as large an area when it reaches B, twice as far away, and over nine times as large an area
FIG. 401. Proof of law of inverse squares
when it reaches C, three times as far away, obviously the intensities at B and at C can be but one fourth and one ninth as great as at A.
The above method of comparing experimentally the intensities of two lights was first used by Count Rumford. The arrangement is therefore called the Rumford photometer (light measurer).
435. Candle power. The last experiment furnishes a method of comparing the light-emitting powers of various sources of light. For example, suppose that the four candles at B are replaced by a gas flame, and that for the condition of equal illumination upon the screen the two distances BC and AC are the same as above, namely, 2 to 1. We should then know that the gas flame, which is able to produce the same illumination at a distance of two feet as a candle at a distance of one foot, has a light-emitting power equal to four candles. In general, then, the candle powers of any two sources which produce equal illumination on a given screen are directly proportional to the squares of the distances of the sources from the screen.
It is customary to express the intensities of all sources of light in terms of candle power, one candle power being defined as the amount of light emitted by a sperm candle inch in
diameter and burning 120 grains (7.776 grams) per hour. The candle power of an ordinary gas flame burning 5 cubic feet per hour is from 16 to 25, depending on the quality of the gas.
A standard candle at a distance of 1 foot gives an intensity of illumination called a foot-candle. A 100-candle-power lamp, for example, at a distance of 1 foot gives an intensity of illumination of 100 foot-candles; at 2 feet, of 25 foot-candles; at 5 feet, of 4 foot-candles; and at 10 feet, of 1 foot-candle.
436. Bunsen's photometer. Let a drop of oil or melted paraffin be placed in the middle of a sheet of unglazed white paper to render it translucent. Let the paper be held near a window and the side away from the window observed. The oiled spot will appear lighter than the remainder of the paper. Then let the paper be held so that the side nearest the window may be seen. The oiled spot will appear darker than the rest of the paper. We learn, therefore, that when the paper is viewed from the side of greater illumination, the oiled spot appears dark; but when it is viewed from the side of lesser illumination, the spot appears light. If, then, the two sides of the paper are equally illuminated, the spot ought to be of the same brightness when viewed from either side. Let the room be darkened and the oiled paper placed between two gas flames, two electric lights, or any two equal sources of light. It will be observed that when the paper is held closer to one than the other, the spot will appear dark when viewed from the side next the closer light; but if it is then moved until it is nearer the other source, the spot will change from dark to light when viewed always from the same side. It is always possible to find some position for the oiled paper at which the spot either disappears altogether or at least appears the same when viewed from either side. This is the position at which the illuminations from the two sources are equal. Hence, to find the candle power of any unknown source it is only necessary to set up a candle on one side and the unknown source on the other, as in Fig. 402, and to move the spot A to the position of equal illumination. The candle power of the unknown source at C will then be the square of the distance from C' to A, divided by the square of the distance from B to A.
FIG. 402. Bunsen's photometer
This arrangement is known as the Bunsen photometer.
QUESTIONS AND PROBLEMS
1. Distinguish between candle power, intensity of light, and intensity of illumination.
2. How many candles will be required to produce the same intensity of illumination at 2 m. that is produced by 1 candle at 30 cm.?
3. A 500-candle-power lamp is placed 50 m. from a darkly shaded place along the street. At what distance would a 100-candle-power lamp have to be to produce the same intensity of illumination?
4. If a 2-candle-power light at a distance of 1 ft. gives enough illumination for reading, how far away must a 32-candle-power lamp be placed to make the same illumination? How strong a lamp should be used at a distance of 8 ft. from the book?
5. A Bunsen photometer placed between an arc light and an incandescent light of 32 candle power is equally illuminated on both sides when it is 10 ft. from the incandescent light and 36 ft. from the arc light. What is the candle power of the arc?
6. A 5-candle-power and a 30-candle-power source of light are 2 m. apart. Where must the oiled disk of a Bunsen photometer be placed in order to be equally illuminated on the two sides by them?
7. If the sun were at the distance of the moon from the earth, instead of at its present distance, how much stronger would sunlight be than at present? The moon is 240,000 mi. and the sun 93,000,000 mi. from the earth.
8. If a gas flame is 300 cm. from the screen of a Rumford photometer, and a standard candle 50 cm. away gives a shadow of equal intensity, what is the candle power of the gas flame?
9. Will a beam of light going from water into flint glass be bent toward or away from the perpendicular drawn into the glass?
10. When light passes obliquely from air into carbon bisulphide it is bent more than when it passes from air into water at the same angle. Is the speed of light in carbon bisulphide greater or less than in water? 11. If light travels with a velocity of 186,000 miles per sec. in air, what is its velocity in water, in crown glass, and in diamond? (See table of indices of refraction, p. 371.)
IMAGES FORMED BY LENSES
437. Focal length of a convex lens. Let a convex lens be held in the path of a beam of sunlight which enters a darkened room, where it is made plainly visible by means of chalk dust or smoke. The beam will be found to converge to a focus F, as shown in Fig. 403.
The explanation is as follows: The waves from the sun or any distant object are without any appreciable curvature when they strike the lens; that is, they are so-called plane waves (see Fig. 403). Since the speed of light is less in glass than in air, the central portion of these waves is retarded more than the outer portions in passing through the lens. Hence, on emerging from the lens the waves are concave instead of plane, and close in to a center or focus at F.
A second way of looking at the phenomenon is to think of the "rays" which strike the lens as being bent by it, in accordance with the laws given in § 423, so that they all pass through the point F.
The line through the point C (the optical center) of the lens, perpendicular to its faces, is called the principal axis.
The point F at which rays parallel to the principal axis (incident plane waves) are brought to a focus is called the 3. principal focus.
The distance CF from the center of the lens to the prin cipal focus is called the focal length (f) of the lens.
FIG. 403. Principal focus F and focal length CF of a convex lens
The plane FFF" (Fig. 404) in which plane waves (parallel rays) coming to the lens from slightly different directions, as from the top and bottom of a distant house, all have their foci F', F", etc. is called the focal plane of the lens.
FIG. 404. Focal plane of a convex lens
Since the curvature of
any arc is defined as the reciprocal of its radius (see footnote, p. 370), the curvature which a lens impresses on an incident
7 plane wave is equal to
Moreover, no matter what the curvature of an incident wave may be, the lens will always
change the curvature by the same amount, 7•
Let the focal length of a convex lens be accurately determined by measuring the distance from the middle of the lens to the image of a distant house.
438. Conjugate foci. If a point source of light is placed at F (Fig. 403), it is obvious that the light which goes through the lens must exactly retrace its former path; that is, its
FIG. 405. Conjugate foci
waves will be rendered plane or its rays parallel by the lens.
curvature (since the curvature of an arc is defined as
the reciprocal of its radius), which is less than their former Since the lens was able to subtract all the