The luminous flux emitted from any practical source of light is not distributed uniformly in all direc tions. In the case of an electric lamp of the vacuum type, for in stance, the flux emitted within a cone of given solid angle is greater when the axis of the cone is horizontal than when it is vertical. It follows that some quantity is needed to express the light-giving power of a source in a specified direction. The nat ural quantity to use for this purpose is the angular flux density in the direction considered. This will be clear from fig. 2. If BLC be a cone of very small solid angle w, having its apex at L, the position of a source of light, then the angular flux density in any direction such as LA is equal to the flux emitted by L within the cone BLC, divided by the solid angle w, the ratio being taken in the limit when co becomes vanishingly small. This ratio is termed the luminous intensity of the source L in the direction LA. It will be seen that the definition of luminous intensity is anal ogous to that of pressure at a point, viz., the ratio of the force exerted on a given surface containing the point to the area of that surface, the ratio being taken in the limit when the area is van ishingly small. The formal definition of luminous intensity is as follows :—The luminous intensity of a point-source in any direc tion is the luminous flux, per unit solid angle, emitted by that source in that direction. (The flux emanating from a source whose dimensions are negligible in comparison with the distance from which it is observed may be considered as coming from a point.) The relation between luminous flux and luminous intensity being thus defined, it is possible to choose units for these two quan tities such that their magnitudes are related rationally to each other. It so happens that, for historical reasons, the primary photometric unit is that of luminous intensity. This unit is the international candle, the magnitude of which was originally de fined as the luminous intensity, in the horizontal direction, of a candle of specified dimension's burning at a specified rate. Many years ago this form of standard was found to be unsatisfactory, and it was replaced by flame lamps of various kinds, viz., the Hefner in Germany, the pentane in this country and in America, and the Carcel lamp in France. These standards have been gen erally abandoned, however, and the magnitude of the unit is pre served by means of specially constructed electric lamps deposited at the various national laboratories throughout the world, viz., the Laboratoire Central d'Electricite, Paris; the National Physical Laboratory, Teddington, England; and the Bureau of Standards, Washington. The formal definition of the "candle" is as follows: The unit of luminous intensity is the international candle, such as resulted from agreement effected between the three national standardising laboratories of France, Great Britain and the United States in 1909. The unit of luminous flux is now very simply ob tained from that of luminous in tensity by considering an ideal source of light which has a uni form luminous intensity of one candle in all directions. The lu men is the amount of flux emitted by such a source within a cone of unit solid angle.
The relationship is shown pic torially in fig. 3 where the ideal point source is imagined at the centre of a sphere of unit radius.
Since the solid angle of any cone with its apex at the centre of such a sphere is numerically equal to the area of the spherical surface cut off by the edge of the cone, it follows that the cone shown shaded in the diagram embraces unit solid angle, so that the flux emitted by the source within this cone is 1 lumen. Since the area of the whole sphere is 4 r times the square on the radius, it f ol lows a uniform point source of i candle emits altogether 4 7r lumens. Since the total flux emitted by a source is independ
ent of the way in which that flux is distributed, the total number of lumens given by a source is equal to the average value of its luminous intensity, measured in all directions in space, multiplied by the factor 4 r. The luminous intensity of a source, when ex pressed in international candles, is termed the candle-power of the source. The average value of the candle-power measured in all directions perpendicular to the geometrical axis of the source is termed the mean horizontal candle-power (M.H.C.P.), while the average candle-power measured in all directions in space is termed the mean spherical candle-power (M.S.C.P.). It will be seen that, for any source, M.S.C.P. X4r = (flux output in lumens).
Illumination: The Inverse Square and Cosine Laws.— Illumination is measured by the amount of luminous flux which reaches unit area of an illuminated surface. From this definition and the fact of the rectilinear propagation of light there follow at once the two basic laws of photometry, viz., the inverse square and the cosine laws. They will be most clearly understood by reference to fig. 2. If BLC be an elementary cone of light emitted from a source L, the area intercepted by this cone on any plane such as XY varies (i.) as the square of the distance (d) of L from the plane, and (ii.) as the secant of the angle (0) which the nor mal to XY makes with the axis LA of the cone. Since the illu mination produced by a given amount of flux varies inversely as the area over which that flux is distributed, it will be seen at once that the illumination of a surface varies (i.) inversely as the square of its distance from the source illuminating it (this is the inverse square law), and (ii.) directly as the cosine of the angle between the normal to the surface and the light rays (this is the cosine law).
Now let the average luminous intensity of the source L in all directions lying within the cone BLC be I candles, and let the area intercepted by this cone on the plane XY be s. Then the solid angle of the cone is equal to (s cos
and the flux F emitted within the cone is I (s cos
lumens. Since, by the definition, the illumination, E, of the surface is equal to F/S, it follows that It will be seen that the magnitude of the unit in which E is meas ured must depend on the unit used for d, i.e., on the unit of length. If d be expressed in feet, the unit in which E is measured is termed the foot-candle and, clearly, one foot-candle is the il lumination produced when an area of one square foot receives one lumen of flux. Similarly, if d be expressed in metres, the unit of E is the metre-candle or the lux, and one lux is equivalent to an illumination of one lumen per square metre. It will be seen that I ft.-candle = 10.76 lux.
The last of the four fundamental photometric quantities is brightness, which is thus defined: The brightness in a given direction of a surface emitting light is the quotient of the luminous intensity measured in that direction, by the area of this surface projected on a plane perpendicular to the direction con sidered. The unit of brightness is the candle per unit area of surface. Thus if a flame of uniform brightness has an apparent area of one square inch when viewed in a given direction, and if the luminous intensity in that direction be 10 candles, the bright ness is 1 o candles per square inch. Similarly, if a surface has an area of two square metres and if its brightness when viewed nor mally is one candle per square metre, it has a luminous intensity of 2 candles in the direction of the normal.