SHADOW. When an opaque body is placed between a screen and a luminous source, it casts a "shadow" on the screen. If the source be a point, the outline of the shadow is to be found by drawing straight lines from the luminous point so as to envelop the opaque body. These lines form a cone. The points of contact form a line on the opaque body separating the illuminated from the non-illuminated portion of its surface. Similarly, when these lines are produced to meet the screen, their points of inter section with it form a line which separates the illuminated from the non-illuminated parts of the screen. This line is called the boundary of the geometrical shadow, and its construction is based on the assumption that light trav els (in homogeneous media) in straight lines and suffers no devi ation on meeting an obstacle. But a deviation, termed diffrac tion, does occur, and conse quently the complete theory of shadows involves considerations based on the nature of the rays themselves ; this aspect is treated in LIGHT.
When there are more luminous points than one, we have only to draw separately the geometri cal shadows due to each of the sources, and then superpose them.
A new consideration now comes in. There will be, in general, portions of all the separate geomet rical shadows which overlap one another in some particular regions of the screen. In such regions we still have full shadow; but around them there will be other regions, some illuminated by one of the sources alone, some by two, etc., until finally we come to the parts of the screen which are illuminated directly by all the sources. There will still be a definite boundary of the parts wholly unilluminated, i.e., the true shadow or umbra, and also a definite boundary of the parts wholly illuminated. The region be tween these boundaries—i.e., the partially illumined portion— is called the penumbra.
Fig. 1 represents the shadow of a circular disk cast by four equal luminous points arranged as the corners of a square—the disk being large enough to admit of a free overlapping of the separate shadows. The amount of want of illumination in each portion of the penumbra is roughly indicated by the shading. The separate shadows are circular, if the disk is parallel to the screen. If we suppose the number of sources to increase indefi
nitely, so as finally to give the appearance of a luminous surface as the source of light, it is obvious that the degrees of darkness at different portions of the penumbra will also increase indefinitely; i.e., there will be a gradual increase of brightness in the penumbra from total darkness at the edge next the geometrical shadow to full illumination at the outer edge.
Thus we see at once why the shadows cast by the sun or moon are in general so much less sharp than those cast by the electric arc. For, practically, at moderate distances the arc appears as a mere luminous point. But if we place a body at a distance of a foot or two only from the arc, the shadow cast will have as much of penumbra as if the sun had been the source. The breadth of the penumbra when the source and screen are nearly equidistant from the opaque body is equal to the diameter of the luminous source. The notions of the penumbra and umbra are important in considering eclipses (q.v.). When the eclipse is total, there is a real geometrical shadow—very small compared with the penumbra (for the apparent diameters of the sun and moon are nearly equal, but their distances are as 370 : I) ; when the eclipse is annular, the shadow is all penumbra. In a lunar eclipse the earth is the shadow-casting body, and the moon is the screen, and we observe things according to our first point of view.
Suppose, next, that the body which casts the shadow is a large one, such as a wall, with a hole in it. If we were to plug the hole, the whole screen would be in geometrical shadow. Hence the illumination of the screen by the light passing through the hole is precisely what would be cut off by a disk which fits the hole, and the complement of fig. I, in which the light and shade are interchanged, would give therefore the effect of four equal sources of light shining on a wall through a circular hole. The umbra in the former case becomes the fully illuminated portion, and vice versa. The penumbra remains the penumbra, but it is now darkest where before it was brightest, and vice versa.