BOOK IV

LIGHT

CHAPTER I

RECTILINEAR PROPAGATION--REFLECTION

322. Scope of the Subject. The word light is used both in a subjective and an objective sense. Thus when our eye is subjected to certain conditions, we experience a sensation which we call light, while the physical cause of this sensation is also called light. In our examination of the phenomena we shall, however, find that bodies which emit that form of radiation which produces the sensation of light, in general, also emit other forms of radiation which, while physically of the same nature as light, yet do not produce the sensation of light. Thus the flame of a Bunsen burner is almost invisible, still, if we hold our hand on a thermometer near the flame, we find that heat energy is being radiated out all the time. By introducing a piece of fine platinum wire into the flame, the wire will be raised to a white heat, and will produce light radiation. The flame still continues to produce radiant heat, but now, in addition, some radiation is also produced, which can be recognised by our eye. Thus while visible radiant energy will be called light, we shall in this portion of the book also examine the phenomena due to invisible radiant energy, which is physically of the same nature as the light radiation, but for the recognition of which we have no special sense organ.

323. Rays-Geometrical Optics-Physical Optics. We shall see that in an isotropic medium light is propagated in straight lines, and is due to a wave-motion. A line drawn so as to represent the direction of propagation of the light proceeding from a luminous body is called a ray.1

Starting from the observed fact that light travels in straight lines, and the laws of reflection and refraction, it is possible to deduce a number of interesting results which have a direct bearing on optical phenomena by mere mathematical or geometrical methods. The subject of light con

1 The word ray is also used to signify "kind of radiation." Thus we speak of heat rays, red rays, blue rays, &c., meaning radiant heat and radiation which produces the sensation of red, blue, &c.

sidered in this way is generally called Geometrical Optics. In geometrical optics no inquiry is made as to the nature or cause of light, neither is any explanation forthcoming of the rectilinear propagation of light. These matters come within the scope of another branch of the subject, called Physical Optics. In the following pages we shall commence by using the methods of geometrical optics. Having in this way obtained a certain familiarity with some of the simpler phenomena, we shall then be in a position to consider the physical nature of these phenomena.

324. Rectilinear Propagation of Light-Shadows. The fact that under ordinary circumstances light is propagated in straight lines is taken for granted by every one in common life, for we always assume that a body exists in the direction of the rays of light which enter our eye. The simplest proof of the rectilinear propagation is afforded by the formation of shadows, for it is found that the edge of the shadow of a body formed by a point source of light, such as a pin-hole in a dark

shutter, the edge of the object and the source of light are all in a straight line.

In order to find the form of the shadow cast by a point source, we have only to draw a number of straight lines from the source, so that they all touch the edge of the object. Where these lines meet the screen will be points on the edge of the shadow. If the source of light has an appreciable magnitude, however, we do not get a simple shadow of uniform blackness with a sharp outline. Let AB (Fig. 280) be a luminous object, say the sun, and CD the body that casts the shadow, say the moon. Then if we consider a point, A, of the luminous body, the shadow cast by this point on a screen at EF would be at HK. In the same way the shadow cast by the point B would be GI. All intermediate points would cast shadows situated between G and K. It will thus be seen that HI will be the only part of the screen which is completely in shadow, i.e. screened from the whole of the luminous object. This part of the shadow is therefore called the umbra. The rest of the shadow is not completely dark, but gets darker and darker from the outside to the

edge of the umbra. This part of the shadow is called the penumbra. In the case of the moon and earth, it is only when the earth enters within the cone CMD that a total eclipse takes place; when it enters within the penumbra the eclipse is only partial, since from any point within the penumbra straight lines can be drawn touching the object, which will intersect the source of light, and so part of the source will be vis.ble from any such point.

325. The Pin-hole Camera.-The working of the pin-hole camera depends on the rectilinear propagation of light. If a small hole is made in an opaque screen, and a luminous object is placed on one side, and a white screen on the other, an inverted image of the luminous object will be formed on the screen. Each luminous point of the object A and B (Fig. 281) will form a small round patch of light on the screen; and if the hole is so small that these patches of light do not very much overlap, they will build up an image of the object, which, as is shown in the figure, is inverted. It is important to note that the image will be formed,

whatever may be the relative distance of the object and screen from the pin-hole, so that in this particular we have an important difference between the image formed in this way and that produced by a lens or mirror (§§ 337, 338). If a second pin-hole were made near the first, say at P, a second image would be produced, which would partly overlap the first image. In the same way, if a number of holes were made surrounding 0, instead of a definite image we should simply have a blur produced by the partial superposition of all the images. This explains why it is that it is only when the pin-hole is small that any sharp image is obtained, for a large hole is the equivalent of a number of pin-holes close together.

326. Assumptions as to the Nature of Light.—We shall in a subsequent section describe experiments to prove that light travels with a finite velocity, and others which show that it is of the nature of a wavemotion. Since, however, for the full grasping of these experiments a knowledge of the laws of reflection and refraction and of the elementary properties of mirrors and lenses is required, the description of them is for the present postponed. We shall, nevertheless, in the following sections

assume that light consists of a wave-motion, and that the velocity with which the light waves move is different in different media, and on these assumptions we shall construct explanations of the simple phenomena of reflection and refraction. Thus when considering the properties of mirrors and lenses we shall not only use the method of the older geometrical optics, namely, the method of rays, but in addition we shall sometimes take as our starting-point the wave-front (§ 272) at any given instant, and then by Huyghens's construction (§ 273) we shall trace out the form of the wave-front at subsequent times. These two methods of viewing the phenomena are essentially the same, for the rays are everywhere at right angles to the corresponding wave-fronts; but it is nevertheless of use when employing the method of rays to have in our mind's eye the corresponding wave-fronts.

327. Curvature of a Surface.-If we have a disturbance produced at a point within an isotropic medium, the wave-fronts will be spheres with the point as centre. If, however, the medium is not isotropic, the form of the wave-fronts will in general be different. In the following pages we shall almost exclusively deal with spherical or plane wavefronts. If we have a surface which is not a sphere, and at any point on this surface draw a sphere touching this surface, then for parts of the surface in the immediate neighbourhood of this point we may suppose the surface replaced by that of the sphere. In the same way we can draw a circle to touch any plane curve, and for points near the point of contact the circle will coincide with the curve.

Let AB (Fig. 282) be a portion of a curve, and at the points A and B draw two tangents to the curve.

Now

B

D

A

T2

at the point A the direction of the curve is that of the tangent AT1, while at B the direction of the curve is BT2. Hence, when we pass from A to B the direction of the curve changes by an angle ADT, or 0. Let the length of the curve between A and B be S, then the rate of change in direction with distance measured along the curve is 0/s, and this is called the curvature of the curve between A and B. If the curve is a circle with its centre at C, the angle ACB is equal to the angles TDT, for the radii are at right angles to the tangents. Let r be the radius of the circle, then the length of the arc AB is re (§ 14). Hence the curvature of the circle is

0r0 or 1/r.

FIG. 282.

Thus the curvature of a circle is numerically equal to the reciprocal of the radius.

In the case of any other curve, if the tangent circle is drawn at any point the curvature of this circle is the reciprocal of the radius, and as the circle coincides with the curve at the given point, this also measures the curvature of the curve in the immediate neighbourhood of this point.

When in the place of plane curves we are dealing with surfaces, the same method is employed to measure the curvature, namely, the curvature at any point is equal to the reciprocal of the radius of the sphere which touches the surface at that point.

The radius of the tangent circle or sphere is called the radius of curvature of the curve or surface respectively, while the centre of the circle or sphere is called the centre of curvature.

In the case of a wave-front we have to distinguish two cases, namely, according as the direction in which the wave is moving is towards the concave or convex surface of the wave. We shall take the curvature of a wave to be positive when it is moving towards the centre of the tangent sphere. Thus the curvature of the spherical wave-surface produced by a disturbance at a point in an isotropic medium is negative.

328. Images. If a wave has a positive curvature it is moving towards the centre of a circle, and if the medium between the wave-front and the centre of the circle is isotropic the wave will converge on this centre, so that at a certain instant the wave-front will be reduced to a point. Under these circumstances the wave is said to come to a real focus, or to produce a real image at the point.

If by reflection or refraction a wave-front of negative curvature is produced such that the centre of curvature does not coincide with the point where the wave was originated, the wave will travel as if it came from this centre, which is called a virtual focus, or virtual image.

Since the rays are always at right angles to the wave-fronts, a spherical wave-front of positive curvature corresponds to a pencil of rays which converge towards a point, this point being the centre of curvature of the wave-front. Thus a real image is produced when the rays of light which have started from a luminous point are, by reflection or refraction, caused to pass through a second point, this point being the real image.

In the same way, when the rays proceed as if they came from a point other than the actual source from which they do proceed, this point is called a virtual image.

In the case of a real image the waves and the rays actually pass through the image, while in the case of a virtual image the waves never actually pass through the image; they, however, proceed as if they had been produced at the image, and had then moved out in ever-widening spheres in an isotropic medium. In a virtual image also the rays never actually pass through the image; their direction, however, is such that if they were prolonged backwards they would pass through the virtual image.

329. Laws of Reflection. The fact that bodies which are not themselves luminous are, when illuminated, visible in all directions,