CHAPTER XLI

REFLECTION

Diffuse and regular reflection, 568. Law of regular reflection, 569. Huygens' principle, 570. Proof of the law of regular reflection, 571. The ray and wave methods, 572. The image in a plane mirror, 573. Virtual and real images, 574. The concave mirror, 575. The principal focus, 576. Conjugate foci, 577. Wave construction for a concave mirror, 578. The graphical construction of real images, 579. Virtual images formed by a concave mirror, 580. Law of size of image, 581. Formula for position of object and image, 582. Derivation of the formula, 583. Principal focus of a convex

lar. When a sheet of white paper is placed in a beam of light, the light is reflected or scattered in all directions. It is said to be diffusely reflected. The light from the page of this book and from all or nearly all the objects in an ordinary room reaches the eye by diffuse reflection; but when a beam of light falls on a mirror, it is all, or practically all, reflected in some one direction. Regular reflection occurs only from smooth surfaces, such as those of glass, mercury, water, and polished metals.

569. Law of regular reflection. In Fig. 360, MM is the trace of a mirror placed perpendicularly to the plane of the figure, AO the direction the incident light is traveling, OB the direction of the regularly reflected light, and ON a line perpendicular, or normal, to the mirror MM. The angle of incidence is the angle AON, and the angle of reflection is the angle NOB.

In the case of regular reflection the angle of reflection is equal to the angle of incidence, and the two angles lie in the same plane. It is rather a simple matter to prove by direct experiment that this law is true. However, a proof based on the wave theory of light will be given. But it will be necessary first to explain a principle that we must use.

570. Huygens' principle. A method due to Huygens affords a simple means of finding the change in the position of any wave. Let the given wave be AB (Fig. 361). It is required to find the position of this wave at some subsequent time. A number of points on this wave-P1, P2, etc. are chosen.

A'

From each of these is drawn an arc of a circle the radius of which has the

A

same length in all cases. Huygens' principle states that the new position of the wave is represented by a line

P

P

disturbance is propagated from this

point in all directions. P1, P2, P3, etc., in Fig. 361, are centers of disturbance. But these disturbances interfere (sect. 329 and Chapter XLVI) with one another in such a way that the resultant disturbance is found only along the surface which is tangent to the wavelets sent out from each of these centers.

571. Proof of the law of regular reflection. In Fig. 362, AB represents an incident wave, and AC the mirror. If the mirror did not interfere, the wave AB would, after a certain time, reach a position DC. A number of points M1, M2, and M, are taken on the surface of the mirror. At different times these points become centers of disturbances which, if the mirror were not there, would reach DC at the same time and, in accordance with

3

N

2

B

F

1

Huygens' principle, form the wave DC. An arc of a circle with a radius equal to AD is drawn with A as the center, another with M1 as the center and with a radius equal to the distance of M1 from CD (that is, the distance M1N1), another with M2 as a center and with a radius equal to M2N2, and another with M3 as a center and with a radius of M3N3. Since the line FC is tangent to these arcs, it gives, by Huygens' principle, the position of the reflected wave. To show that the angle of reflection is equal to the angle of incidence, it is necessary merely to show that the two lines BC and AF, which represent the directions the incident and reflected waves are traveling, make equal angles with the mirror AC. Since the line AF is equal to AD by construction, and since AD is equal to BC, AF is equal to BC. The side AC is common to the triangles ABC and AFC. Since the two triangles have

A

M

M

M

N

N

D

FIG. 362

two sides equal, and since they both contain right angles, they must be equal; hence the angle BCA is equal to the angle FAC. But the angle of incidence is not the angle the line BC makes with the reflecting surface: it is the angle that BC makes with the normal to the surface. However, it should be entirely clear to the student that if the angles BCA and FAC are equal, the angles of incidence and reflection must be equal.

It should be obvious that since this proof applies to any type of wave-motion, the law of regular reflection applies not only to light-waves but to all kinds of waves.

572. The ray and wave methods. Since light usually travels in straight lines, it is often simpler to deal only with a line which indicates the direction the light is traveling. Such a line is often called a ray of light, but the term has no other physical significance. Light does not consist of bundles of rays: it is a wave

motion. While the ray method is often very simple, it has the disadvantage of tending to conceal the true mode of propagation. In the next section a simple problem will be worked out by both

2 2

1

I

FIG. 363

2

surface at P1 equal to the angle made by OP1 with the same normal. P1R1 then represents the ray reflected at P1. In a similar manner the line P2R2 represents the ray reflected at P2. The two rays P1R1 and P2R2 appear to come from the point I, which is located by continuing these lines back through the mirror. I is called the image of O. If a large number of rays were drawn from O, the reflected rays would all appear to come from the image I. It will now be proved (1) that the line Ol joining the object

mirror. It follows from the construction and from the law of reflection that the angles OPN, NPR, MOP, and MO'P are all equal. It then follows, rather simply, that the triangle MOP is equal to MO'P; hence OM is equal to MO'.

But this is true no matter where the point P is taken on the surface. If another ray is drawn from O to some point on the surface, and if the reflected ray is drawn backward until it cuts the line OM, it will pass through the point O'; hence O' must be the image of the object. Since O' is a point on the perpendicular line OM, the line connecting the image and the object is perpendicular to the surface of the mirror.

2. The second fact, that the image lies just as far behind the mirror as the object is in front, is proved as soon as it is shown that the point O' is the position of the image; for it has been shown that OM is equal to MO'.

It is not necessary to go into great detail to explain the wave method, for it can be understood by the aid of Fig. 365. The source of the waves is at O. If the reflecting surface MN did not interfere, a wave would reach the position ABC. But the wave is turned back by the surface and travels just as far back as

0

it would otherwise go forward. The method of finding the reflected wave B' can be seen in the figure. Several points on the mirror are chosen, and circles are drawn which are tangent to the wave ABC. At some time each of the centers of these circles is a center of disturbance, and the wavelet emitted travels upward the same distance it would go downward if the mirror MN were not there. The reflected wave B' is drawn tangent to these circles. The wave B' in traveling back from the surface appears to come from the geometrical center of the wave, the point I, which is therefore the image of O. From symmetry it can be seen that the center I of the wave B' must be just as far below the plane as the source O (the center of the wave B) is above it.

574. Virtual and real images. The images formed in plane mirrors are called virtual images, to distinguish them from real images, where the light actually reaches the image. The images formed