Obrázky stránek
PDF
ePub

There are always two principal foci for a lens, one on each side; for light can travel in either direction through the lens.

M

N

604. The optical center. In a thin lens a certain point has the property that the rays which pass through it are not appreciably bent by the lens. This point is called the optical center. If the lens is a symmetrical one, such as a double-convex or doubleconcave lens, the optical center is at the geometrical center. However, the statement that a ray which passes through the optical center is not bent is an approximation, and is true only for thin lenses. If the lens is thick, there are points, called nodal points (Mand N in Fig. 405), which have the following properties: Any incident ray which travels toward the nearer one of these points is so refracted that it emerges traveling parallel to its original direction, in a line which, if drawn back into the lens, would pass through the other nodal point. The path of such a ray is shown in the figure. In thin lenses the two

L

FIG. 405

F

F

I

FIG. 406. Real image formed by a converging lens

points practically coincide in a point called the optical center. In a thin lens all rays which pass through the optical center emerge from the lens without having been appreciably bent.

605. Graphical construction of images. In most cases it is comparatively easy to locate graphically the position of an image formed by a thin lens. In Fig. 406, O is the object. From the upper end of the object are drawn three rays the paths of which are known. First, a ray which is parallel to the axis will, after refraction, go through the principal focus F2 as shown. A second ray is drawn straight through the optical center of the lens. As stated in the last section, such a ray is not appreciably bent. A third ray passes through the principal focus F1; because it goes through that point, it travels parallel to the axis after emerging from the lens. These three rays meet at a point forming the image of the point from which they started. The image is real and inverted. In actual practice it is not necessary to draw three rays: any two of the three are sufficient. The object and the image of Fig. 406 are at conjugate foci, and their positions may hence

be interchanged.

I

F

F

In Fig. 407 is given the graphi- FIG. 407. Virtual image formed by

cal construction for an object in

side the principal focus. Here, as

a converging lens

before, the image of only one point of the object is found. To attempt more would only cause confusion. Three rays are drawn as before. The ray parallel to the axis is bent down so that it passes through the principal focus F2; the second is drawn straight through the optical center of the lens; and the third, which has the same direction as if it came from F1, travels, after emergence, in a direction parallel to the axis. It is seen in the figure that the three

emerging rays are divergent and will never meet. However, they appear to come from the virtual image 1. This virtual image is erect (that is, right side up) and

0

F

magnified.

F

The two graphical constructions FIG. 408. Virtual image formed

which have just been given show

(1) that a real image is obtained

by a diverging lens

when the object is outside the principal focus, and (2) that a virtual image is obtained when the object is inside the principal focus. It is easy to verify these statements by simple experiments with a lens. Diverging lenses, except when used in conjunction with other lenses, never form real images. No matter where the object is, the image is always virtual. This may be proved by the graphical method (Fig. 408). In this type of lens the principal foci, F, and

1

F2, are virtual and must be treated differently from those of Figs. 406 and 407. In this figure the ray drawn from O parallel to the axis is bent so as to appear to come from F1; the ray drawn from O toward F2 will be parallel, after emergence, to the axis. In this case the emerging rays are divergent, and appear to come from an image at I. The image is virtual, erect, and diminished.

2

606. The quantitative law of position of image. There is a law, similar to that for mirrors, which gives the quantitative relation between the distances of the object, the image, and the principal focus from the lens. If p is the distance of the object from the lens, q the distance of the image from the lens, and f the distance of the principal focus from the lens, then

[blocks in formation]

As a numerical illustration suppose that a lens which has a focal length (the distance from the lens to the principal focus) of 30 centimeters is placed 40 centimeters from the object. Where will the image be? In this case p= 40 and f=30. Substituting these values in equation (22), we obtain

hence

and

1 1 1
+-=
;
30

40

[blocks in formation]

Hence the image will be located 120 centimeters from the lens. The two points-one 40 centimeters and the other 120 centimeters from the lens-are conjugate foci. If the object is placed 120 centimeters away, the image will be found at the 40-centimeter point. This can easily be proved by equation (22).

The student should verify the following propositions for the case of a lens with a focal length of 30 centimeters:

[blocks in formation]

In the last case the object is at one of the principal foci.

It has been pointed out that when the object is inside the principal focus, the image is virtual. In terms of the notation of equation (22), this means that is less than f. In general, we have, from equation (22),

[blocks in formation]

From this relation it can be seen that if p is less than f and hence 1/p is larger than 1/f, the right-hand side will always be negative. Hence, in the use of equation (22) a negative value of q means a virtual image.

607. Derivation of the quantitative law. The relation given by equation (22),

11
+
Pq

1

will now be derived. In Fig. 409, AB is the object and EG the image. The parallel ray AC passes, after refraction, through the principal focus F. The ray AD passes through the optical center of the lens without

[blocks in formation]

The left-hand members of the last two equations are both equal to the same ratio; hence they must be equal to each other:

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small]

From this, by a simple transposition, we have the desired equation (equation (22)),

111
+
P
f

[ocr errors]

It should be noticed that in the construction of Fig. 409 it was assumed that the lens was thin. In general, the methods explained in this chapter apply only to thin lenses.

608. The quantitative law applied to diverging lenses. In a diverging lens the principal focus is a virtual one (see Fig. 401). In section 606 it was pointed out that when the image is virtual, the distance q is a negative quantity. For a similar reason the focal length f of a diverging lens is negative. Since the lenses are often rated in terms of their focal lengths, a diverging lens is often called a negative lens.

The quantitative law given by equation (22) can be applied to diverging lenses if the focal length is taken as negative.*

* It is best to form the habit of always writing the quantitative law in the form in which it appears in equation (22). If the image or principal focus is virtual, the negative sign should be used when the numerical value is inserted in the equation; in other words, let the negative sign always go with the numerical quantity and not with the algebraic quantity. This simple rule will enable one to avoid much confusion.

« PředchozíPokračovat »