13. A concave lens has a focal length of 16 cm.

What

is the conjugate focus of a point 10 cm. from the lens?

32. Images. To construct the image of an object formed by a lens the following construction will be found the simplest :

R

L

B

FIG. 55.

Let Z be the lens, Fand F' its principal foci; I one extremity of the object. To find I', the image of I. We choose the ray parallel to the principal axis and the ray passing through O, the optical center. Draw the path of the ray IABF'I' by the construction given on page 131. Draw the straight line IOI'. The intersection of these two rays determines the position of the image of I. In a similar manner locate the image of the other extremity of the object.

NOTES.-1. Except where great accuracy is desired or in very large drawings, the path IABI' may be determined merely by estimating the refraction. It will be noticed that the construction need be applied only to the incident ray. The emergent ray will always pass through the principal focus.

2. The ray IOI' really suffers a slight lateral displacement. This is negligible if the lens is thin.

3. Remember that in all cases the ray parallel to the axis and the ray passing through the optical center are sufficient to locate the image.

SIZE OF IMAGE AND OBJECT.-The triangles IRO and I'SO (Fig. 55) are similar (mutually equiangular). Therefore

or, the linear dimensions of object and image are to each other as their distances from the lens.

LVIII. IMAGES.

1. Draw a diagram to show the size and position of the image of an object distant twice the focal length from a lens.

2. Show by diagram that if the object be at the principal focus no image will be formed.

3. Draw a diagram to show the image of an object at a considerable distance from a lens.

4. Show by a diagram that if the object be within the principal focus the emergent rays will diverge, and their intersection will be on the same side as the object, the image thus being virtual.

5. Apply the construction to the determination of the size and location of the image (real or virtual ?) formed by a double-concave lens.

6. If the image of a candle-flame is formed 20 cm. from a lens when the flame itself is 10 cm. from the lens, how large is the image compared with the flame ?

7. At what distance from a convex lens must an object be placed so that the image may be half the size of the object?

8. What is the ratio of the size of the object to the size of the image in Prob. 8, page 135?

9. Given a lens of focal length 30 cm., where must an object be placed that the image may be―

(a) The same size as the object?

(b) Half as large?

(c) Twice as large ?

10. When an object is placed 2 in. from a lens of focal length 3 in., where is the image formed and what is its length if the object is 1 in. long?

11. What is the magnifying power of a lens of 30 cm. focal length when the object is 15 cm. from the lens?

12. The image of a clock-face is thrown upon a screen. The time is 12.30. Make a drawing of the image as seen

by an observer looking from the lens. (Harvard Entrance Examination.)

LIX. OPTICAL INSTRUMENTS.

Draw diagrams to show the principles of the following optical instruments:

1. Crystalline lens and retina of the eye.

2. Simple magnifying-glass or microscope.

3. A simple form of compound microscope. Let L (Fig. 56) be the eyepiece, O the object-glass, and AB the object. Trace the rays through O to form a real inverted image of

L

FIG. 56.

AB between the lenses and thence through L to form the virtual image of the real image.

4. A simple form of a refracting telescope (object at long distance from large object-glass and small eyepiece magnifying the small real image formed by object-glass).

5. Simple, single-lens, photographic camera.

6. Stereopticon or magic lantern (a light at the principal focus of large lens (condenser), object in the path of the emergent parallel rays, system of small lenses (or one powerful lens) at such a distance as to form real, inverted, and magnified image on a distant screen.

7. Galilean telescope or opera-glass (large objective of comparatively short focal length; double-concave lens making divergent the rays from a distant object which would (if the concave lens were not interposed) converge at the focus of the objective.

CHAPTER XI.

MAGNETISM.

DEFINITIONS.-Unit pole; unit field; lines of magnetic force; flow or flux of force; magnetic potential; electromagnet; magnetomotive force; permeability; paramagnetic; diamagnetic ; reluctance; terrestrial magnetism; magnetic poles of the earth.

MAGNETIC POLES AND FIELDS.

33. Laws of Magnetic Poles.—

(a) Like magnetic poles repel each other; unlike magnetic poles attract each other.

(b) The force between two given poles is inversely proportional to the square of the distance.

(c) The force exerted at a given distance between two poles is directly proportional to the product of the strengths of the poles. The last two laws are summarized as follows: When q and q' are the strengths of the poles respectively, d the distance between them, and F the force in dynes between them, then

F=

LX. MAGNETIC POLES AND FIELDS.

1. Show from the formula just given what is meant by "unit pole."

SUGGESTION. Suppose q, q', and d each equal unity.

2. What force in dynes will be exerted on a unit pole by a pole of strength 20 at a distance of 5 cm. ?

3. Two equal magnetic poles repel each other with a

force of 100 dynes when they are 5 cm. apart. What is the strength of each (in unit-pole-strengths)?

4. If two poles attract each other with a force of 300 dynes when 3 cm. apart, what will be the force between them at 10 cm. ?

5. It is found that a force of 2 gm. must be applied to keep two magnet poles from approaching each other when they are 2 cm. apart. It is known that the strength of one pole is to the strength of the other as 7 is to 3. Find strength of each pole. (1 gram = 980 dynes.)

6. A magnetic pole of strength 30 is attracted by a second pole of strength 40 placed exactly 4 cm. east of it, and by a third pole of strength 10 placed 2 cm. exactly west. In which direction and with what.force will the first pole move?

7. Two magnets are known to have equal strengths. When the N. pole of one is placed at a distance of .5 cm. from the S. pole of the other, it is found that a force of 6 gm. is required to hold them apart. Assuming that the magnets are so long that the effect of the more distant poles may be disregarded, find the strength of the poles. (1 gram 980 dynes.)

8. Two poles of strengths 17 and 18 repel each other with a force of 306 dynes. How far apart are they?

NOTE I. The intensity of a magnetic field at any point is measured by the force exerted on a unit magnetic pole at the point. Therefore the intensity of the field about a magnetic pole varies inversely as the square of the distance of the point from the pole and directly as the strength of that pole.

NOTE II.—In all these problems the field is considered as created only by one pole of a magnet, and by no other magnetic influences. The conditions can be approximately realized in practice by one pole of a very long slender magnet removed to a considerable distance from other magnetic influences. Even then, however, the earth-magnetism is always present.

9. The strength of a certain magnet pole is 50, What