strength of the pole and on the strength of the field in which it is placed.

The strength or intensity of a magnetic field at any point is the force in dynes on a unit magnet pole placed at that point.

Thus the earth field has a strength 0.5 at a point where a magnetic pole of unit strength is acted on with a force of 0.5 dynes.

When a pole of strength m is at a point where the strength of field is H, it is acted on by a force of Hm dynes.

In any field of force the two poles of a magnetic needle are urged in opposite directions. The direction in which the north pole tends to move is known as the positive direction of the line of force at that point.

a

+Hm

+m

490. Magnetic Moment.-Suppose that in a magnetic field of strength H, a magnetic needle is placed in such a position that the line joining its poles makes an angle a with the lines of force of the field. Let m represent the strength, or number of units in its north pole, and -m the strength of its south pole. Then the north pole is urged with a force Hm in the positive direction of the lines of force of the field and the south pole experiences an equal force in the opposite direction. These equal and parallel forces constitute a couple whose moment is Hml sin a, where is the distance in centimeters between the poles mm' of the magThe quantities m and I belong to the magnet and their product ml is known as the magnetic moment of the magnet, and is represented by M.

-m

net.

-Hm

FIG. 261.

Thus the magnetic moment of a magnet may be defined as the product of the strength of one of its poles by the distance between them.

The couple which acts on the magnet may then be expressed by the formula,

HM sin a

491. Period of Oscillation of a Magnet in a Magnetic Field.If a magnetic needle in a field of force is disturbed from its position of rest, it will vibrate to and fro just as a pendulum

E

A

oscillates in the field of the earth's attraction. The period of one complete oscillation of a pendulum has been shown to be

where K is the moment of inertia of the pendulum and mgh is a quantity which when multiplied by the sine of the angle of inclination of the pendulum gives the moment of force which at that instant urges it toward its equilibrium position.

In case of the oscillating needle the mechanical conditions involved are the same, except that the couple causing the motion is due to magnetism instead of to gravitation. The factor HM is the quantity which when multiplied by sin a gives the couple acting to turn the magnet; it therefore plays the same part in this case as the factor mgh in case of the pendulum.

Hence the period of oscillation of a magnet in a field where the strength is H units, is

where K is the moment of inertia of the magnet and M is its magnetic moment. It should be observed that in this case as in that of the pendulum the formula gives the period when the arc is exceedingly small. With large arcs of vibration the period is longer.

From the above formula it is clear that the stronger the field of force at a point where a magnetic needle is placed the more rapidly it will oscillate when set in vibration. This is the explanation of the rapid quivering of a compass needle when brought near the pole of a magnet.

Problem.

1. Find the ratio of the strengths of field at two places when a certain magnetic needle oscillates n times per sec. at one place and n' times per second at the other.

492. Strength of Field at a Point Near a Magnet.-The direction and intensity of the force near a magnet may be calculated as follows. m be the strengths of the two poles of the magnet, and let r be the distance from the point P to the

Let m and
Let m and

north pole of the magnet and let r' be its distance from the south pole. Then if a unit pole were at P it would be subject

m

m

to a force in the direction a, and to a force in the direction r2 (r')2

b. Laying off distances a and b proportional to the amounts of these two forces, the resultant force will be represented on the same scale by the diagonal of the parallelogram on a and b. The resultant is of course tangent to the line of force at P.

It is sometimes desirable to calculate the force due to a magnet at a point P in line with the axis of the magnet as shown in Fig. 263. Let m be the strength of each of the poles, r the distance of P from the center of the magnet, and I the distance of either pole from the center of the magnet. Then by Coulomb's law the force on a unit north pole placed at P due to the north pole is and is directed toward the right. That due to

m

(r-l)2

the other pole is

m

(r+1) 2

and is toward the left. The resultant

force at P is then toward the right and may be written

If is small compared with r the terms involving 12 and 14 in the denominator may be neglected, as they are insignificant compared with r', so approximately

where M = 2ml, the magnetic moment of the magnet.

Problem.

1. Calculate in a similar manner the strength of field at a point on a line drawn through the center of the magnet at right angles to its axis.

TERRESTRIAL MAGNETISM

493. Declination of the Magnetic Needle. The compass needle, mounted so as to rotate in a horizontal plane, does not in general point directly

ལ་ ང་

FIG. 264-Declination of compass.

north, but a few degrees east or west of north, and this deviation is called its declination. In observing the declination it will not do to assume that the magnetic axis of the needle is in the same direction as its axis of figure. If the magnetic axis is as represented by the dotted line in figure 264 then the apparent declination is in the first case too small. If the needle is now turned over and suspended with the opposite side upward, it will give too great an apparent declination. The mean of the two will be the true declination. This direction is called the magnetic meridian.

494. Dip or Inclination.-It was observed by Hartmann (1489-1564) that if a needle was balanced before being magnetized the north end would dip downward after magnetization. The so-called dipping needle was first made by Norman, a London instrument maker, who mounted a needle on a horizontal axis so that it could swing freely in a vertical circle. The needle was then carefully balanced so that it would stand in any position before magnetization. But after it was magnetized it was observed that the north pole pointed downward some 70°

below the horizontal if the plane in which it turned was north and south by the compass.

To diminish friction the cylindrical pinions on the ends of the axis of the dipping needle usually rest on horizontal plates of polished agate, on which they roll as the needle turns.

In this case also the needle must be reversed, the side toward the east being turned toward the west, to guard against error due to the axis of the needle not being in line with the direction of its magnetization. To guard against any want of balance in the needle, it should be magnetized over again with its poles reversed, and the dip again observed. If the needle is well constructed, the mean of these four observations will be the required dip or inclination.

495. Resultant Direction of Magnetic Force. The dipping needle gives the direction of the resultant magnetic force at any

n

FIG. 266.-Lines of force of the earth.

point. It is found that near the equator the needle is horizontal, as it is taken north its north pole points downward by an amount which increases steadily till at some point northwest of Hudson Bay it points vertically downward. That point is called the north magnetic pole, and near there a horizontal compass needle would have no directive tendency and would be useless. North of that point the north pole of the

compass needle would point south.

In vessels that change their latitude greatly the compass needle must be provided with a little sliding weight or counterpoise to correct its dipping tendency. South of the equator the south pole of the needle dips downward.

Figure 266 shows the probable form of the lines of magnetic force around the earth; of course the direction of these lines of force is known only at the earth's surface. The magnetic condition of the interior of the earth is entirely unknown.

The declination or deviation of the resultant force from the