To indicate the polarity a positive or negative sign is placed before the numerical measure of the strength. Positive and negative numbers are opposite to one another, just as are north poles which emit lines of force, and south poles which absorb them. It is usual to attribute positive values to north poles, and negative values to south poles.

Now on multiplying together two quantities which are both positive or both negative we obtain a positive product, while we know that two poles of the same name repel one another. And again, two quantities of which one is positive and the other negative give on multiplication a negative product, while poles of unlike name attract one another. Thus our formula (2) gives the sense as well as the magnitude of the force, provided we reckon a force of repulsion positive and a force of attraction negative.

In magnetic investigations an important part is played by the so-called unit pole, that is the north pole of a thin bar-magnet, whose length is so great that the field due to one pole is not sensibly disturbed by the presence of the other, and which is magnetised to such an extent that the strength of its pole is exactly equal to one absolute unit.

Such a pole may be approximately realised in practice by suitably magnetising a long thin knitting-needle, by stroking it with a magnet. The strength of pole will then be m=1 absolute unit.

65. Dimensions of strength of pole.-Since we know the dimensions of force in terms of the fundamental units, we may deduce those of strength of pole from the formula (2), which expresses a relation amongst quantities of the nature force, length, strength-of-pole. In the special case where the strengths of the two poles concerned are equal and equal to m, the relation (2) becomes

r2f=m•m or m=rf1.

The force ƒ has the dimensions [L] [M][T]-2, so that its square root has the dimensions [L] [M][T]', while r has

of course the dimension [L]. Hence the dimensions of strength-of-pole are

[L]'[M]'[T]-'

These fractional exponents show that if we are to speak of magnetic masses we shall encounter some difficulties of conception. Such masses would not be related in any simple way to masses of ordinary matter, which can be measured in grams.

66. Determination of strength of pole in absolute measure. We may conveniently proceed as in § 58, measuring in dynes the force which one pole of the bar to be examined exerts upon one pole of a similar bar magnetised to the same extent; the force in question being determined by means of the balance, by finding how many grams have a weight equal to it in amount. If we measure the distance between the mutually acting poles, we can calculate from the law of inverse square what the attraction would be at a distance of 1 cm. This attraction is numerically equal to the product of the strengths of the two equal poles, and thus its square root gives the value required.

In experiment 35, let the thickness d of the glass plate be determined; this is the quantity r of our formula, the distance between the mutually acting poles. From ƒ and r we obtain the value of m.

Numerical example.-Thickness of glass plate d=0·184 cm., and accordingly strength of pole m=d√ƒ12=about 2 absolute units; i.e. m=2 cm. gr. sec.1

67. Magnetic moment.-When the places on a magnet from which lines of force proceed or to which they return are approximately points, the forces exerted on the magnet by external magnetic fields will have these points for their points of application. The rotatory influence of the field upon the magnet is determined by the distance between the two poles, as will be shown in § 74 below.

The strength of either pole m, multiplied by the distance a between the two poles, is called the magnetic moment M of the bar magnet; or

If the magnet is very long and thin (its length being about forty times its diameter) we may assume that its poles are at its two ends.

In this case a=l, the length of the bar.

For bar magnets which are shorter and thicker, the distance between the poles is about of the length of the bar: a=l.

The knitting-needles, the strength of whose poles were determined in § 66 as about 2 absolute units, had a length of 17 cm. Their magnetic moments are thus equal to 34 units: M=34 absolute units of magnetic moment.

The dimensions of magnetic moment are:

[L]3 [M] '[T] ̄1 × [L]=[L] * [M]' T-1

The moment per unit mass of a bar is called its specific magnetism; in ordinary steel bars it seldom exceeds 40 absolute units. In the above example, where the mass of the needle is 9.23 grams, there are only three or four units of specific magnetism.

68. Conception and measurement of strength of field.—If we make exception of the few special cases where we are dealing with long and thin bar magnets, the action of a magnetic field cannot be referred to poles situated at determinate points; there are always more extended regions from which lines of force proceed. It will be best, then, to deal directly with the distribution of lines of force in the field, without attempting to refer the magnetic influence to definite centres. So far we have only inferred from the lines of force the relative magnitudes of the magnetic forces at different places. If we suppose, however, that a unit pole +1 is brought into the field at a point where the strength is to be determined, we shall be able to obtain the strength of field in absolute measure. The pole indeed produces a slight disturbance of the lines of force throughout the field; but we assume that the lines of force are very numerous, so that the few lines introduced by the comparatively weak pole +1 produce but little disturbance except quite near to the pole itself. The corresponding [This, however, would in no way invalidate the measurement.]

negative pole -1 (of the same bar magnet) is supposed to be far enough off to be without appreciable influence. The pole +1 experiences a force due to the tension along the lines of force and the pressure across them, and this force, measured in dynes, gives the direction and magnitude of the magnetic force at the point in question. At a given point in a given field the force exerted on a pole of strength m will be m times as great as that on the pole +1. The magnetic force within any field may thus be completely determined if we adopt the following definition of the absolute unit of measurement :

The strength of field, or intensity of field, at any point is measured by the force in dynes exerted upon a unit pole placed at that point.

Following MAXWELL, we shall denote this quantity by H.

Experiment 37.-Suspend from the shorter scale of a hydrostatic balance one of the bars mentioned in § 64, and add weights to restore equilibrium. Approximate to this bar from below the pole of a vertically supported bar magnet, and let the weights which must now be added to restore equilibrium be P1, Ph... grams, corresponding to the distances r1, 2, . . . between the attracting poles. The field intensities at points along the axis of the bar magnet and at distances r1, r2, from the pole are thus 981 p1, 981 P2,

units.

We shall find later that the intensity of the earth's magnetic field in this part of the world is about 0.5 such units. Thus the north-seeking pole +1 will be acted upon by a force of about half a dyne directed obliquely downwards (parallel to the axis of the dipping-needle), the south-seeking pole -1 being acted upon by an equal force directed obliquely upwards. Since a dyne is approximately equal to the weight of a milligram, the forces here specified will be equal to the weight of about half a milligram, and therefore very small compared with the weight of the needle itself. We may say then, generally, that the earth's magnetic field gives rise to much smaller forces than does its gravitational field.

69. Mechanical force exerted upon a pole of determinate strength by a field of given intensity.-Our definition of field

intensity leads at once to the value of the force exerted on a magnetic pole of any strength m. We shall suppose that, as in the case of the pole + 1, the field is sensibly unaltered by the introduction of the pole m.' If the former pole experiences a force of H dynes (the measure of the given field-intensity), the force acting on the pole m will be f mh dynes, in the direction of the lines of force.

=

70. Dimensions of field intensity. We estimate the intensity of a field by the effect produced upon a pole of determinate strength introduced into the field. From ƒ = 5 m it follows that = f/m. The dimensions of field-intensity are to be found by dividing those of force by those of strength of pole. Thus the dimensions of are: [L] [M] [T]2 ÷ [L] 1 [M] ' [T]'' = [L]-* [M]* [T]-'

Corresponding to the relation f=m for a magnetic field, we have in the case of gravity the weight p=gμ, where g is the acceleration of a freely falling body, and μ is the mass of the body whose weight is p. Thus the acceleration g measures the intensity of the earth's gravitational field.

The difference in the dimensions of g and H is due to the difference between the quantities with which the force ƒ is associated; the gram in the one case, the unit pole in the other.

71. Determination of field-intensity from the number of lines of force. The introduction of a unit pole into the field and the measurement of the force exerted upon it is a fiction like the unit pole itself, and is only intended to make clear the conception of field-intensity. The practical application of the principle, quite apart from other difficulties, would always be upset by the circumstance already mentioned; the introduction of the unit pole deforms the field, so that we never measure it as it really is."

For the comparative estimation of field-intensities throughout any given region, we have already made use of the number of lines of force which cut through an area of 1 cm.2 taken perpendicularly to their direction. It was Faraday who developed this method of counting the lines.

Cf. footnote, p. 88.

2 But see footnote, p. 88.