torsion head respectively, that must exist where we are dealing with repulsions, are indicated by the opposite directions of graduation. In this figure the needle ns answers to the needle a b of § 1. Supposing now that we wish to compare the strength of two magnetic poles.

We start with needle and torsion index both at o°. There is a hole in the glass top to the instrument in such a position that we can lower the one or the other pole of the magnets we are comparing into the place occupied by the pole of the needle n s when this lies over o°. The needle is swung to one side, and the first pole, which we will call 1, lowered. As we always take care to lower a pole similar to that pole of the needle whose place at o it takes, we have the needle deflected until the moment of the force of repulsion about the axis of suspension of the needle just balances the moment of the couple due to the twisting of the wire.

As a rule we shall find it advisable to turn the torsion head, and so to twist the wire, until the needle is forced back to some angle 0° of deflexion from zero, this angle being much less than the original angle of deflexion. One reason for this is that if we cause to be small, we may neglect the restoring couple due to the earth's field.

Let us suppose that the torsion head has been turned through (n1 × 360 + a1). Then the total twist on the wire will be (n1 × 360 + a1 + 0)°; and the couple that it exerts will be measured by k (n1 × 360 + «¡ + H), where k is some constant that depends upon the nature and dimensions of the wire.

Now this couple is balanced by the moment about O due to the action of the pole u, upon the needle ns; and this moment will be measured by the product μ, h. In this product μ, represents the pole strength of the first magnet; while h is some quantity depending upon the magnetic moment of the needle n s and upon the angle of deflexion, and so will be constant while is constant and while n s remains unaltered.

Since the two are in equilibrium, we have

Repeating the process with the second magnet pole μ, and turning the torsion head through such a number of degrees that remains constant, we have

Notes.-(i.) The reader should notice that we here compare primarily the two moments acting upon the needle ʼn s in the two cases respectively.

But if the magnetic moment of ns, and also the angle 0, remain unaltered, it follows that we are really comparing the two field-strengths due to the pole μ, and μ2 respectively.

And finally, since these field-strengths must be proportional to the polestrengths μ, and μ, respectively when the conditions of distance and situation remain constant, it follows that we are comparing μ, and μ., as was desired.

(ii.) We assume that the magnetic moment of the needle ns is constant. As pointed out at the end of § 3, this is not true absolutely; and under certain conditions the error, due to the alteration in this magnetic moment, may be great.

This method may also be employed to observe the distribution of evident magnetism along a bar-magnet; since we may, without otherwise altering the process, lower the bar-magnet so that the different portions of it may be successively opposite to the pole n of the needle.

It will be found that the evident magnetism is distributed along a good bar-magnet somewhat as here represented by the

diagram. In this the height of any ordinate such as a a' represents the strength of polarity at the point a in the magnet.

If we find the centres of gravity of the two areas included between the axis of the bar-magnet, the ordinates at its two ends, and the curve; and if through these centres of gravity we draw lines perpendicular to the axis of the magnet, meeting it at P and P', these points P and P' may be called the principal poles of the bar. If we consider any external point that is so remote from the bar that straight lines drawn from it to the extremities of the latter are practically parallel, then the action at this point will be the same as if the polarities of the bar were concentrated at the two points P and P' respectively. But for points that are relatively near to the magnet, the action is not so simple.

In using the torsion balance it must not be forgotten that the needle n s is under the influence of the earth's field. This obliges us to place the instrument in such a position that the needle, when under the earth's action only, stands at the zero mark, or oo. When the needle is displaced through an angle the earth's field gives a restoring couple that is proportional to sin; and this is added to the torsion couple.

If be made small, we may often neglect this couple in comparison with the large couple due to torsion; since the smaller is, the smaller is the earth's couple, and the larger is the torsion couple. Or we may find experimentally what angle of torsion is required to displace the needle through ° under the earth's action only. If the angle of torsion required is 3o, then we must add 3 to the angle through which the torsion head was turned in the experiment.

§ 3. Method of Oscillations.-The figure represents a uniform magnetic field of strength H, and a needle n s of magnetic moment m placed in this field.

n

If disturbed from this position of rest, the needle will oscillate more or less quickly; the number of uscillations per second depending upon the field

strength H, the magnetic moment m, and mass and shape of the needle.

Now it can be shown that in all such cases of oscillation(whether of magnetic needles or of pendula or of elastic rods)

the time of a single oscillation is independent of the angle of oscillation, provided that this is very small. If this angle do not exceed a few degrees we may consider that this law is true. Hence we can speak of 'the rate of oscillation,' without specifying the extent of oscillation in degrees. In the case we are considering it can be shown by mechanics that for the same needle

The product H m is proportional to the square of the number of oscillations per second.-This gives us a means of comparing, or of measuring, two field-strengths; for we may oscillate in the two respectively a needle of constant magnetic moment m, counting the number of oscillations per second.-We may also compare two pole-strengths; since, at constant distance, the field strengths will be proportional to these.

In this method we must reckon in the earth's action; since we cannot here make it relatively so small as to be negligible.

We so arrange matters that the lines of force of the field that we are considering act in the same direction as those due to the earth; e. so that the two fields, acting upon the needle, are simply added.

Let the needle under the earth's action only, or in the field h, oscillate O times per second; under the earth's action + the first field, or in the field h + H1, let the number be O,; and under the earth's action + the second field, or in the field h + H2, let the number be Og.

Then by the mechanical law quoted above we have

where k is some constant involving the mass and dimensions of the needle.

Subtracting (i.) and (ii.) we have m H, = k (0,2 - 0,2)

land subtracting (i.) and (iii.) we have m H,= k (O22 — 0 ̧2).

From this we obtain by division that

This gives us also the ratio of the two pole-strengths, if the

distance be constant.

Experiment.—Such experiments are best made with a short massive needle ; the oscillations being counted for a sufficient length of time to give accurately the number per second.

Notes.i.) The needle must be so short that there is no practical difference in the field-strengths at its two ends.

(ii.) The reader must notice that we assume that m is constant in fields of different strengths. This is unfortunately not true; as m will, as a rule, change when the field-strength changes. But the error is not great provided that the fields are so weak that the magnetism of the needle is found not to be permanently affected.

Comparison of the magnetic moments of two needles.-This method enables us to compare the magnetic moments of two needles. For if we oscillate them in the same field H, e.g. that due to the earth, we have

(mHk. 0,2
\m' H = k'. O22

for the two respectively. From this it follows that

Here, the constants k and k' depend upon the masses and dimensions of the two needles respectively; and may be readily calculated, provided that the needles are of simple forms. If they are identical in mass and in dimensions, we have simply that

§ 4. Laws of Magnetism.-There are two fundamental laws in magnetism.

I. Like poles repel, unlike poles attract, one another.

This simple observed fact needs no comment.

II. The force between two poles varies inversely as the square

of the distance between them.

and

It may also be stated that the force between two poles μ is proportional to the product p x μ. But this is hardly a separate law; since the magnitude of μ and μ' would be found by measuring forces of repulsion; and hence we should be reasoning in a circle were we to give this as a third independent law.

The second law is very important, and requires experimental proof.