pendicular to the lines of force which pass through it. Suppose that the number of lines of force which pass through this area is n', then it can be proved, as a consequence of the law of force between two quantities of magnetism, that the strength of the field at any point of this second small area a' is numerically equal to the ratio n'\a'.

The field of force can thus be mapped out by means of the lines of force, and the intensity of the field at each point determined by their aid.

The intensity is numerically equal to the number of lines of force passing through any small area of an equipotential surface divided by the number of square centimetres in that area, provided that the lines of force have originally been drawn in the manner described above.1

For an explanation of the method of mapping a field of force by means of lines of force, see Maxwell's Elementary Electricity, chaps. v. and vi. and Cumming's Electricity, chaps. ii. and iii. The necessary propositions may be summarised thus (leaving out the proofs) :

(1) Consider any closed surface in the field of force, and imagine the surface divided up into very small elements, the area of one of which is ; let F be the resultant force at any point of σ, resolved normally to the surface inwards; let Fo denote the result of adding together the products Fσ for every small elementary area of the closed surface. Then, if the field of force be due to matter, real or imaginary, for which the law of attraction or repulsion is that of the inverse square of the distance,

ΣFσ = 4 TM,

where M is the quantity of the real or imaginary matter in question contained inside the closed surface.

(2) Apply proposition (1) to the case of the closed surface formed by the section of a tube of force cut off between two equipotential surfaces. [A tube of force is the tube formed by drawing lines of force through every point of a closed curve.]

Suppose σ and o' are the areas of the two ends of the tube, F and F the forces there; then Fσ = F'o'.

(3) Imagine an equipotential surface divided into a large number of very small areas, in such a manner that the force at any point is inversely proportional to the area in which the point falls. Then a being the measure of an area and F the force there, Fo is constant for every element of the surface.

(4) Imagine the field of force filled with tubes of force, with the elementary areas of the equipotential surface of proposition (3) as bases. These tubes will cut a second equipotential surface in a series of elementary areas. Let F' be force at o, then by propositions (2) and

On the magnetic potential due to a single pole.-—The force between two magnetic poles of strengths m and m', at a distance r, centimetres apart is, we have seen, a repulsion of mm'r,2dynes. Let us suppose the pole m' moved towards through a small distance. Let A (fig. 42) be the position of m, P1, P2 the

A

2

FIG. 42.

P3 PP

two positions of m'. Then A P2 P, is a straight line, and A P1 =1. Let A P2 = 72, P1 P2 = r 1 — r 2.

Then, if, during the motion, from P, to P2, the force remained constant and of the same value as at P1, the work done would be

while if, during the motion, the force had retained the value which it has at P2, the work would have been

Thus the work actually done lies between these two values. But since these fractions are both very small, we may neglect the difference between r1 and r2 in the denominators. Thus the denominator of each may be

(3) Fo' is constant for every small area of the second equipotential surface, and equal to Fσ, and hence Fσ is constant for every section of every one of the tubes of force; thus Fo= K.

I

(5) By properly choosing the scale of the drawing, κ may be made equal to unity. Hence F or the force at any point is equal to the number of tubes of force passing through the unit of area of the equipotential surface which contains the point.

(6) Each tube of force may be indicated by the line of force which forms, so to speak, its axis. With this extended meaning of the term 'line of force' the proposition in the text follows. The student will notice that, in the chapter referred to, Maxwell very elegantly avoids the analysis here indicated by accepting the method of mapping the electrical field as experimentally verified, and deducing from it the law of the inverse square.

written 12 instead of r12 and r2 respectively. The two expressions become the same, and hence the work done is

or

2

Similarly the work done in going from P, to a third point, P3, is

And hence we see, by adding the respective elements together, that the work done in going from a distance to a distance is

r

Hence the work done in bringing the pole m from infinity to a distance from the pole m is mm'/r. But the potential due to m at a distance r, being the work done in bringing up a unit pole from beyond the influence of the pole m, will be found by dividing this by m'; it is therefore equal to m/r.

Again, it follows from the principle of conservation of energy that the work done in moving a unit pole from any one point to any other is independent of the path, and hence the work done in moving the unit pole from any point whatever at a distance to any point at a distance from the pole m is

For a single pole of strength m, the equipotential surfaces are clearly a series of concentric spheres, with m as centre; the lines of force are radii of these spheres.

If we have a solenoidal magnet of strength m, and r1, r1⁄2 be the distances of any point, P (fig. 43), from the positive

and negative poles N and s of the magnet, then the potential at P due to the north pole is m/r1, and that due to the south pole is -m/ra; hence the potential at p due to the magnet is

N

FIG. 43.

where c is a constant quantity, and the lines of force are at right angles to these surfaces. To find the resultant magnetic force at p we have to compound a repulsion of m/r, along N P with an attraction of m/r, along Ps, using the ordinary laws for the composition of forces.

Let us now consider the case in which the lines of force in the space in question are a series of parallel straight lines. uniformly distributed throughout the space.

The intensity of the field will be the same throughout; such a distribution constitutes a uniform magnetic field.

The earth is magnetic, and the field of force which it produces is practically uniform in the neighbourhood of any point provided that there be no large masses of iron near, and the lines of force are inclined to the horizon in these latitudes at an angle of about 67°.

On the Forces on a Magnet in a Uniform Field.

We proceed to investigate the forces on a solenoidal magnet in a uniform field.

Let us suppose the magnet held with its axis at right angles to the lines of force, and let / be the distance between its poles, m the strength of each pole, and H the intensity

of the field. The north pole is acted on by a force m H at right angles to the axis of the magnet, the south pole by an equal, parallel, but opposite force m H. These two forces constitute a couple; the distance between the lines of action, or arm of the couple, is , so that the moment of the couple is ml H. If the axis of the magnet be inclined at an angle 0 to the lines of force, the arm of the couple will be ml sin 0, and its moment m/H sin 0. In all cases the couple will depend on the product ml.

DEFINITION OF MAGNETIC MOMENT OF A MAGNET.— The product of the strength of either pole into the distance between the poles, is called the magnetic moment of a solenoidal magnet. Let us denote it by м; then we see that if the axis of the magnet be inclined at an angle to the lines of force, the couple tending to turn the magnet so that its axis shall be parallel to the lines of force is M H sin 0. Thus the couple only vanishes when is zero; that is, when the axis of the magnet is parallel to the lines of force.

But, as we have said, the actual bar magnets which we shall use in the experiments described below are not strictly solenoidal, and we must therefore consider the behaviour, in a uniform field, of magnets only approximately solenoidal.

If we were to divide a solenoidal magnet into an infinitely large number of very small, equal, similar, and similarly situated portions, each of these would have identical magnetic properties; each would be a small magnet with a north pole of strength m and a south pole of strength -m.

If we bring two of these elementary magnets together so as to begin to build up, as it were, the original magnet, the north pole of the one becomes adjacent to the south pole of the next; we have thus superposed, a north pole of strength. m and a south pole of strength -m; the effects of the two at any distant point being thus equal and opposite, no external action can be observed. We have therefore a magnet equal in length to the sum of the lengths of the other two, with two poles of the same strength as those of either.