magnets the magnetisation differs from uniformity. No two single points can strictly be taken as centres of force completely representing the action of the magnet. For many practical purposes, however, a well-made bar magnet may be treated as solenoidal with sufficient accuracy; that is to say, its action may be regarded as due to two poles or centres of force, one near each end of the magnet.

The following are the laws of force between two magnetic poles:

(1) There is a repulsive force between any two like magnetic poles, and an attractive force between any two unlike poles.

(2) The magnitude of the force is in each case numerically equal to the product of the strength of the poles divided by the square of the distance between them.

This second law is virtually a definition of the strength of a magnetic pole.

In any magnet the strength of the positive pole is equal in magnitude, opposite in sign, to that of the negative pole. If the strength of the positive pole be m, that of the negative pole is -m. Instead of the term 'strength of pole,' the term 'quantity of magnetism' is sometimes used. We may say, therefore, that the uniformly and longitudinally magnetised thin cylindrical bar behaves as if it had quantities m and -m of magnetism at its two ends, north and south respectively; we must, however, attach no properties to magnetism but those observed in the poles of magnets. then, we have two magnetic poles of strengths m and m', or two quantities of magnetism m and m', at a distance of centimetres apart, there is a force of repulsion between them which, if m and m' are measured in terms of a proper unit, is

mm'r dynes.

If,

If one of the two m or m' be negative, the repulsion becomes an attraction.

The C.G.S. unit strength of pole is that of a pole which

repels an equal pole placed a centimetre away with a force of one dyne."

In practice it is impossible to obtain a single isolated pole; the total quantity of magnetism in any actual magnet, reckoned algebraically, is always zero.

DEFINITION OF MAGNETIC FIELD.-A portion of space throughout which magnetic effects are exerted by any distribution of magnetism is called the magnetic field due to that distribution.

Let us consider the magnetic field due to a given distribution of magnetism. At each point of the field a pole of strength is acted on by a definite force. The Resultant Magnetic Force at each point of the field is the force which is exerted at that point on a positive pole of unit strength placed there.

This is also called the Strength of the Magnetic Field at the point in question.

If H be the strength of the field, or the resultant magnetic force at any point, the force actually exerted at that point on a pole of strength m is m II.

The magnetic force at each point of the field will be definite in direction as well as in magnitude.

DEFINITION OF LINE OF MAGNETIC FORCE.-If at any point of the field a straight line be drawn in the direction of the magnetic force at that point, that straight line will be a tangent to the Line of Magnetic Force which passes through the point. A Line of Magnetic Force is a line drawn in such a manner that the tangent to it at each point of its length is in the direction of the resultant magnetic force at that point.

A north magnetic pole placed at any point of a line of force would be urged by the magnetic force in the direction of the line of force.

As we shall see shortly, a small magnet, free to turn about its centre of gravity, will place itself so that its axis is in the direction of a line of force.

A surface which at each point is at right angles to the

line of force passing through that point is called a level surface or surface of equilibrium, for since the lines of force are normal to the surface, a north magnetic pole placed anywhere on the surface will be urged by the magnetic forces perpendicularly to the surface, either inwards or outwards, and might therefore be regarded as kept in equilibrium by the magnetic forces and the pressure of the surface. Moreover, if the pole be made to move in any way over the surface, since at each point of its path the direction of its displacement is at right angles to the direction of the resultant force, no work is done during the motion.

DEFINITION OF MAGNETIC POTENTIAL.-The magnetic potential at any point is the work done against the magnetic forces in bringing up a unit magnetic pole from the boundary of the magnetic field to the point in question.

The work done in transferring a unit magnetic pole from one point to another against magnetic forces is the difference between the values of the magnetic potential at those points. Hence it follows that the magnetic potential is the same at all points of a level surface. It is therefore called an equipotential surface.

Let us suppose that we can draw an equipotential surface belonging to a certain configuration of magnets, and that we know the strength of the magnetic field at each point of the surface. Take a small element of area, a square centimetres in extent, round any point, and through it draw lines of force in such a manner that if H be the strength of the magnetic field at the point, the number of lines of force which pass through the area a is H a.

Draw these lines so that they are uniformly distributed over this small area.

Do this for all points of the surface.

Take any other point of the field which is not on this equipotential surface; draw a small element of a second equipotential surface round the second point and let its area be a' square centimetres. This area will, of course, be per

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 σ; let F be the resultant force at any point of σ, resolved normally to the surface inwards; let Fσ 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,

Σε σ = 4π Μ,

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 o'. 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 m through a small distance. Let A (fig. 42) be the position of m, P1, P2 the

FIG. 42.

two positions of m'. Then A P2 P, is a straight line, and A P11. Let A P2 = ̃1⁄2, P1 P2 = r] — 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) F'o' 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 Fσ = K.

(5) By properly choosing the scale of the drawing,

σ

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.