Magnetic Potential

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 equi-potential 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 perpendicular 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⁾

(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): 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 ΣFσ 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 = 4πM, where M is the quantity of the real or imaginary matter in question contained inside the closed surface. 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 σ' are the areas of the two ends of the tube, F and F' the forces there; then Fσ = Fσ'. 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 σ being the measure of an area and F the force there, Fσ is constant for every element of the surface. 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 σ'. Let F' be force at σ', then by propositions (2) and (3) F'σ' 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 = κ. By properly choosing the scale of the drawing, κ may be made equal to unity. Hence F = 1/σ, 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. 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.