ƒ = m1m2 / d²
(1).
The force is one of attraction or repulsion, according as the sign of the product m1m2 is negative or positive. The poles at the ends of an infinitely thin uniform magnet, or magnetic filament, would act as definite centres of force. An actual magnet may generally be regarded as a bundle of magnetic filaments, and those portions of the surface of the magnet where the filaments terminate, and so-called “free magnetism” appears, may be conveniently called poles or polar regions. A more precise definition is the following: When the magnet is placed in a uniform field, the parallel forces acting on the positive poles of the constituent filaments, whether the filaments terminate outside the magnet or inside, have a resultant, equal to the sum of the forces and parallel to their direction, acting at a certain point N. The point N, which is the centre of the parallel forces, is called the north or positive pole of the magnet. Similarly, the forces acting in the opposite direction on the negative poles of the filaments have a resultant at another point S, which is called the south or negative pole. The opposite and parallel forces acting on the poles are always equal, a fact which is sometimes expressed by the statement that the total magnetism of a magnet is zero. The line joining the two poles is called the axis of the magnet.
Magnetic Field.—Any space at every point of which there is a finite magnetic force is called a field of magnetic force, or a magnetic field. The strength or intensity of a magnetic field at any point is measured by the force in dynes which a unit pole will experience when placed at that point, the direction of the field being the direction in which a positive pole is urged. The field-strength at any point is also called the magnetic force at that point; it is denoted by H, or, when it is desired to draw attention to the fact that it is a vector quantity, by the block letter H, or the German character ℌ. Magnetic force is sometimes, and perhaps more suitably, termed magnetic intensity; it corresponds to the intensity of gravity g in the theory of heavy bodies (see Maxwell, Electricity and Magnetism, § 12 and § 68, footnote). A line of force is a line drawn through a magnetic field in the direction of the force at each point through which it passes. A uniform magnetic field is one in which H has everywhere the same value and the same direction, the lines of force being, therefore, straight and parallel. A magnetic field is generally due either to a conductor carrying an electric current or to the poles of a magnet. The magnetic field due to a long straight wire in which a current of electricity is flowing is at every point at right angles to the plane passing through it and through the wire; its strength at any point distant r centimetres from the wire is
H = 2i / r,
(2)
i being the current in C.G.S. units.[4] The lines of force are evidently circles concentric with the wire and at right angles to it; their direction is related to that of the current in the same manner as the rotation of a corkscrew is related to its thrust. The field at the centre of a circular conductor of radius r through which current is passing is
H = 2πi / r,
(3)
the direction of the force being along the axis and related to the direction of the current as the thrust of a corkscrew to its rotation. The field strength in the interior of a long uniformly wound coil containing n turns of wire and having a length of l centimetres is (except near the ends)