# Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/59

405.]

LINES OF MAGNETIC INDUCTION.

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Since the distribution of magnetic induction is solenoidal, the induction through any surface bounded by a closed curve depends only on the form and position of the closed curve, and not on that of the surface itself.

404.] Surfaces at every point of which

 ${\displaystyle la+mb+nc=0\,}$ (19)

are called Surfaces of no induction, and the intersection of two such surfaces is called a Line of induction. The conditions that a curve, s, may be a line of induction are

 ${\displaystyle {\frac {1}{a}}{\frac {dx}{ds}}={\frac {1}{b}}{\frac {dy}{ds}}={\frac {1}{c}}{\frac {dz}{ds}}.}$ (20)

A system of lines of induction drawn through every point of a closed curve forms a tubular surface called a Tube of induction.

The induction across any section of such a tube is the same. If the induction is unity the tube is called a Unit tube of induction.

All that Faraday[1] says about lines of magnetic force and magnetic sphondyloids is mathematically true, if understood of the lines and tubes of magnetic induction.

The magnetic force and the magnetic induction are identical outside the magnet, but within the substance of the magnet they must be carefully distinguished. In a straight uniformly magnetized bar the magnetic force due to the magnet itself is from the end which points north, which we call the positive pole, towards the south end or negative pole, both within the magnet and in the space without.

The magnetic induction, on the other hand, is from the positive pole to the negative outside the magnet, and from the negative pole to the positive within the magnet, so that the lines and tubes of induction are re-entering or cyclic figures.

The importance of the magnetic induction as a physical quantity will be more clearly seen when we study electromagnetic phenomena. When the magnetic field is explored by a moving wire, as in Faraday's Exp. Res. 3076, it is the magnetic induction and not the magnetic force which is directly measured.

### The Vector-Potential of Magnetic Induction.

405.] Since, as we have shewn in Art. 403, the magnetic induction through a surface bounded by a closed curve depends on

1. * Exp. Res., series xxviii.