Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/63

CHAPTER III.

MAGNETIC SOLENOIDS AND SHELLS^[1].

On Particular Forms of Magnets.

407.] If a long narrow filament of magnetic matter like a wire is magnetized everywhere in a longitudinal direction, then the product of any transverse section of the filament into the mean intensity of the magnetization across it is called the strength of the magnet at that section. If the filament were cut in two at the section without altering the magnetization, the two surfaces, when separated, would be found to have equal and opposite quantities of superficial magnetization, each of which is numerically equal to the strength of the magnet at the section.

A filament of magnetic matter, so magnetized that its strength is the same at every section, at whatever part of its length the section be made, is called a Magnetic Solenoid.

If m is the strength of the solenoid, ds an element of its length, r the distance of that element from a given point, and ε the angle which r makes with the axis of magnetization of the element, the potential at the given point due to the element is

${\frac {mds\cos \epsilon }{r^{2}}}={\frac {m}{r^{2}}}{\frac {dr}{ds}}ds.$

Integrating this expression with respect to s, so as to take into account all the elements of the solenoid, the potential is found to be

$V=m\left({\frac {1}{r_{1}}}-{\frac {1}{r_{2}}}\right),$

r₁ being the distance of the positive end of the solenoid, and r₂ that of the negative end from the point where V exists.

Hence the potential due to a solenoid, and consequently all its magnetic effects, depend only on its strength and the position of

1. See Sir W. Thomson's 'Mathematical Theory of Magnetism,' Phil. Trans., 1850, or Reprint.