Page:Elementary Text-book of Physics (Anthony, 1897).djvu/280

266 We designate by $$m'$$ the pole of the second magnet, by $$2l'$$ its length, and by $$\theta$$ the complement of the angle made by its axis with the line joining the centres of the magnets. On these assumptions, the force acting on the north pole of the second magnet is $$\frac{2m'M}{r^3},$$ and the force acting on its south pole is $$-\frac{2m'M}{r^3}\cdot$$ These two forces constitute a couple with an arm $$2l' \, \cos \theta,$$ and the moment of this couple is where $$M'$$ represents the magnetic moment of the second magnet. This moment of couple varies from $$\frac{2MM'}{r^3}$$ if the magnets are at right angles to each other, to zero if they are in the same straight line.

243. The Magnetic Shell.—A magnetic shell may be defined as an infinitely thin sheet of magnetizable matter, magnetized transversely; so that any line in the shell normal to its surfaces may be looked on as an infinitesimally short and thin magnet. These imaginary magnets have their like poles contiguous. The product of the intensity of magnetization at any point in the shell into the thickness of the shell at that point is called the strength of the shell at that point, and is denoted by the symbol $$j0$$

Since we may substitute for the magnetic arrangement an imaginary distribution of magnetism over the surfaces of the shell, we may define the strength of the shell as the product of the surface-density and the thickness of the shell.

The dimensions of the strength of a magnetic shell follow at once from this definition. We have $$[j]$$ equal to the dimensions of intensity of magnetization multiplied by a length. Therefore $$[j] = M^{\frac{1}{2}}L^{\frac{1}{2}}T^{-1}.$$

We obtain first the potential of such a shell of infinitesimal