Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/312

(3) The case of an ellipsoid uniformly magnetized parallel to a given line has been discussed in Art. 437.

If the ellipsoid is coiled with wire in parallel and equidistant planes, the magnetic force within the ellipsoid will be uniform.

(4) A Cylindric Magnet or Solenoid.

676.] If the body is a cylinder having any form of section and bounded by planes perpendicular to its generating lines, and if $V_{1}$ is the potential at the point ($x$, $y$, $z$) due to a plane area of surface-density unity coinciding with the positive end of the solenoid, and $V_{2}$ the potential at the same point due to a plane area of surface-density unity coinciding with the negative end, then, if the cylinder is uniformly and longitudinally magnetized with intensity unity, the potential at the point ($x$, $y$, $z$) will be

If the cylinder, instead of being a magnetized body, is uniformly lapped with wire, so that there are $n$ windings of wire in unit of length, and if a current, $\gamma$, is made to flow through this wire, the magnetic potential outside the solenoid is as before,

but within the space bounded by the solenoid and its plane ends

The magnetic potential is discontinuous at the plane ends of the solenoid, but the magnetic force is continuous.

If $r_{1}$, $r_{2}$, the distances of the centres of inertia of the positive and negative plane end respectively from the point ($x$, $y$, $z$), are very great compared with the transverse dimensions of the solenoid, we may write

where $A$ is the area of either section.

The magnetic force outside the solenoid is therefore very small, and the force inside the solenoid approximates to a force parallel to the axis in the positive direction and equal to $4\pi n\gamma$.

If the section of the solenoid is a circle of radius $a$, the values of $V_{1}$ and $V_{2}$ may be expressed in the series of spherical harmonics given in Thomson and Tait's Natural Philosophy, Art. 546, Ex. II.,

⁠	$V=2\pi \left\{-rQ_{1}+a{\frac {1}{2}}{\frac {r^{2}}{a}}Q_{2}-{\frac {1\cdot 1}{2\cdot 4}}{\frac {r^{4}}{a^{3}}}Q_{4}+{\frac {1\cdot 1\cdot 3}{2\cdot 4\cdot 6}}{\frac {r^{6}}{a^{5}}}Q_{6}+\mathrm {\&c.} \right\}$ when $r<a$,	(14)
	$V=2\pi \left\{{\frac {1}{2}}{\frac {a^{2}}{r}}-{\frac {1\cdot 1}{2\cdot 4}}{\frac {a^{4}}{r^{3}}}Q_{2}+{\frac {1\cdot 1\cdot 3}{2\cdot 4\cdot 6}}{\frac {a^{6}}{r^{5}}}Q_{4}-\mathrm {\&c.} \right\}$ when $r>a$.	(15)