Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/334

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302

CIRCULAR CURRENTS.

[696.

	$\omega ={}$	$-2\pi \left\{1+{\frac {r}{c}}Q_{1}(\theta )+\mathrm {\&c.} +(-)^{8}{\frac {1.3.(2s-1)}{2.4.2s}}{\frac {r^{2s+1}}{c^{2s+1}}}Q_{2s+1}(\theta )\right\}$ ,	(8)
⁠
	$\omega ^{\prime }={}$	$\;\;2\pi \left\{{\frac {1}{2}}{\frac {c^{2}}{r^{2}}}Q_{1}(\theta )+\mathrm {\&c.} +(-)^{8}{\frac {1.3\ldots (2s+1)}{2.4\ldots (2s+2)}}{\frac {c^{2s+2}}{r^{2s+2}}}Q_{2s+1}(\theta )\right\}$ ,	(8′)

where the orders of all the harmonics are odd^[1].

On the Potential Energy of two Circular Currents.

Fig. 47. 696.] Let us begin by supposing the two magnetic shells which are equivalent to the currents to be portions of two concentric spheres, their radii being $c_{1}$ and $c_{2}$ , of which $c_{1}$ is the greater (Fig. 47). Let us also suppose that the axes of the two shells coincide, and that $a_{1}$ the angle subtended by the radius of the first shell, and $a_{2}$ the angle subtended by the radius of the second shell at the centre $C$ .

Let $\omega _{1}$ be the potential due to the first shell at any point within it, then the work required to carry the second shell to an infinite distance is the value of the surface-integral

M=-\iint {\frac {d\omega _{1}}{dr}}dS

extended over the second shell. Hence

M=\int _{\mu _{2}}^{1}{\frac {d\omega _{1}}{dr}}2\pi c_{2}^{2}d\mu _{2}

,

4\pi ^{2}\sin ^{2}\alpha _{1}c_{2}^{2}\left\{{\frac {1}{c_{1}}}Q^{\prime }(\alpha _{1})\int _{\mu _{2}}^{1}Q(\alpha _{2})d\mu _{2}+\mathrm {\&c.} +{\frac {c_{2}^{i-1}}{c^{i}}}Q_{i}^{\prime }(\alpha _{1})\int _{\mu _{2}}^{1}Q(\alpha _{2})d\mu _{2}\right\}

,

or, substituting the value of the integrals from equation (2), Art. 694,

M=4\pi ^{2}\sin ^{2}\alpha _{1}\sin ^{2}\alpha _{2}c_{2}^{2}\left\{{\frac {1}{2}}{\frac {c_{2}}{c_{1}}}Q_{1}^{\prime }(\alpha _{1})Q_{1}^{\prime }(\alpha _{2})+\mathrm {\&c.} +{\frac {1}{i(i+1)}}{\frac {c_{2}^{i}}{c_{1}^{i}}}Q_{i}^{\prime }(\alpha _{1})Q_{i}^{\prime })a_{2})\right.

↑ The value of the solid angle subtended by a circle may be obtained in a more direct way as follows.—

The solid angle subtended by the circle at the point $Z$ in the axis is easily shewn to be

\omega =2\pi \left(1-{\frac {z-c\cos \alpha }{HZ}}\right)

.

Expanding this expression in spherical harmonics, we find

	$\omega$	${}=2\pi \left\{(\cos \alpha -1)+(Q_{1}(\alpha )\cos \alpha -Q_{0}(\alpha )){\frac {z}{c}}+\mathrm {\&c.} +(Q_{1}(\alpha )\cos \alpha -Q_{i-1}(\alpha )){\frac {z^{i}}{c^{i}}}+\mathrm {\&c.} \right\}$ ,
	$\omega ^{\prime }$	${}=2\pi \left\{(Q_{0}(\alpha )\cos \alpha -Q_{a}(\alpha )){\frac {c}{z}}+\mathrm {\&c.} +(Q_{i}(\alpha )\cos \alpha -Q_{i+1}(\alpha )){\frac {c^{i+1}}{z^{i+1}}}+\mathrm {\&c.} \right\}$ ,

for the expansions of

\omega

for points on the axis for which

z

is less than

c

or greater than

c

respectively. Remembering the equations (42) and (43) of Art. 132 (vol. i. p. 165), the coefficients in these equations are evidently the same as those we have now obtained in a more convenient form for computation.

[1] The value of the solid angle subtended by a circle may be obtained in a more direct way as follows.—
The solid angle subtended by the circle at the point $Z$ in the axis is easily shewn to be

$\omega =2\pi \left(1-{\frac {z-c\cos \alpha }{HZ}}\right)$ .
Expanding this expression in spherical harmonics, we find

$\omega$ ${}=2\pi \left\{(\cos \alpha -1)+(Q_{1}(\alpha )\cos \alpha -Q_{0}(\alpha )){\frac {z}{c}}+\mathrm {\&c.} +(Q_{1}(\alpha )\cos \alpha -Q_{i-1}(\alpha )){\frac {z^{i}}{c^{i}}}+\mathrm {\&c.} \right\}$ ,

$\omega ^{\prime }$ ${}=2\pi \left\{(Q_{0}(\alpha )\cos \alpha -Q_{a}(\alpha )){\frac {c}{z}}+\mathrm {\&c.} +(Q_{i}(\alpha )\cos \alpha -Q_{i+1}(\alpha )){\frac {c^{i+1}}{z^{i+1}}}+\mathrm {\&c.} \right\}$ ,
for the expansions of $\omega$ for points on the axis for which $z$ is less than $c$ or greater than $c$ respectively. Remembering the equations (42) and (43) of Art. 132 (vol. i. p. 165), the coefficients in these equations are evidently the same as those we have now obtained in a more convenient form for computation.

[1]