Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/343

 

From the general equation of $M$, Art. 703,

${\frac {d^{2}M}{dx^{2}}}+{\frac {d^{2}M}{dy^{2}}}-{\frac {1}{a+x}}{\frac {dM}{dx}}=0$,

we obtain another set of conditions,

$2A_{2}+2A_{2}^{\prime }=A_{1}$,

$2A_{2}+A_{2}^{\prime }+6A_{3}+2A_{3}^{\prime }=2A_{2}$;

$n(n-1)A_{n}+(n+1)nA_{n+1}+1.2A_{n}^{\prime }+1.2A_{n+1}^{\prime }=nA_{n}$,

$(n-1)(n-2)A_{n}^{\prime }+n(n-1)A_{n+1}^{\prime }+2.3A_{n}^{\prime \prime }+2.3A_{n+1}^{\prime \prime }=(n-2)A_{n}^{\prime }$,

&c.;

$4A_{2}+{}$	$A_{1}={}$	$2B_{2}+2B_{2}^{\prime }-B_{1}=4A_{2}^{\prime }$,
$6A_{3}+{}$	$3A_{2}={}$	$2B_{2}^{\prime }+6B_{3}+2B_{3}^{\prime }=6A_{3}^{\prime }+3A_{2}^{\prime }$,
$(2n-1)A_{n}+(2n+2)A_{n+1}$
		${}=n(n-2)B_{n}+(n+1)nB_{n+1}+1.2B_{n}^{\prime }+1.2B_{n+1}^{\prime }$,

Solving these equations and substituting the values of the coefficients, the series for $M$ becomes

$M=4\pi a\log {\frac {8a}{r}}$	$\left\{1+{\frac {1}{2}}{\frac {x}{a}}+{\frac {x^{2}+3y^{2}}{16a^{2}}}-{\frac {x^{3}+3xy^{2}}{32a^{3}}}+\mathrm {\&c.} \right\}$
	$+4\pi a\left\{-2-{\frac {1}{2}}{\frac {x}{a}}+{\frac {3x^{2}-y^{2}}{16a^{2}}}-{\frac {x^{3}-6xy^{2}}{48a^{3}}}+\mathrm {\&c.} \right\}$.

To find the form of a coil for which the coefficient of self-induction is a maximum, the total length and thickness of the wire being given.

706.] Omitting the corrections of Art. 705, we find by Art. 673

$L=4\pi n^{2}a\left(\log {\frac {8a}{R}}-2\right)$,

where $n$ is the number of windings of the wire, $a$ is the mean radius of the coil, and $R$ is the geometrical mean distance of the transverse section of the coil from itself. See Art. 690. If this section is always similar to itself, $R$ is proportional to its linear dimensions, and $n$ varies as $R^{2}$. Since the total length of the wire is $2\pi an$, $a$ varies inversely as $n$. Hence

${\frac {dn}{n}}=2{\frac {dR}{R}}$, and ${\frac {da}{a}}=-2{\frac {dR}{R}}$,

and we find the condition that $L$ may be a maximum

$\log {\frac {8a}{R}}={\frac {7}{2}}$.