Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/342

 

is nearly the same as if the wire had been straight, we can calculate the difference between the induction through the circle whose radius is $a-c$, and the circle $A$ by the formula

Hence we find the value of the induction between $A$ and $a$ to be

approximately, provided $r$ is small compared with $a$.

705.] Since the mutual induction between two windings of the same coil is a very important quantity in the calculation of experimental results, I shall now describe a method by which the approximation to the value of $M$ for this case can be carried to any required degree of accuracy.

We shall assume that the value of $M$ is of the form

where	⁠	$A={}$		$a+A_{1}x+A_{2}{\frac {x^{2}}{a}}+A_{2}^{\prime }{\frac {y^{2}}{a}}+A_{3}+A_{3}^{\prime }{\frac {xy^{2}}{a^{2}}}+\mathrm {\&c.}$,	⁠
and		$B={}$	$-2$	$a+B_{1}x+B_{2}{\frac {x^{2}}{a}}+B_{2}^{\prime }+{\frac {y^{2}}{a}}+B_{3}{\frac {x^{3}}{a^{2}}}+B_{3}^{\prime }{\frac {xy^{2}}{a^{2}}}+\mathrm {\&c.}$,

where $a$ and $a+x$ are the radii of the circles, and $y$ the distance between their planes.

We have to determine the values of the coefficients $A$ and $B$. It is manifest that only even powers of y can occur in these quantities, because, if the sign of $y$ is reversed, the value of $M$ must remain the same.

We get another set of conditions from the reciprocal property of the coefficient of induction, which remains the same whichever circle we take as the primary circuit. The value of $M$ must therefore remain the same when we substitute $a+x$ for $a$, and $-x$ for $x$ in the above expression.

We thus find the following conditions of reciprocity by equating the coefficients of similar combinations of $x$ and $y$,

$A_{1}$	${}=1-A_{1}$,	⁠	$B_{1}$	${}=1-2-B_{1}$,
$A_{3}$	${}=-A_{2}-A_{3}$,	⁠	$B_{3}$	${}={\frac {1}{3}}-{\frac {1}{2}}A_{1}+{}$	$A_{2}-B_{2}-B_{3}$,
$A_{3}^{\prime }$	${}=-A_{2}^{\prime }-A_{3}^{\prime }$,		$B_{3}^{\prime }$	${}={}$	$A_{2}^{\prime }-B_{2}^{\prime }-B_{3}^{\prime }$;
$(-)^{n}A_{n}=A_{2}+(n-2)A_{3}+{\frac {(n-2)(n-3)}{1.2}}A_{4}+\mathrm {\&c.} +A_{n}$,
$(-)^{n}B_{n}=-{\frac {1}{n}}+{\frac {1}{n-1}}A_{1}-{\frac {1}{n-2}}A_{2}+\mathrm {\&c.} +(-1)^{n}A_{n-1}$
		${}+B_{2}+(n-2)B_{3}+{\frac {(n-2)(n-3)}{1.2}}B_{4}+\mathrm {\&c.} +B_{n}$.