Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/356

 

The volume of this layer is

$dV=Nx^{3}\,dx=Y^{2}\,dl$,

(9)

where $dl$ is the length of wire in this layer.

This gives us $dl$ in terms of $dx$. Substituting this in equations (3) and (4), we find

	$dG$	${}=N{\frac {dx}{Y^{2}}}$,		(10)
	$dR$	${}=N{\frac {\rho }{\pi }}{\frac {x^{2}dx}{Y^{2}y^{2}}}$,		(11)

where $dG$ and $dR$ represent the portions of the values of $G$ and of $R$ due to this layer of the coil.

Now if $E$ be the given electromotive force,

$E=\gamma (R+r)$,

where $r$ is the resistance of the external part of the circuit, independent of the galvanometer, and the force at the centre is

$\gamma G=E{\frac {G}{R+r}}$.

We have therefore to make ${\frac {G}{R+r}}$ a maximum, by properly adjusting the section of the wire in each layer. This also necessarily involves a variation of $Y$ because $Y$ depends on $y$.

Let $G_{0}$ and $R_{0}$ be the values of $G$ and of $R+r$ when the given layer is excluded from the calculation. We have then

${\frac {G}{R+r}}={\frac {G_{0}+dG}{R_{0}+dR}}$,

(12)

and to make this a maximum by the variation of the value of $y$ for he given layer we must have

${\frac {{\frac {d}{dy}}.dG}{{\frac {d}{dy}}.dR}}={\frac {G}{R+r}}$.

(13)

Since $dx$ is very small and ultimately vanishes, ${\frac {G_{0}}{R_{0}}}$ will be sensibly, and ultimately exactly, the same whichever layer is excluded, and we may therefore regard it as constant. We have therefore, by (10) and (11),

${\frac {x^{2}}{y^{2}}}\left(1+{\frac {Y}{y}}{\frac {dy}{dY}}\right)={\frac {\rho }{\pi }}{\frac {R+r}{G}}=\mathrm {constant}$.

(14)

If the method of covering the wire and of winding it is such that the proportion between the space occupied by the metal of