Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/357

 

the wire bears the same proportion to the space between the wires whether the wire is thick or thin, then

${\frac {Y}{y}}{\frac {dy}{dY}}=1$,

and we must make both $y$ and $Y$ proportional to $x$, that is to say, the diameter of the wire in any layer must be proportional to the linear dimension of that layer. If the thickness of the insulating covering is constant and equal to $b$, and if the wires are arranged in square order,

and the condition is

In this case the diameter of the wire increases with the diameter of the layer of which it forms part, but not in so high a ratio.

If we adopt the first of these two hypotheses, which will be nearly true if the wire itself nearly fills up the whole space, then we may put

$y=\alpha x$,⁠ $Y=\beta y$,

where $\alpha$ and $\beta$ are constant numerical quantities, and

$G=N{\frac {1}{\alpha ^{2}\beta ^{2}}}\left({\frac {1}{a}}-{\frac {1}{x}}\right)$,

$R=N{\frac {\rho }{\pi }}{\frac {1}{\alpha ^{4}\beta ^{2}}}\left({\frac {1}{a}}-{\frac {1}{x}}\right)$,

where $a$ is a constant depending upon the size and form of the free space left inside the coil.

Hence, if we make the thickness of the wire vary in the same ratio as $x$, we obtain very little advantage by increasing the external size of the coil after the external dimensions have become a large multiple of the internal dimensions.

720.] If increase of resistance is not regarded as a defect, as when the external resistance is far greater than that of the galvanometer, or when our only object is to produce a field of intense force, we may make $y$ and $Y$ constant. We have then

$G={\frac {N}{Y^{2}}}(x-a)$,

$R={\frac {1}{3}}{\frac {N}{Yy^{2}}}{\frac {\rho }{\pi }}(x^{3}-a_{1}^{3})$,

where $a$ is a constant depending on the vacant space inside the coil. In this case the value of $G$ increases uniformly as the dimensions of the coil are increased, so that there is no limit to the value of $G$ except the labour and expense of making the coil.