Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/355

 

It follows from this that the outer surface of any layer of wire ought to have a constant value of $x$, for if $x$ is greater at one place than another a portion of wire might be transferred from the first place to the second, so as to increase the force at the centre of the galvanometer.

The whole force due to the coil is $\gamma G$, where

the integration being extended over the whole length of the wire, $x$ being considered as a function of $l$.

719.] Let $y$ be the radius of the wire, its transverse section will be $\pi y^{2}$. Let $\rho$ be the specific resistance of the material of which the wire is made referred to unit of volume, then the resistance of a length $l$ is ${\frac {l\rho }{\pi y^{2}}}$, and the whole resistance of the coil is

where $y$ is considered a function of $l$. Let $Y^{2}$ be the area of the quadrilateral whose angles are the sections of the axes of four neighbouring wires of the coil by a plane through the axis, then $Y^{2}l$ is the volume occupied in the coil by a length $l$ of wire together with its insulating covering, and including- any vacant space necessarily left between the windings of the coil. Hence the whole volume of the coil is

where $Y$ is considered a function of $l$. But since the coil is a figure of revolution

or, expressing $r$ in terms of $x$, by equation (2),

Now $2\pi \int _{0}^{\pi }(\sin \theta )^{\frac {5}{2}}\,d\theta$ is a numerical quantity, call it $N$, then

where $V_{0}$ is the volume of the interior space left for the magnet.

Let us now consider a layer of the coil contained between the surfaces $x$ and $x+dx$.