Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/315

Rh That this may be a maximum, $$x$$ and $$z$$ being given, and $$y$$ variable,Rh

This equation gives the best relation between the depths of the primary and secondary coil for an induction-machine without an iron core.

If there is an iron core of radius $$z$$, then $$G$$ remains as before, but

If $$y$$ is given, the value of $$z$$ which gives the maximum value of $$g$$ is RhWhen, as in the case of iron, $$\kappa$$ is a large number, $$z = \frac{2}{3}y,$$ nearly.

If we now make $$x$$ constant, and $$y$$ and $$z$$ variable, we obtain the maximum value of $$G g$$ when Rh

The coefficient of self-induction of a long solenoid whose outer and inner radii are $$x$$ and $$y$$, and having a long iron core whose radius is $$z$$, is Rh

680.] We have hitherto supposed the wire to be of uniform thickness. We shall now determine the law according to which the thickness must vary in the different layers in order that, for a given value of the resistance of the primary or the secondary coil, the value of the coefficient of mutual induction may be a maximum.

Let the resistance of unit of length of a wire, such that $$n$$ windings occupy unit of length of the solenoid, be $$\rho n^2$$.

The resistance of the whole solenoid is Rh

The condition that, with a given value of $$R$$, $$G$$ may be a maximum is $$\frac{dG}{dr} = C \frac{dR}{dr}$$, where $$C$$ is some constant.

This gives $$n^2$$ proportional to $$\frac{1}{r}$$, or the diameter of the wire of the exterior coil must be proportional to the square root of the radius.

In order that, for a given value of R, g may be a maximumRh