Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/339

 

Second Expression for $M$.

An expression for $M$, which is sometimes more convenient, is got by making $c_{1}={\frac {r_{1}-r_{2}}{r_{1}+r_{2}}}$ in which case

To draw the Lines of Magnetic Force for a Circular Current.

702.] The lines of magnetic force are evidently in planes passing through the axis of the circle, and in each of these lines the value of $M$ is constant.

Calculate the value of $K_{\theta }={\frac {\sin \theta }{(F_{\sin \theta }-E_{\sin \theta })^{2}}}$ from Legendre's tables for a sufficient number of values of $\theta$.

Draw rectangular axes of $x$ and $z$ on the paper, and, with centre at the point $x={\frac {1}{2}}a(\sin \theta +\mathrm {cosec} \,\theta )$, draw a circle with radius ${\frac {1}{2}}a(\mathrm {cosec} \,\theta -\sin \theta )$. For all points of this circle the value of $c_{1}$ will be $\sin \theta$. Hence, for all points of this circle,

Now $A$ is the value of $x$ for which the value of $M$ was found. Hence, if we draw a line for which $x=A$, it will cut the circle in two points having the given value of $M$.

Giving $M$ a series of values in arithmetical progression, the values of $A$ will be as a series of squares. Drawing therefore a series of lines parallel to $z$, for which $x$ has the values found for $A$, the points where these lines cut the circle will be the points where the corresponding lines of force cut the circle.

If we put $m=4\pi a$, and $M=nm$, then

We may call $n$ the index of the line of force.

The forms of these lines are given in Fig. XVIII at the end of this volume. They are copied from a drawing given by Sir W. Thomson in his paper on 'Vortex Motion^[1]'.

703.] If the position of a circle having a given axis is regarded as defined by $b$, the distance of its centre from a fixed point on the axis, and $a$, the radius of the circle, then $M$, the coefficient of induction of the circle with respect to any system whatever

 

1. Trans. R. S., Edin., vol. xxv. p. 217 (1869).