Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/341

 

Coefficient of Induction of Two Parallel Circles when the Distance between the Arcs is Small compared with the Radius of either Circle.

704.] We might deduce the value of $M$ in this case from the expansion of the elliptic integral already given when its modulus is nearly unity. The following method, however, is a more direct application of electrical principles.

First Approximation.

Let $A$ and $a$ be the radii of the circles, and b the distance between their planes, then the shortest distance between the arcs is

$r$ = ${\sqrt {(A-a)^{2}+b^{2}}}$.

Fig. 49. We have to find $M_{1}$, the magnetic induction through the circle $A$, due to a unit current in $a$ on the supposition that $r$ is small compared with $A$ or $a$.

We shall begin by calculating the magnetic induction through a circle in the plane of $a$ whose radius is $a-c$, $c$ being a quantity small compared with $a$ (Fig. 49).

Consider a small element $ds$ of the circle $a$. At a point in the plane of the circle, distant $\rho$ from the middle of $ds$, measured in a direction making an angle $\theta$ with the direction of <$ds$, the magnetic force due to $ds$ is perpendicular to the plane, and equal to

${\frac {1}{\rho ^{2}}}\sin \theta \,ds$.

If we now calculate the surface-integral of this force over the space which lies within the circle $a$, but outside of a circle whose centre is $ds$ and whose radius is $c$, we find it

$\int _{0}^{\pi }\int _{c}^{2a\sin \theta }{\frac {1}{\rho ^{2}}}\sin \theta \,ds\,d\theta \,d\rho =\{\log 8a-\log c-2\}\,ds$.

If $c$ is small, the surface-integral for the part of the annular space outside the small circle $c$ may be neglected.

We then find for the induction through the circle whose radius is $a-c$, by integrating with respect to $ds$,

$M_{ac}=4\pi a\{\log 8a-\log c-2\}$,

provided $c$ is very small compared with $a$.

Since the magnetic force at any point, the distance of which from a curved wire is small compared with the radius of curvature,