Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/310

Rh If $$N$$ is the whole number of windings, and if $$\gamma$$ is the strength of the current in each winding,

Hence the magnetic force within the coil is

673.] Let us next find the method of coiling the wire in order to produce within the sphere a magnetic potential of the form of a solid zonal harmonic of the second degree,

12= jl*(f cos 2 0-i).

Here = ~A (f cos 2 0-i).

i.

If the whole number of windings is $$N$$ the number between the pole and the polar distance $$\theta$$ is $$\frac{1}{2} N\sin^2 \theta$$.

The windings are closest at latitude 45°. At the equator the direction of winding changes, and in the other hemisphere the windings are in the contrary direction.

Let $$\gamma$$ be the strength of the current in the wire, then within the shell

Let us now consider a Conductor in the form of a plane closed curve placed anywhere within the shell with its plane perpendicular to the axis. To determine its coefficient of induction we have to find the surface-integral of $$\frac{d\Omega}{dz}$$ over the plane bounded by the curve, putting $$\gamma = 1$$.

Now 12 = ~ 2 N (z 2 - i (x* 4 /)),

O d

d& STT ,

and - = 2 Nz. dz 5 a 2

Hence, if $$S$$ is the area of the closed curve, its coefficient of induction is

If the current in this conductor is $$\gamma^\prime$$, there will be, by Art. 583, a force $$Z$$, urging it in the direction of $$z$$, where and, since this is independent of $$x$$, $$y$$, $$z$$, the force is the same in whatever part of the shell the circuit is placed.

674.] The method given by Poisson, and described in Art. 437,