Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/365

two suspended coils are similar, but with the current flowing in opposite directions, the effect of terrestrial magnetism on the position of the arm of the torsion-balance will be completely eliminated.

Fig. 57.

727.] If the suspended coil is in the shape of a long solenoid, and is capable of moving parallel to its axis, so as to pass into the interior of a larger fixed solenoid having the same axis, then, if the current is in the same direction in both solenoids, the suspended solenoid will be sucked into the fixed one by a force which will be nearly uniform as long as none of the extremities of the solenoids are near one another.

728.] To produce a uniform longitudinal force on a small coil placed between two equal coils of much larger dimensions, we should make the ratio of the diameter of the large coils to the distance between their planes that of $2$ to ${\sqrt {3}}$. If we send the same current through these coils in opposite directions, then, in the expression for $\omega$, the terms involving odd powers of $r$ disappear, and since $\sin ^{2}\alpha ={\frac {4}{7}}$ and $\cos ^{2}\alpha ={\frac {3}{7}}$, the term involving $r^{4}$ disappears also, and we have

$\omega ={\frac {8}{7}}{\sqrt {\frac {3}{7}}}\pi n\gamma \left\{3{\frac {r^{2}}{c^{2}}}Q_{2}(\theta )+{\frac {11}{7}}{\frac {r^{6}}{c^{6}}}Q_{6}(\theta )+\mathrm {\&c.} \right\}$,

which indicates a nearly uniform force on a small suspended coil. The arrangement of the coils in this case is that of the two outer coils in the galvanometer with three coils, described at Art. 715. See Fig. 51.