Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/394

 

Let $M$ be the coefficient of induction between the galvanometer coil and the suspended magnet. It is of the form

$M=G_{1}g_{1}Q_{1}(\theta )+G_{2}g_{2}Q_{2}+\mathrm {\&c.}$

(4)

where $G_{1}$, $G_{2}$, &c. are coefficients belonging to the coil, $g_{1}$, $g_{2}$, &c. to the magnet, and $Q_{1}(\theta )$, $Q_{2}(\theta )$ &c., are zonal harmonics of the angle between the axes of the coil and the magnet. See Art. 700. By a proper arrangement of the coils of the galvanometer, and by building up the suspended magnet of several magnets placed side by side at proper distances, we may cause all the terms of $M$ after the first to become insensible compared with the first. If we also put $\phi ={\frac {\pi }{2}}-\theta$, we may write

$M=Gm\sin \phi$,

(5)

where $G$ is the principal coefficient of the galvanometer, $m$ is the magnetic moment of the magnet, and $\phi$ is the angle between the axis of the magnet and the plane of the coil, which, in this experiment, is always a small angle.

If $L$ is the coefficient of self-induction of the coil, and $R$ its resistance, and $\gamma$ the current in the coil,

�� ��(6)

��or L- + R-/ + Gmcos<t> -= 0. (7)

dt dt

The moment of the force with which the current $\gamma$ acts on the magnet is $\gamma {\frac {dM}{d\phi }}$, or $Gm\gamma \cos \phi$. The angle $\phi$ is in this experiment so small, that we may suppose $\cos \phi =1$.

Let us suppose that the equation of motion of the magnet when the circuit is broken is

$A{\frac {d^{2}\phi }{dt^{2}}}+B{\frac {d\phi }{dt}}+C\phi =0$,

(8)

where $A$ is the moment of inertia of the suspended apparatus, $B{\frac {d\phi }{dt}}$ expresses the resistance arising from the viscosity of the air and of the suspension fibre, &c., and $C\phi$ expresses the moment of the force arising from the earth s magnetism, the torsion of the suspension apparatus, &c., tending to bring the magnet to its position of equilibrium.

The equation of motion, as affected by the current, will be

$A{\frac {d^{2}\phi }{dt^{2}}}+B{\frac {d\phi }{dt}}+C\phi =Gm\gamma$.

(9)