Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/412

 

Fig. 65. passes a vertical axis which carries the magnet $M$, the axis of which revolves in a horizontal plane between the coils $L^{\prime }L$. The coil $L$ has a large coefficient of self-induction, and is fixed. The suspended coil $L^{\prime }$ is protected from the currents of air caused by the rotation of the magnet by enclosing the rotating parts in a hollow case.

The motion of the magnet causes currents of induction in the coil, and these are acted on by the magnet, so that the plane of the suspended coil is deflected in the direction of the rotation of the magnet. Let us determine the strength of the induced currents, and the magnitude of the deflexion of the suspended coil.

Let $x$ be the charge of electricity on the upper surface of the condenser $C$, then, if $E$ is the electromotive force which produces this charge, we have, by the theory of the condenser,

$x=CE$.

(1)

We have also, by the theory of electric currents,

$R{\dot {x}}{\frac {d}{dt}}(L{\dot {x}}+M\cos \theta )+E=0$,

(2)

where $M$ is the electromagnetic momentum of the circuit $L^{\prime }$, when the axis of the magnet is normal to the plane of the coil, and $\theta$ is the angle between the axis of the magnet and this normal.

The equation to determine $x$ is therefore

$CL{\frac {d^{2}x}{dt^{2}}}+CR{\frac {dx}{dt}}+x=CM\sin \theta {\frac {d\theta }{dt}}$.

(3)

If the coil is in a position of equilibrium, and if the rotation of the magnet is uniform, the angular velocity being $n$,

$\theta =nt$.

(4)

The expression for the current consists of two parts, one of which is independent of the term on the right-hand of the equation, and diminishes according to an exponential function of the time. The other, which may be called the forced current, depends entirely on the term in $\theta$, and may be written

$x=A\sin \theta +B\cos \theta$.

(5)