Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/235

The electromagnet A is capable of rotating about the horizontal axis BB', within a ring which itself revolves about a vertical axis.

Let A, B, C be the moments of inertia of the electromagnet about the axis of the coil, the horizontal axis BB', and a third axis CC' respectively.

Let θ be the angle which CC' makes with the vertical, φ azimuth of the axis BB', and ψ a variable on which the motion of electricity in the coil depends.

Then the kinetic energy of the electromagnet may be written

$2T=A{\dot {\phi }}^{2}\sin ^{2}\theta +B{\dot {\theta }}^{2}+C{\dot {\phi }}^{2}\cos ^{2}\theta +E({\dot {\phi }}\sin \theta +{\dot {\psi }})^{2},$

where E is a quantity which may be called the moment of inertia of the electricity in the coil.

If Θ is the moment of the impressed force tending to increase θ, we have, by the equations of dynamics,

$\Theta =B{\frac {d^{2}\theta }{dt^{2}}}-\{(A-C){\dot {\phi }}^{2}\sin \theta \cos \theta +E{\dot {\phi }}\cos \theta ({\dot {\phi }}\sin \theta +{\dot {\psi }})\}.$

By making Ψ, the impressed force tending to increase ψ, equal to zero, we obtain

${\dot {\phi }}\sin \theta =\gamma ,\,$

a constant, which we may consider as representing the strength of the current in the coil.

If C is somewhat greater than A, Θ will be zero, and the equilibrium about the axis BB' will be stable when

$\sin \theta ={\frac {E\gamma }{(C-A){\dot {\phi }}}}.$

This value of θ depends on that of γ, the electric current, and is positive or negative according to the direction of the current.

The current is passed through the coil by its bearings at B and B', which are connected with the battery by means of springs rubbing on metal rings placed on the vertical axis.

To determine the value of θ, a disk of paper is placed at C, divided by a diameter parallel to BB' into two parts, one of which is painted red and the other green.

When the instrument is in motion a red circle is seen at C when θ is positive, the radius of which indicates roughly the value of θ. When is negative, a green circle is seen at C.

By means of nuts working on screws attached to the electromagnet, the axis CC' is adjusted to be a principal axis having its moment of inertia just exceeding that round the axis A, so as