Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/139

459.] a non-magnetic body from the same wire and observing its time of oscillation, then if $$A$$ is the moment of inertia of this body, and $$T$$ the time of a complete vibration, $\tau=\frac{4\pi^2A}{T^2}$.

The chief objection to the use of the torsion balance is that the zero-reading $$a_0$$ is liable to change. Under the constant twisting force, arising from the tendency of the magnet to turn to the north, the wire gradually acquires a permanent twist, so that it becomes necessary to determine the zero-reading of the torsion circle afresh at short intervals of time.

Bifilar Suspension.

459.] The method of suspending the magnet by two wires or fibres was introduced by Gauss and Weber. As the bifilar suspension is used in many electrical instruments, we shall investigate it more in detail. The general appearance of the suspension is shewn in Fig. 16, and Fig. 17 represents the projection of the wires on a horizontal plane.

$$AB$$ and $$A'B'$$ are the projections of the two wires.

$$AA'$$ and $$BB'$$ are the lines joining the upper and the lower ends of the wires.

$$a$$ and $$b$$ are the lengths of these lines.

$$\alpha$$ and $$\beta$$ their azimuths.

$$W'$$ and $$W$$ the vertical components of the tensions of the wires.

$$Q$$ and $$Q'$$ their horizontal components.

$$h$$ the vertical distance between $$AA'$$ and $$BB'$$.

The forces which act on the magnet are its weight, the couple arising from terrestrial magnetism, the torsion of the wires (if any) and their tensions. Of these the effects of magnetism and of torsion are of the nature of couples. Hence the resultant of the tensions must consist of a vertical force, equal to the weight of the magnet, together with a couple. The resultant of the vertical components of the tensions is therefore along the line whose projection is $$O$$, the intersection of $$AA'$$ and $$BB'$$, and either of these lines is divided in $$O$$ in the ratio of $$W'$$ to $$W$$.

The horizontal components of the tensions form a couple, and are therefore equal in magnitude and parallel in direction. Calling either of them $$Q$$, the moment of the couple which they form is

Rh

where $$PP'$$ is the distance between the parallel lines $$AB$$ and $$A'B'$$.