Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/379

749.] Let $$M$$ be the magnetic moment, and $$A$$ the moment of inertia of the magnet and suspended apparatus, RhIf the time of the passage of the current is very small, we may integrate with respect to $$t$$ during this short time without regarding the change of $$\theta$$, and we find RhThis shews that the passage of the quantity $$Q$$ produces an angular momentum $$MGQ\cos \theta_0$$ in the magnet, where $$\theta_0$$ is the value of $$\theta$$ at the instant of passage of the current. If the magnet is initially in equilibrium, we may make $$\theta_0 = 0$$.

The magnet then swings freely and reaches an elongation $$\theta_1$$. If there is no resistance, the work done against the magnetic force during this swing is $$MH (1 - \cos \theta_1 )$$.

The energy communicated to the magnet by the current is RhEquating these quantities, we find

But if $$T$$ be the time of a single vibration of the magnet, where $$H$$ is the horizontal magnetic force, $$G$$ the coefficient of the galvanometer, $$T$$ the time of a single vibration, and $$\theta_1$$ the first elongation of the magnet.

749.] In many actual experiments the elongation is a small angle, and it is then easy to take into account the effect of resistance, for we may treat the equation of motion as a linear equation.

Let the magnet be at rest at its position of equilibrium, let an angular velocity $$\nu$$ be communicated to it instantaneously, and let its first elongation be $$\theta_1$$.