Quantity of Electricity and Angular Velocity of the Magnetic Needle

Let K be the moment of inertia of the needle (p. 144), and suppose that it begins to move with an angular velocity ω. Then it is shown in books on Dynamics, (see also Maxwell, 'Matter and Motion', p. 56), that the moment of momentum of the needle is Kω, and the kinetic energy Kω/2.

Now, by the second law of motion, the moment of momentum is equal to the moment of the impulse produced by the passage of the electricity, and, by the principle of the conservation of energy, the kinetic energy is equal to the work which is done by the earth's horizontal force in reducing the needle to instantaneous rest at the extremity of its first swing. Let M be the magnetic moment of the galvanometer needle, G the galvanometer constant, Q the total quantity of electricity which passes, and β the angle through which the magnet is deflected. The moment of the force produced on the needle by a current γ is MGγ, and if this current flow for a time, τ, the impulse is MGγτ; but γτ is the total quantity of electricity which flows through, and this has been denoted by Q.

Thus the impulse is MGQ, and if the time of transit be so short that we may assume that all the electricity has passed through the coils before the needle has appreciably moved from its position of rest - in practice with a suitable galvanometer this condition is satisfied - this impulse is equal to the moment of momentum, or Kω.

Thus

Kω = MGQ

[1]