Prove that the deflection of a ballistic galvanometer is act

Prove that the deflection of a ballistic galvanometer is actually dependent only upon the amount of charge flowing through and not upon the rate of coil rotation. (Some ancient Versions of a Weber, White, and Manning general Physics text might Provide Clues.)

Solution

A galvanometer specially designed to measure the quantity of charge displaced in a current of short duration (such as in the charging or discharging of a capacitor). It can be shown that the deflection of the coil is proportional to the quantity of charge passing through it. It is typically a sensitive moving-coil mirror galvanometer with a coil that has a relatively large moment of inertia and a long period of vibration as compared with a galvanometer designed for current measurement.

When a short burst of current passes through the galvanometer, there is very little rotation of the galvanometer coil until after the current has ceased. An angular impulse is delivered to the coil that is proportional to the quantity of charge q = Idt that passed through. The angular impulse depends on the torque, which is proportional to the current, and upon the time interval during which the torque is applied. This angular impulse delivered to the coil produces an equal angular momentum of the coil. The coil then proceeds to swing until its initial kinetic energy has been converted to potential energy of the suspension. The process is completely analogous to what takes place when a ballistic pendulum is struck by a bullet.