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# 10.4: Ballistic Galvanometer and the Measurement of Magnetic Field

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

A galvanometer is similar to a sensitive ammeter, differing mainly in that when no current passes through the meter, the needle is in the middle of the dial rather than at the left hand end. A galvanometer is used not so much to measure a current, but rather to detect whether or not a current is flowing, and in which direction. In the ballistic galvanometer, the motion of the needle is undamped, or as close to undamped as can easily be achieved. If a small quantity of electricity is passed through the ballistic galvanometer in a time that is short compared with the period of oscillation of the needle, the needle will jerk from its rest position, and then swing to and fro in lightly damped harmonic motion. (It would be simple harmonic motion if it could be completely undamped.) The amplitude of the motion, or rather the extent of the first swing, depends on the quantity of electricity that was passed through the galvanometer. It could be calibrated, for example, by discharging various capacitors through it, and making a table or graph of amplitude of swing versus quantity of electricity passed.

Now, if we have a small coil of area $$A$$, $$N$$ turns, resistance $$R$$, we could place the coil perpendicular to a magnetic field $$B$$, and then connect the coil to a ballistic galvanometer. Then, suddenly (in a time that is short compared with the oscillation period of the galvanometer), remove the coil from the field (or rotate it through $$90^\circ$$ ) so that the flux through the coil goes from $$AB$$ to zero. While the flux through the coil is changing, and EMF will be induced, equal to , $$NA\dot B$$ and consequently a current will flow momentarily through the coil of magnitude

$I=\dfrac{NA\dot B}{R+r}, \label{10.4.1}$

where r is the resistance of the galvanometer. Integrate this with respect to time, with initial condition $$Q = 0\text{ when }t = 0$$, and we find for the total quantity of electricity that flows through the galvanometer

$Q=\dfrac{NAB}{R+r}.\label{10.4.2}$

Since $$Q$$ can be measured from the amplitude of the galvanometer motion, the strength of the magnetic field, $$B$$ is determined.

I mentioned that the ballistic galvanometer differs from that of an ordinary galvanometer or ammeter in that its motion is undamped. The motion of the needle in an ordinary ammeter is damped, so that the needle doesn't swing violently whenever the current is changed, and so that the needle moves promptly and purposefully towards its correct position. How is this damping achieved?

The coil of a moving-coil meter is wound around a small aluminium frame called a former. When the current through the ammeter coil is changed, the coil – and the former – swing round; but a current is induced in the former, which gives the former a magnetic moment in such a sense as to oppose and therefore dampen the motion. The resistance of the former is made just right so that critical damping is achieved, so that the needle reaches its equilibrium position in the least time without overshoot or swinging. The little aluminium former does not look as if it were an important part of the instrument – but in fact its careful design is very important!