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# Work done in Turning the Magnetic Needle

Suppose first that the magnet consists of two poles, each of strength m, at a distance 2l apart. Let ACB (fig. 77) be the position of equilibrium of the magnet, A'CB' the position of instantaneous rest, and let the angle BCB'=β.

Draw A'D, B'E at right angles to ACB.

Then the work done against the earth's magnetic field H, during the displacement, is mH(AD+BE).

Now,

Hence the work done

The whole magnet may be considered as made up of a series of such magnetic poles, and if we indicate by Σ the result of the operation of adding together the effects on all the separate poles, the total work will be

From the definition of the magnetic moment (p. 356), it can readily be shown that

Hence the total work will be

And this work is equal to the kinetic energy produced by the impulse, that is to Kω2/2.

So that

Thus from [1]

Thus

But if T be the time of a complete oscillation of the needle, and if we suppose that there is no appreciable damping, i.e. that the amplitude of any swing of the needle differs but very slightly in magnitude from that of the preceding, then since the couple acting on the magnet when displaced through a small angle θ is, approximately, MHθ,

Hence substituting for K/M we find from [2]

If the consecutive swings decrease appreciably, then it follows, from the complete mathematical investigation (Maxwell, 'Electricity and Magnetism', §749), that we must replace sin(0.5β) in the above formula by (1+λ/2)sin(0.5β), where λ is quantity known as the logarithmic decrement, and depends on the ratio of the amplitudes of the consecutive vibrations in the following manner:

If c1 be the amplitude of the first and cn that of the nth vibration when the magnet, after being disturbed, is allowed to swing freely, then (Maxwell, 'Electricity and Magnetism', §736)

Thus we get finally

We have used the symbol H for the intensity of the field in which the magnet hangs, though that field need not necessarily be produced by the action of the earth's magnetism alone; we may replace H/G by k, the reduction factor of the galvanometer under the given conditions. Then, if k be known for the galvanometer used, and T, β and λ be determined by observation, we have all the quantities requisite to determine the quantity of electricity which has passed through. A galvanometer adapted for such a measurement is known as a ballistic galvanometer. In such a one, the time of swing should be long and the damping small. These requisites are best attained by the use of a heavy needle, supported by a long torsionless fibre of silk. For accurate work the deflexions should be observed by the use of a scale and telescope, as described in §23.

We shall in the following sections describe some experiments in which we require to use the above formula to obtain the results desired.

Last Update: 2011-03-19