= Let M magnetic moment of the galvanometer needle; w = angular velocity of the needle at any instant ; = The quantity of electricity Q = cr where c is the current strength, and the time during which it flows. Moment of the force on the needle due to c = MGc The total impulse given to the needle MGCT = Iw (1) If the pole strength of the needle is m, and its length 27, the work done in deflecting the needle through an angle is But the work done on the needle = kinetic energy in needle α Now, combining equations (1) and (2), we get HT Multiplying this result by 10 gives the number of coulombs, since there are 10 coulombs in 1 C.G.S. unit of quantity. 225. Correction for Damping.-In the above calculation we have assumed that the only retarding force acting on the needle is that due to the controlling-field H. There are, however, other causes which affect the amplitude or the swing of the needle. Of these the most important are (1) the induced currents set up in the galvanometer coil, due to the motion of the needle inside it, in accordance with Lenz's law; (2) the resistance to the motion of the needle due to the friction of the air; (3) the temporary alteration in M due to the field produced by the current in the galvanometer coil; and (4) the torsion of the suspending fibre. The effect of (3) may be made small by sending discharges through the galvanometer, which will only produce small swings; and (4) may be allowed for in the same manner as described in par. 138. The effects of (1) and (2), which tend to diminish the amplitude of the swings, are usually termed the "damping" effects, and may be determined experimentally. If the needle of a ballistic galvanometer is set swinging, the amplitude of the swings gradually diminishes, on account of this damping action, till finally the needle is brought to rest. If the damping is not too great, then it is found that the amplitudes of successive swings diminish in a geometrical series, calling a1, a2, ɑз, a, successive swings, then απ Ալ I == (0)' - ()' = ()' = (“;) + = = or, calling o, amplitude of the mth swing = n-m = C = C log, is the log of the constant ratio, and is known as the logarithmic decrement (A) ; · 226. In order to correct any swing for damping we can now proceed as follows. Let a represent the observed swing of the ballistic galvanometer needle. If there had been no damping, the swing would have been greater, say a', so that the effect of the damping has been, during a half-period of a complete swing of the needle, to reduce the swing from a value a' to a value o ; therefore, since λ but when expanded gives 1 + + etc., and since A is 2 small it is not necessary to go further up the series than the second term; also- So that, in order to correct the observed swing of a ballistic galvanometer needle for damping, we must multiply by (x + 1). We may therefore write the full formula for 2 This simple correction for damping may be applied in all cases when does not exceed o'5, without introducing any sensible error. MEASUREMENT of λ. 227. From what has just been said, it will be seen that we may determine λ from the observation of a series of swings of the needle. It must, however, be borne in mind that λ is not a constant for all conditions under which the galvanometer may be used; since it depends partly on the induced currents set up in the coil in accordance with Lenz's law, and these depend on the resistance of the circuit in which they are induced, the damping must obviously vary with the resistance of the galvanometer circuit. The measurement of A should therefore be made for a number of cases, the galvanometer circuit resistance being specified in each case, starting with the galvanometer on open circuit and ending with it short circuited. A curve may then be plotted showing the relation between A and the resistance of the galvanometer circuit. From such a curve the value of A for any particular experiment may be deduced. To measure the galvanometer needle should be set swinging, either by means of a magnet, which is afterwards removed from its vicinity, or else by passing a current momentarily through the galvanometer coils. The successive swings to right and left of the scale zero are then noted by two observers. Let these be a1, a, a, a, ... an. These should be tabulated in two parallel columns, and for convenience a zigzag line connecting them, thus— If the zero lies between a, and a, then the arithmetical sum of a + a gives the amplitude of the first swing, or, in general, calling readings to the left hand of zero minus, and to the right hand plus- The algebraic diff. of a, a, is the amplitude of the 1st swing The logarithmic decrement may be calculated from the above, taking various combinations of swings, and using the relation 228. In order to determine the logarithmic decrement of a ballistic galvanometer, the needle was adjusted so that the spot of light was at the zero on the scale. The needle was then started swinging by bringing a magnet near the galvano |