749.] TRANSIENT CURRENTS. 347 Let M be the magnetic moment, and A the moment of inertia of the magnet and suspended apparatus, d20 Α + MH sin 0 = MG y cos 0. (2) If the time of the passage of the current is very small, we may integrate with respect to t during this short time without regarding the change of 0, and we find do A dt = MG cos 0, fy dt + C = MGQ cos 0, +C. (3) This shews that the passage of the quantity Q produces an angular momentum MGQ cos 0, in the magnet, where 0 is the value of at the instant of passage of the current. If the magnet is initially in equilibrium, we may make 0,= 0. The magnet then swings freely and reaches an elongation 01. If there is no resistance, the work done against the magnetic force during this swing is MH (1—cos 01). The energy communicated to the magnet by the current is But if T be the time of a single vibration of the magnet, where H is the horizontal magnetic force, & the coefficient of the galvanometer, T the time of a single vibration, and 0, the first elongation of the magnet. 749.] In many actual experiments the elongation is a small angle, and it is then easy to take into account the effect of resistance, for we may treat the equation of motion as a linear equation. Let the magnet be at rest at its position of equilibrium, let an angular velocity v be communicated to it instantaneously, and let its first elongation be 01. which gives the first elongation in terms of the quantity of electricity in the transient current, and conversely, where T is the observed time of a single vibration as affected by the actual resistance of damping. When A is small we may use the approximate formula HT (17) Method of Recoil. 750.] The method given above supposes the magnet to be at rest in its position of equilibrium when the transient current is passed through the coil. If we wish to repeat the experiment we must wait till the magnet is again at rest. In certain cases, however, in which we are able to produce transient currents of equal intensity, and to do so at any desired instant, the following method, described by Weber *, is the most convenient for making a continued series of observations. * Resultate des Magnetischen Vereins, 1838, p. 98. 750.] METHOD OF RECOIL. 349 Suppose that we set the magnet swinging by means of a transient current whose value is Qo. If, for brevity, we write When it returns through the point of equilibrium in a negative direction its velocity will be When the magnet returns to the point of equilibrium, its velocity will be (23) Now let an instantaneous current, whose total quantity is -Q, be transmitted through the coil at the instant when the magnet is at the zero point. It will change the velocity v into v2-v, where If Q is greater than Qe-2, the new velocity will be negative and equal to MG The motion of the magnet will thus be reversed, and the next elongation will be negative, The magnet is then allowed to come to its positive elongation 01 = −01eλ = d1 = (25) (26) and when it again reaches the point of equilibrium a positive current whose quantity is Q is transmitted. This throws the magnet back in the positive direction to the positive elongation. or, calling this the first elongation of a second series of four, (27) (28) Proceeding in this way, by observing two elongations + and -, then sending a positive current and observing two elongations -- and, then sending a positive current, and so on, we obtain a series consisting of sets of four elongations, in each of which If n series of elongations have been observed, then we find the logarithmic decrement from the equation Σ (7) - Σ (0) = e^, and Q from the equation (31) KQ (1+e) (2n-1) = Σ„(a−b—c+d) (1+e−2 1) — (α2 — b1)—(d„—c„) e-21. (32) Fig. 60. The motion of the magnet in the method of recoil is graphically represented in Fig. 60, where the abscissa represents the time, and the ordinate the deflexion of the magnet at that time. See Art. 760. Method of Multiplication. 751.] If we make the transient current pass every time that the magnet passes through the zero point, and always so as to increase the velocity of the magnet, then, if 01, 02, &c. are the successive elongations, (33) (34) The ultimate value to which the elongation tends after a great many vibrations is found by putting 01 =—0n-1, whence we find 0 = + 1 1 e KQ. (35) If A is small, the value of the ultimate elongation may be large, but since this involves a long continued experiment, and a careful determination of A, and since a small error in A introduces a large error in the determination of Q, this method is rarely useful for 751.] MISTIMING THE CURRENT. 351 numerical determination, and should be reserved for obtaining evidence of the existence or non-existence of currents too small to be observed directly. In all experiments in which transient currents are made to act on the moving magnet of the galvanometer, it is essential that the whole current should pass while the distance of the magnet from the zero point remains a small fraction of the total elongation. The time of vibration should therefore be large compared with the time required to produce the current, and the operator should have his eye on the motion of the magnet, so as to regulate the instant of passage of the current by the instant of passage of the magnet through its point of equilibrium. To estimate the error introduced by a failure of the operator to produce the current at the proper instant, we observe that the effect of a force in increasing the elongation varies as and that this is a maximum when = 0. Hence the error arising from a mistiming of the current will always lead to an underestimation of its value, and the amount of the error may be estimated by comparing the cosine of the phase of the vibration at the time of the passage of the current with unity. |