778.] CONDENSER COMPARED WITH COIL. 377 resistance of the rest of the system between the electrodes of the condenser, and E the electromotive force due to the connexions with the battery. Hence t Q = (Qo+EC) e R2 — EC, where Q is the initial value of Q. T (7) If is the time during which contact is maintained during each discharge, the quantity in each discharge is By making c and y in equation (4) large compared with ẞ, a, or a, the time represented by RC may be made so small compared with 7, that in calculating the value of the exponential expression we may use the value of C in equation (5). We thus find where R is the resistance which must be substituted for the condenser to produce an equivalent effect. R2 is the resistance of the rest of the system, T is the interval between the beginning of a discharge and the beginning of the next discharge, and is the duration of contact for each discharge. We thus obtain for the corrected value of C in electromagnetic measure IV. Comparison of the Electrostatic Capacity of a Condenser with the Electromagnetic Capacity of Self-induction of a Coil. 778.] If two points of a conducting circuit, between which the resistance is R, are connected with the electrodes of a condenser whose capacity is C, then, when an electromotive force acts on the circuit, part of the current, instead of passing through the resistance R, will be employed in charging the condenser. The current through R will therefore rise to its final value from zero in a gradual manner. It appears from the Fig. 64. S mathematical theory that the manner in which the current through R rises from zero to its final value is expressed by a formula of exactly the same kind as that which expresses the value of a current urged by a constant electromotive force through the coil of an electromagnet. Hence we may place a condenser and an electromagnet on two opposite members of Wheatstone's Bridge in such a way that the current through the galvanometer is always zero, even at the instant of making or breaking the battery circuit. In the figure, let P, Q, R, S be the resistances of the four members of Wheatstone's Bridge respectively. Let a coil, whose coefficient of self-induction is L, be made part of the member AH, whose resistance is Q, and let the electrodes of a condenser, whose capacity is C, be connected by pieces of small resistance with the points F and Z. For the sake of simplicity, we shall assume that there is no current in the galvanometer G, the electrodes of which are connected to F and H. We have therefore to determine the condition that the potential at F may be equal to that at H. It is only when we wish to estimate the degree of accuracy of the method that we require to calculate the current through the galvanometer when this condition is not fulfilled. Let be the total quantity of electricity which has passed through the member AF, and z that which has passed through FZ at the time t, then a-z will be the charge of the condenser. The electromotive force acting between the electrodes of the condenser is, by Ohm's law, Rdz is C. dt so that if the capacity of the condenser Let y be the total quantity of electricity which has passed through the member AH, the electromotive force from A to H must be equal to that from A to F, or Since there is no current through the galvanometer, the quantity which has passed through HZ must be also y, and we find Substituting in (2) the value of x, derived from (1), and comparing with (3), we find as the condition of no current through the galvanometer Ld 779.] CONDENSER COMBINED WITH COIL. 379 The condition of no final current is, as in the ordinary form of Wheatstone's Bridge, QR = SP. (5) The condition of no current at making and breaking the battery connexion is L L = RC. Q (6) Here and RC are the time-constants of the members Q and R Q respectively, and if, by varying or R, we can adjust the members of Wheatstone's Bridge till the galvanometer indicates no current, either at making and breaking the circuit, or when the current is steady, then we know that the time-constant of the coil is equal to that of the condenser. The coefficient of self-induction, L, can be determined in electromagnetic measure from a comparison with the coefficient of mutual induction of two circuits, whose geometrical data are known (Art. 756). It is a quantity of the dimensions of a line. The capacity of the condenser can be determined in electrostatic measure by comparison with a condenser whose geometrical data are known (Art. 229). This quantity is also a length, c. The electromagnetic measure of the capacity is Substituting this value in equation (8), we obtain for the value of v2 с v2 = QR, (8) where c is the capacity of the condenser in electrostatic measure, I the coefficient of self-induction of the coil in electromagnetic measure, and Q and R the resistances in electromagnetic measure. The value of v, as determined by this method, depends on the determination of the unit of resistance, as in the second method, Arts. 772, 773. V. Combination of the Electrostatic Capacity of a Condenser with the Electromagnetic Capacity of Self-induction of a Coil. 779.] Let C be the capacity of the condenser, the surfaces of which are connected by a wire of resistance R. In this wire let the coils Land I be inserted, and let L denote the sum of their capacities of self-induction. The coil L' is hung by a bifilar suspension, and consists of two coils in vertical planes, between which passes a vertical axis which carries the magnet M, the axis of which revolves in a horizontal plane between the coils L'L. The coil L has a large coefficient of self-induction, and is fixed. The sus с Fig. 65. pended coil L' is protected from the currents of air caused by the rotation of the magnet by enclosing the rotating parts in a hollow case. The motion of the magnet causes currents of induction in the coil, and these are acted on by the magnet, so that the plane of the suspended coil is deflected in the direction of the rotation of the magnet. Let us determine the strength of the induced currents, and the magnitude of the deflexion of the suspended coil. Let be the charge of electricity on the upper surface of the condenser C, then, if E is the electromotive force which produces this charge, we have, by the theory of the condenser, x = CE. We have also, by the theory of electric currents, (1) (2) where M is the electromagnetic momentum of the circuit I, when the axis of the magnet is normal to the plane of the coil, and is the angle between the axis of the magnet and this normal. The equation to determine a is therefore If the coil is in a position of equilibrium, and if the rotation of the magnet is uniform, the angular velocity being n, The expression for the current consists of two parts, one of which is independent of the term on the right-hand of the equation, and diminishes according to an exponential function of the time. The other, which may be called the forced current, depends entirely on the term in 0, and may be written x = A sino + B cos 0. (5) 779.] CONDENSER COMBINED WITH COIL. 381 Finding the values of A and B by substitution in the equation (3), we obtain x = MCn RCn cos 0-(1- CL n2) sin 0 R2C2n2+(1-CL n2)2 (6) The moment of the force with which the magnet acts on the coil L', in which the current is flowing, is Integrating this expression with respect to t, and dividing by t, we find, for the mean value of >, If the coil has a considerable moment of inertia, its forced vibrations will be very small, and its mean deflexion will be proportional to →. Let D1, D2, D3 be the observed deflexions corresponding to angular velocities n1, n2, n, of the magnet, then in general Eliminating P and R from three equations of this form, we find If n2 n is such that CLn,2 = 1, the value of will be a minimum D for this value of n. The other values of n should be taken, one greater, and the other less, than The value of CL, determined from this equation, is of the dimensions of the square of a time. Let us call it 72. If C, be the electrostatic measure of the capacity of the condenser, and L the electromagnetic measure of the self-induction of the coil, both C, and I are lines, and the product and m m where 72 is the value of C2L2, determined by this experiment. The experiment here suggested as a method of determining v is of the same nature as one described by Sir W. R. Grove, Phil. Mag., |