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discharging of the condenser. The effect would be likely to be much more marked if the capacity were much greater.
The conditions for a maximum throw of the galvanometer for a given E.M.F. are, moreover, not the same as those for the maximum sensitiveness to steady current. There is usually no difficulty, however, in obtaining sufficient sensitiveness of galvanometer to detect want of proportionality of the arms PQRS of the bridge, so that a change of S by 1 in 200 will thus give a deflexion of convenient magnitude. It is, however, very advantageous to have a galvanometer whose period of oscillation can be varied at pleasure, so as to give about the same throw as steady deflexion for the chosen value of dS. One advantage which the method has is that there is no necessity to know the galvanometer or battery resistance. In the ordinary ballistic-galvanometer method it is very doubtful if the resistance of the galvanometer, unless it is measured at the time, is known within 1 to 2 per cent.
Further, the two essential parts of the determination and a may be made in quick succession, without any shifting of contacts or connexions and with the battery-current flowing continuously, leaving only the period of the galvanometer and its damping correction to be afterwards determined.
The method may be of service in the simultareous determination of the resistance of, and joint capacity and inductance of, a submarine cable or telephone- or telegraph-line.
XVII. Electrical Notes. By ARTHUR SCHUSTER, F.R.S.* II. On the Measurement of Resistance.
NUMBER of physical instruments, like the bolometer and platinum thermometer, depend on an accurate measurement of electric resistance. The question not unfrequently therefore arises, what the limit of accuracy is with which, with a given galvanometer and resistance, the measurement can be made.
Whatever arrangement we adopt, the sensitiveness is always increased by an increase of electromotive force; and the limit is reached either when we have put into action all the electromotive force at our disposal, or when the currents become so strong that there is danger of overheating one or other of the resistances. The overheating may either damage the insulation or produce disturbances in the way of thermoelectric currents, or be the cause of other irregularities. For a given resistance, we may generally fix on some current * Communicated by the Author.
which is the greatest that can with safety be used. ordinary laboratory practice our battery-power is generally sufficient, so that it is the heating of the conductors which puts the limit to a measurement of resistance; and I propose to discuss the question from this point of view. It may be answered in a very simple manner. Imagine any network of conductors, and let i be the current which passes through the resistance to be measured (p). Consider a small change of resistance Sp. This change will alter the currents, and, amongst others, that passing through the galvanometer which forms the measuring instrument. The change in the currents is the same as if an electromotive force numerically equal to isp was introduced into the branch of which p forms part. In order that this electromotive force should affect the galvanometer, which we take to have a resistance g, there must be some one mesh of the network of conductors which contains both p and g. and g. If there should be no other resistances in that mesh, and if p does not form part of any other mesh, the current Sy in the galvanometer produced by the change of resistance Sp will be given by
If the above conditions are not satisfied, and they never can be completely, the current through the galvanometer will be smaller than the value given. It will be shown that we may make such experimental arrangements as will render equation (1) approximately correct; at any rate, that equation will give us a lower limit for the change of resistance dp which produces an appreciable effect, if for i we substitute the maximum current which can pass the resistance. If this is denoted by im, and if dy stands for the smallest current the galvanometer can detect, we obtain
This represents a somewhat important proposition:With a given resistance and galvanometer, the ratio of the smallest change of resistance which can be detected to the sum of the given and galvanometer resistance is equal to the ratio of the smallest current which can be detected by the galvanometer to the maximum current which can be sent through the resistances.
If we have a choice of galvanometers and take account of the fact that for the same type of instrument the deflexion varies as the square root of the resistance, equation (2) shows
that the best galvanometer-resistance is that for which p=g, and in that case
With a given conductor and type of galvanometer, the smallest change per unit-resistance which can be measured is given by twice the ratio of the smallest current which can be detected by means of a galvanometer having the same resistance as the one to be measured to the greatest current which can be sent through the galvanometer.
For the smallest current which can be observed with a galvanometer of given type we may write dy=a/(√g). Inserting th's in equation (3) and observing that the equation holds only when p=g, we obtain
The highest percentage accuracy with which a given resistance can be measured is directly proportional to the square root of the maximum electric work which can be done on it without overheating.
Sometimes the galvanometer-resistance is given but p may be varied, as when we wish to design a bolometer or platinum thermometer. Equation (2) shows that there will be an advantage in taking p as large as possible provided we do not thereby reduce the value of in. This can be done by increasing the length of the conductor without diminishing the cross section.
The case which most frequently occurs is the one in which we have a certain, but not unlimited, choice of galvanometer. Each laboratory will probably be provided with a low and a high resistance galvanometer; but we cannot for each individual measurement construct one which has exactly the right resistance. It is useful to realize, therefore, how much we lose in sensitiveness by working with a galvanometer which has not exactly the right resistance.
Substituting again a/(g) for dy, the smallest observable change of resistance dp then becomes
Phil. Mag. S. 5. Vol. 39. Nọ. 237. Feb. 1895.
If we further put g=np, where p is equal to the best galvanometer-resistance,
times that of the obtainable maximum. And
the reduction is the same whether the galvanometer-resistance is np or p/n. The following table giving the values of 2√n
n+1 may be useful.
It is seen that if the galvanometer-resistance is five times too great or too small, the sensitiveness is reduced by 25 per cent. Except for very special purposes, such a reduction is not of great importance; but I think sufficient consideration is not paid, in ordinary laboratory practice, to the proper choice of galvanometer-resistance. The tendency is to use a galvanometer of too great resistance, probably because the makers take more trouble with the more expensive instruments, and in consequence they are more easily adjusted and less liable to get out of order. With the same instrument it is convenient to be able to vary the resistance; and this can be done by using the pattern of a differential galvanometer, and by dividing each of the two coils again into two parts. The resistance may be varied in that case in the ratio of 1 to 16. If we had the command of three galvanometers each having four coils respectively of 01, 4, and 1000 ohms, we could measure resistances varying between 0005 and 20,000 ohms, and always be within 25 per cent. of the greatest obtainable sensitiveness so far as the galvanometer is concerned. If we are only able to have one galvanometer, we can obtain a useful combination by choosing one of four coils of about
10 ohms resistance each. By joining them in multiple arc or series, we may change the internal resistance from 2.5 to 40 ohms, and conduct measurements of resistances lying between half an ohm and 200 ohms with considerable accuracy.
We have obtained a value for the smallest change of resistance which it is possible to measure with any arrangement, but have not discussed how closely we may approximate to the theoretical value by any of the known methods. The simplest arrangement, in many ways, is that in which a differential galvanometer is used, as in fig. 1. The two nearly equal resistances to be compared are p and p', while G and G' denote the two coils of the galvanometer. If balance is obtained with a current im passing through the resistances, it is required to determine the change Sp which will produce the smallest observable change of current dy in the galvanometer. Equation (1) will be applicable if the internal resistance of the battery b is small compared to p+g; for in that case a small change of resistance Sp will not disturb appreciably the current in G', and will produce the maximum effect in its own branch.
In order that this arrangement shall be workable, it is necessary, however, that the maximum current im which can pass p shall also be able to pass through the galvanometer without overheating it. Calling Ym the greatest current which the instrument will stand, we have as a necessary condition that Ym im.
As a second case, we take a differential galvanometer, but place it in parallel circuit with the resistance p. The arrangement is diagrammatically represented in fig. 2. In order that an electromotive force
iop isp shall produce a current in p+ g the galvanometer-branch G, it is necessary that b should be large compared with the combined resistance of and 9. The current through the ip galvanometer will be P; and hence