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THE ABSOLUTE MEASUREMENT OF CAPACITY.
By EDWARD B. Rosa and FREDERICK W. GROVER.
1. THE METHOD EMPLOYED. The usual method of determining the capacity of a condenser in electromagnetic measure is Maxwell's bridge method, using a tuning fork or a rotating commutator to charge and discharge the condenser,
which is placed in the fourth arm of a Wheatstone bridge. This is a null method, is adapted to measuring large and small capacities equally well, and requires an accurate knowledge only of a resistance and the rate of the fork or commutator.
The formula for the capacity C of the condenser as given by J. J. Thomson, is as follows:
dla+c+9) in which a, c, and d are the resistances of three arms of a Wheatstone bridge, b and g are battery and galvanometer resistances, respectively, and n is the number of times the condenser is charged and discharged per second.
When the vibrating arm P touches Q the condenser is
charged, and when it touches R it is short circuited and discharged. When Pis not touching Q the arm BD of the bridge is interrupted and a current flows from D to C through the galvanometer; when P touches Q the condenser is charged by a current coming partly through c and partly through g from C' to D. Thus the current through the galvanometer is alternately in opposite directions, and when these opposing currents balance each other there is no deflection of the galvanometer. This is accomplished by varying one of the resistances, a, c, ord. The capacity may then be computed from the formula (1).
a Phil. Trans., 1883.
Under ordinary circumstances the capacity of the condenser is very
a nearly given by the first part of the formula the remainder
ned constituting a correction factor which we may represent by F. Then
(2) ncd If after the experiment we replace the condenser and commutator by an accurately calibrated resistance box, and vary its resistance until the bridge is balanced while d, c, and d remain unchanged, then a 1
and hence cdTMR
F O'= nR
This is not practicable for very small capacities, as R would then be inconveniently large at ordinary values of n. But for capacities of 0.01 microfarad and larger this can usually be done. Of course, if a, c, and d are all known with sufficient accuracy this operation may be omitted.
2. THE CORRECTION FACTOR F.
The correction factor F, as we have called it, shown in equation (1), may be written as follows:
It is, of course, desirable to have this factor as nearly equal to unity
b. d 1+-+
and + should be as large as possible. To this end a should be small, cand d relatively large. Under these circumstances the values of b and g are not very important, although it is generally better to have them small. If b is nearly zero, the first parenthesis of the denominator disappears, as well as one term in the first parenthesis of the numerator, and the expression for F is somewhat simplified. Since c and d are opposite branches of the bridge, their product is constant, and therefore increasing c decreases d; both should be as large as possible, in order to make F as near unity as possible, but there are other reasons to be given later which make it desirable to have d much larger than c.
To show how the value of F varies with the values of the several quantities a, b, c, d, g, we have computed and plotted a number of values in the curves of figs. 3 and 1. In each case a=c=100 ohms. In the curves of fig. 3 the galvanometer resistance is 20 ohms, while the battery resistance is 0), 6, and 12 ohms in the three curves, respectively.
Ordinates show the values of F and abscissas increasing values of d. The value of F falls as low as 0.99 for d=6,000 ohms, running up to 0.995 at about 12,000 ohms, 0.999 for 60,000 ohms, and 0.9996 for d=150,000 ohms. If n=100 per second, these values correspond to capacities of 1.67, 0.83, 0.167, and 0.056 microfarads, respectively.
Evidently F is more nearly unity as the frequency n is smaller and the capacity of the condenser is smaller.
The difference in F due to change in b from 0 to 6 ohms is only 0.0006 at d=10,000 and 0.00006 at d=100,000. In our own experi
ments b was usually either 3 or 6 ohms, and hence the change in F due to changing the battery was very small, and a suitable correction easily made.
The effect of changing the galvanometer resistance is shown in the curves of fig. 4. Here a=c=100 as before, b=6 and g=20, 100, and