215*.] A Resistance Coil is a conductor capable of being easily placed in the voltaic circuit, so as to introduce into the circuit a known resistance. The electrodes or ends of the coil must be such that no appreciable error may arise from the mode of making the connexions. For resistances of con- siderable magnitude it is sufficient that the electrodes should be made of stout copper wire or rod well amalgamated with mercury at the ends, and that the ends should be made to press on flat amalgamated copper surfaces placed in mercury cups.

For very great resistances it is sufficient that the electrodes should be thick pieces of brass, and that the connexions should be made by inserting a wedge of brass or copper into the interval between them. This method is found very convenient.

The resistance coil itself consists of a wire well covered with silk, the ends of which are soldered permanently to the electrodes.

The coil must be so arranged that its temperature may be easily observed. For this purpose the wire is coiled on a tube and covered with another tube, so that it may be placed in a vessel of water, and that the water may have access to the inside and the outside of the coil.

To avoid the electromagnetic effects of the current in the coil the wire is first doubled back on itself and then coiled on the tube, so that at every part of the coil there are equal and opposite currents in the adjacent parts of the wire.

FORMS OF RESISTANCE COILS

When it is desired to keep two coils at the same temperature the wires are sometimes placed side by side and coiled up together. This method isespecially useful when it is more important to secure equality of resistance than to know the absolute value of the resistance, as in the case of the equal arms of Wheatstone’s Bridge (Art. 221).

When measurements of resistance were first attempted, a resistance coil, consisting of an uncovered wire coiled in a spiral groove round a cylinder of insulating material, was much used. It was called a Rheostat. The accuracy with which it was found possible to compare resistances was soon found to be inconsistent with the use of any instrument in which the contacts are not more perfect than can be obtained in the rheostat. The rheostat, however, is still used for adjusting the resistance where accurate measurement is not required.

Resistance coils are generally made of those metals whose resistance is greatest and which vary least with temperature. German silver fulfils these conditions very well, but some specimens are found to change their properties during the lapse of years. Hence for standard coils, several pure metals, and also an alloy of platinum and silver, have been employed, and the relative resistance of these during several years has been found constant up to the limits of modern accuracy∗ .

216*.] For very great resistances, such as several millions of Ohms, the wire must be either very long or very thin, and the construction of the coil is expensive and difficult. Hence tellurium and selenium have been proposed as materials for constructing standards of great resistance. A very ingenious and easy method of construction has been lately proposed by Phillips† . On a piece of ebonite or ground glass a fine pencil-line is drawn. The ends of this filament of plumbago are connected to metallic electrodes, and the whole is then covered with insulating varnish. If it should be found that the resis- tance of such a pencil-line remains constant, this will be the best method of obtaining a resistance of several millions of Ohms.

217*.] There are various arrangements by which resistance coils may be easily introduced into a circuit.

For instance, a series of coils of which the resistances are 1, 2, 4, 8, 16, ∗ † [More recent experiments indicate a small change in resistance in course of time.] Phil. Mag., July, 1870.RESISTANCE BOXES. 201 &c., arranged according to the powers of 2, may be placed in a box in series. Fig. 45. The electrodes consist of stout brass plates, so arranged on the outside of the box that by inserting a brass plug or wedge between two of them as a shunt, the resistance of the corresponding coil may be put out of the circuit. This arrangement was introduced by Siemens.

Each interval between the electrodes is marked with the resistance of the corresponding coil, so that if we wish to make the resistance box equal to 107 we express 107 in the binary scale as 64 + 32 + 8 + 2 + 1 or 1101011. We then take the plugs out of the holes corresponding to 64, 32, 8, 2 and 1, and leave the plugs in 16 and 4. This method, founded on the binary scale, is that in which the smallest number of separate coils is needed, and it is also that which can be most readily tested. For if we have another coil equal to 1 we can test the equality of 1 and 1′ , then that of 1 + 1′ and 2, then that of 1 + 1′ + 2 and 4, and so on.

The only disadvantage of the arrangement is that it requires a familiarity with the binary scale of notation, which is not generally possessed by those accustomed to express every number in the decimal scale. 218*.] A box of resistance coils may be arranged in a different way for the purpose of measuring conductivities instead of resistances.THE COMPARISON OF RESISTANCES.

The coils are placed so that one end of each is connected with a long thick piece of metal which forms one electrode of the box, and the other end is connected with a stout piece of brass plate as in the former case. Fig. 46.

The other electrode of the box is a long brass plate, such that by inserting brass plugs between it and the electrodes of the coils it may be con- nected to the first electrode through any given set of coils. The conductivity of the box is then the sum of the conductivities of the coils. In the figure, in which the resistances of the coils are 1, 2, 4, &c., and the plugs are inserted at 2 and 8, the conductivity of the box is 12 + 18 = 58 , and the resistance of the box is therefore 85 or 1·6.

This method of combining resistance coils for the measurement of frac- tional resistances was introduced by Sir W. Thomson under the name of the method of multiple arcs. (See Art. 158.)

On the Comparison of Resistances.

219*.] If E is the electromotive force of a battery, and R the resistance of the battery and its connexions, including the galvanometer used in measuring the current, and if the strength of the current is I when the battery connexions are closed, and I1 , I2 when additional resistances r1 , r2 are introduced into the circuit, then, by Ohm’s Law, E = IR = I1 (R + r1 ) = I2 (R + r2 ). Eliminating E, the electromotive force of the battery, and R the resistance of the battery and its connexions, we get Ohm’s formula r1 (I − I1 )I2

. r2 (I − I2 )I1THE COMPARISON OF RESISTANCES.

This method requires a measurement of the ratios of I, I1 and I2 , and this implies a galvanometer graduated for absolute measurements. If the resistances r1 and r2 are equal, then I1 and I2 are equal, and we can test the equality of currents by a galvanometer which is not capable of determining their ratios.

But this is rather to be taken as an example of a faulty method than as a practical method of determining resistance. The electromotive force E cannot be maintained rigorously constant, and the internal resistance of the battery is also exceedingly variable, so that any methods in which these are assumed to be even for a short time constant are not to be depended on. 220*.] The comparison of resistances can be made with extreme accuracy by either of two methods, in which the result is independent of variations of R and E. Fig. 47.

The first of these methods depends on the use of the differential galvanome- ter, an instrument in which there are two coils, the currents in which are independent of each other, so that when the currents are made to flow in op- posite directions they act in opposite directions on the needle, and when the ratio of these currents is that of m to n they have no resultant effect on theTHE COMPARISON OF RESISTANCES.

galvanometer needle.

Let I1 , I2 be the currents through the two coils of the galvanometer, then the deflexion of the needle may be written δ = mI1 − nI2 . Now let the battery current I be divided between the coils of the gal- vanometer, and let resistances A and B be introduced into the first and second coils respectively. Let the remainder of the resistance of the coils and their connexions be α and β respectively, and let the resistance of the battery and its connexions between C and D be r, and its electromotive force E. Then we find, by Ohm’s Law, for the difference of potentials between C and D, C − D = I1 (A + α) = I2 (B + β) = E − Ir, and since I1 = E B+β , D I1 + I2 = I, A+α+B+β A+α I2 = E , I =E , D D where D = (A + α)(B + β) + r(A + α + B + β). The deflexion of the galvanometer needle is therefore δ= E {m(B + β) − n(A + α)}, D and if there is no observable deflexion, then we know that the quantity en- closed in brackets cannot differ from zero by more than a certain small quan- tity, depending on the power of the battery, the suitableness of the arrange- ment, the delicacy of the galvanometer, and the accuracy of the observer. Suppose that B has been adjusted so that there is no apparent deflexion. Now let another conductor A′ be substituted for A, and let A′ be adjusted till there is no apparent deflexion. Then evidently to a first approximation A′ = A.