A Treatise on Electricity and Magnetism/Part II/Chapter XII

by James Clerk Maxwell

CHAPTER XII.

ON THE ELECTRIC RESISTANCE OF SUBSTANCES.

359.]There are three classes in which we may place different substances in relation to the passage of electricity through them.

The first class contains all the metals and their alloys, some sulphurets, and other compounds containing metals, to which we must add carbon in the form of gas-coke, and selenium in the crystalline form.

In all these substances conduction takes place without any decomposition, or alteration of the chemical nature of the substance, either in its interior or where the current enters and leaves the body. In all of them the resistance increases as the temperature rises.

The second class consists of substances which are called electrolytes, because the current is associated with a decomposition of the substance into two components which appear at the electrodes. As a rule a substance is an electrolyte only when in the liquid form, though certain colloid substances, such as glass at 100°C, which are apparently solid, are electrolytes. It would appear from the experiments of Sir B. C. Brodie that certain gases are capable of electrolysis by a powerful electromotive force.

In all substances which conduct by electrolysis the resistance diminishes as the temperature rises.

The third class consists of substances the resistance of which is so great that it is only by the most refined methods that the passage of electricity through them can be detected. These are called Dielectrics. To this class belong a considerable number of solid bodies, many of which are electrolytes when melted, some liquids, such as turpentine, naphtha, melted paraffin, &c., and all gases and vapours. Carbon in the form of diamond, and selenium in the amorphous form, belong to this class.

The resistance of this class of bodies is enormous compared with that of the metals. It diminishes as the temperature rises. It is difficult, on account of the great resistance of these substances, to determine whether the feeble current which we can force through them is or is not associated with electrolysis.

On the Electric Resistance of Metals.

360.] There is no part of electrical research in which more numerous or more accurate experiments have been made than in the determination of the resistance of metals. It is of the utmost importance in the electric telegraph that the metal of which the wires are made should have the smallest attainable resistance. Measurements of resistance must therefore be made before selecting the materials. When any fault occurs in the line, its position is at once ascertained by measurements of resistance, and these measurements, in which so many persons are now employed, require the use of resistance coils, made of metal the electrical properties of which have been carefully tested.

The electrical properties of metals and their alloys have been studied with great care by MM. Matthiessen, Vogt, and Hockin, and by MM. Siemens, who have done so much to introduce exact electrical measurements into practical work.

It appears from the researches of Dr. Matthiessen, that the effect of temperature on the resistance is nearly the same for a considerable number of the pure metals, the resistance at 100°C being to that at 0°C in the ratio of 1.414 to 1, or of 1 to 70.7. For pure iron the ratio is 1.645, and for pure thallium 1.458.

The resistance of metals has been observed by Dr. C.W. Siemens^[1] through a much wider range of temperature, extending from the freezing point to 350°C, and in certain cases to 1000°C. He finds that the resistance increases as the temperature rises, but that the rate of increase diminishes as the temperature rises. The formula, which he finds to agree very closely both with the resistances observed at low temperatures by Dr. Matthiessen and with his own observations through a range of 1000°C, is

$r=\alpha T^{\frac {1}{2}}+\beta T\gamma$,

where $T$ is the absolute temperature reckoned from −273°C, and $\alpha ,\,\beta ,\,\gamma$ are constants. Thus, for

Platinum⁠	$r=0.039369T^{\frac {1}{2}}+0.00216407T-0.2413$,
Copper	$r=0.026577T^{\frac {1}{2}}+0.0031443T-0.22751$,
Iron	$r=0.072545T^{\frac {1}{2}}+0.0038133T-1.23971$.

From data of this kind the temperature of a furnace may be determined by means of an observation of the resistance of a platinum wire placed in the furnace.

Dr. Matthiessen found that when two metals are combined to form an alloy, the resistance of the alloy is in most cases greater than that calculated from the resistance of the component metals and their proportions. In the case of alloys of gold and silver, the resistance of the alloy is greater than that of either pure gold or pure silver, and, within certain limiting proportions of the constituents, it varies very little with a slight alteration of the proportions. For this reason Dr. Matthiessen recommended an alloy of two parts by weight of gold and one of silver as a material for reproducing the unit of resistance.

The effect of change of temperature on electric resistance is generally less in alloys than in pure metals.

Hence ordinary resistance coils are made of German silver, on account of its great resistance and its small variation with temperature.

An alloy of silver and platinum is also used for standard coils.

361.] The electric resistance of some metals changes when the metal is annealed; and until a wire has been tested by being repeatedly raised to a high temperature without permanently altering its resistance, it cannot be relied on as a measure of resistance. Some wires alter in resistance in course of time without having been exposed to changes of temperature. Hence it is important to ascertain the specific resistance of mercury, a metal which being fluid has always the same molecular structure, and which can be easily purified by distillation and treatment with nitric acid. Great care has been bestowed in determining the resistance of this metal by W. and C. F. Siemens, who introduced it as a standard. Their researches have been supplemented by those of Matthiessen and Hockin.

The specific resistance of mercury was deduced from the observed resistance of a tube of length $l$ containing a weight $w$ of mercury, in the following manner.

No glass tube is of exactly equal bore throughout, but if a small quantity of mercury is introduced into the tube and occupies a length $\lambda$ of the tube, the middle point of which is distant $x$ from one end of the tube, then the area $s$ of the section near this point will be $s={\frac {C}{\lambda }}$ where $C$ is some constant.

The weight of mercury which fills the whole tube is

$w=\rho \int 's\;dx=\rho C\Sigma \left({\frac {1}{\lambda }}\right){\frac {l}{n}}$,

where $n$ is the number of points, at equal distances along the tube, where $\lambda$ has been measured, and $\rho$ is the mass of unit of volume.

The resistance of the whole tube is

$R=\int {\frac {r}{s}}\;ds={\frac {r}{C}}\Sigma (\lambda ){\frac {l}{n}}$,

where $r$ is the specific resistance per unit of volume.

Hence

$wR=r\rho \Sigma (\lambda )\Sigma \left({\frac {1}{\lambda }}\right){\frac {l^{2}}{n^{2}}}$,

and

$r={\frac {wR}{\rho l^{2}}}{\frac {n^{2}}{\Sigma (\lambda )\Sigma \left({\frac {1}{\lambda }}\right)}}$

gives the specific resistance of unit of volume.

To find the resistance of unit of length and unit of mass we must multiply this by the density.

It appears from the experiments of Matthiessen and Hockin that the resistance of a uniform column of mercury of one metre in length, and weighing one gramme at 0°C, is 13.071 Ohms, whence it follows that if the specific gravity of mercury is 13.595, the resistance of a column of one metre in length and one square millimetre in section is 0.96146 Ohms.

362.] In the following table $R$ is the resistance in Ohms of a column one metre long and one gramme weight at 0°C, and $r$ is the resistance in centimetres per second of a cube of one centimetre, according to the experiments of Matthiessen.^[2]

	Specific gravity			$R$		$r$	Percentage increment of resistance for 1°C at 20°C.
Silver	10	.50	hard drawn	0	.1689	1609	0	.377
Copper	8	.95	hard drawn	0	.1469	1642	0	.388
Gold	19	.27	hard drawn	0	.4150	2154	0	.365
Lead	11	.391	pressed	2	.257	19847	0	.387
Mercury	13	.595	liquid	13	.071	96146	0	.072
Gold 2, Silver 1	15	.218	hard or annealed	1	.668	10988	0	.065
Selenium at 100°C			Crystalline form			6 × 10¹³	1	.00

On the Electric Resistance of Electrolytes.

363.] The measurement of the electric resistance of electrolytes is rendered difficult on account of the polarization of the electrodes, which causes the observed difference of potentials of the metallic electrodes to be greater than the electromotive force which actually produces the current.

This difficulty can be overcome in various ways. In certain cases we can get rid of polarization by using electrodes of proper material, as, for instance, zinc electrodes in a solution of sulphate of zinc. By making the surface of the electrodes very large compared with the section of the part of the electrolyte whose resistance is to be measured, and by using only currents of short duration in opposite directions alternately, we can make the measurements before any considerable intensity of polarization has been excited by the passage of the current.

Finally, by making two different experiments, in one of which the path of the current through the electrolyte is much longer than in the other, and so adjusting the electromotive force that the actual current, and the time during which it flows, are nearly the same in each case, we can eliminate the effect of polarization altogether.

364.] In the experiments of Dr. Paalzow^[3] the electrodes were in the form of large disks placed in separate flat vessels filled with the electrolyte, and the connexion was made by means of a long siphon filled with the electrolyte and dipping into both vessels. Two such siphons of different lengths were used.

The observed resistances of the electrolyte in these siphons being $R_{1}$ and $R_{2}$, the siphons were next filled with mercury, and their resistances when filled with mercury were found to be $R_{1}'$ and $R_{2}'$.

The ratio of the resistance of the electrolyte to that of a mass of mercury at 0°C of the same form was then found from the formula

$\rho ={\frac {R_{1}-R_{2}}{R_{1}'-R_{2}'}}$.

To deduce from the values of $\rho$ the resistance of a centimetre in length having a section of a square centimetre, we must multiply them by the value of $r$ for mercury at 0°C. See Art. 361.

The results given by Paalzow are as follow:—

Mixtures of Sulphuric Acid and Water.
		Temp.	Resistance compared with mercury.
H₂SO₄		⁠15°C⁠	96950
H₂SO₄	+  14 H²O	19°C	14157
H₂SO₄	+  13 H²O	22°C	13310
H₂SO₄	+ 499 H²O	22°C	184773
Sulphate of Zinc and Water.
ZnSO₄	+  23 H²O	23°C	194400
ZnSO₄	+  24 H²O	23°C	191000
ZnSO₄	+ 105 H²O	23°C	354000
Sulphate of Copper and Water.
CuSO₄	+  45 H²O	22°C	202410
CuSO₄	+ 105 H²O	22°C	339341
Sulphate of Magnesium and Water.
MgSO₄	+  34 H²O	22°C	199180
MgSO₄	+ 107 H²O	22°C	324600
Hydrochloric Acid and Water.
HCl	+  15 H²O	23°C	13626
HCl	+ 500 H²O	23°C	86679

365.] MM. F. Kohlrausch and W. A. Nippoldt^[4] have determined the resistance of mixtures of sulphuric acid and water. They used alternating magneto-electric currents, the electromotive force of which varied from 12 to 174 of that of a Grove's cell, and by means of a thermoelectric copper-iron pair they reduced the electromotive force to 1429000 of that of a Grove's cell. They found that Ohm's law was applicable to this electrolyte throughout the range of these electromotive forces.

The resistance is a minimum in a mixture containing about one-third of sulphuric acid.

The resistance of electrolytes diminishes as the temperature increases. The percentage increment of conductivity for a rise of 1°C is given in the following table.

Resistance of Mixtures of Sulphuric Acid and Water at 22°C in terms of Mercury at 0°C.   MM. Kohlrausch and Nippoldt.
Specific gravity at 18°5	Percentage of H2SO4	Resistance at 22°C (Hg=1)	Percentage increment of conductivity for 1°C.
0.9985	0.0	746300	0.47
1.00	0.2	465100	0.47
1.0504	8.3	34530	0.653
1.0989	14.2	18946	0.646
1.1431	20.2	14990	0.799
1.2045	28.0	13133	1.317
1.2631	35.2	13132	1.259
1.3163	41.5	14286	1.410
1.3547	46.0	15762	1.674
1.3994	50.4	17726	1.582
1.4482	55.2	20796	1.417
1.5026	60.3	25574	1.794

On the Electrical Resistance of Dielectrics.

366.] A great number of determinations of the resistance of gutta-percha, and other materials used as insulating media, in the manufacture of telegraphic cables, have been, made in order to ascertain the value of these materials as insulators.

The tests are generally applied to the material after it has been used to cover the conducting wire, the wire being used as one electrode, and the water of a tank, in which the cable is plunged, as the other. Thus the current is made to pass through a cylindrical coating of the insulator of great area and small thickness.

It is found that when the electromotive force begins to act, the current, as indicated by the galvanometer, is by no means constant. The first effect is of course a transient current of considerable intensity, the total quantity of electricity being that required to charge the surfaces of the insulator with the superficial distribution of electricity corresponding to the electromotive force. This first current therefore is a measure not of the conductivity, but of the capacity of the insulating layer.

But even after this current has been allowed to subside the residual current is not constant, and does not indicate the true conductivity of the substance. It is found that the current continues to decrease for at least half an hour, so that a determination of the resistance deduced from the current will give a greater value if a certain time is allowed to elapse than if taken immediately after applying the battery.

Thus, with Hooper's insulating material the apparent resistance at the end of ten minutes was four times, and at the end of nineteen hours twenty-three times that observed at the end of one minute. When the direction of the electromotive force is reversed, the resistance falls as low or lower than at first and then gradually rises.

These phenomena seem to be due to a condition of the gutta-percha, which, for want of a better name, we may call polarization, and which we may compare on the one hand with that of a series of Leyden jars charged by cascade, and, on the other, with Ritter's secondary pile, Art. 271.

If a number of Leyden jars of great capacity are connected in series by means of conductors of great resistance (such as wet cotton threads in the experiments of M. Gaugain), then an electromotive force acting on the series will produce a current, as indicated by a galvanometer, which will gradually diminish till the jars are fully charged.

The apparent resistance of such a series will increase, and if the dielectric of the jars is a perfect insulator it will increase without limit. If the electromotive force be removed and connexion made between the ends of the series, a reverse current will be observed, the total quantity of which, in the case of perfect insulation, will be the same as that of the direct current. Similar effects are observed in the case of the secondary pile, with the difference that the final insulation is not so good, and that the capacity per unit of surface is immensely greater.

In the case of the cable covered with gutta-percha, &c., it is found that after applying the battery for half an hour, and then connecting the wire with the external electrode, a reverse current takes place, which goes on for some time, and gradually reduces the system to its original state.

These phenomena are of the same kind with those indicated by the 'residual discharge' of the Leyden jar, except that the amount of the polarization is much greater in gutta-percha, &c. than in glass.

This state of polarization seems to be a directed property of the material, which requires for its production not only electromotive force, but the passage, by displacement or otherwise, of a considerable quantity of electricity, and this passage requires a considerable time. When the polarized state has been set up, there is an internal electromotive force acting in the substance in the reverse direction, which will continue till it has either produced a reversed current equal in total quantity to the first, or till the state of polarization has quietly subsided by means of true conduction through the substance.

The whole theory of what has been called residual discharge, absorption of electricity, electrification, or polarization, deserves a careful investigation, and will probably lead to important discoveries relating to the internal structure of bodies.

367.] The resistance of the greater number of dielectrics diminishes as the temperature rises.

Thus the resistance of gutta-percha is about twenty times as great at 0°C as at 24°C. Messrs. Bright and Clark have found that the following formula gives results agreeing with their experiments. If $r$ is the resistance of gutta-percha at temperature $T$ centigrade, then the resistance at temperature $T+t$ will be

$R=r\times 0.8878^{t}$,

the number varies between 0.8878 and 0.9.

Mr. Hockin has verified the curious fact that it is not until some hours after the gutta-percha has taken its temperature that the resistance reaches its corresponding value.

The effect of temperature on the resistance of india-rubber is not so great as on that of gutta-percha.

The resistance of gutta-percha increases considerably on the application of pressure.

The resistance, in Ohms, of a cubic metre of various specimens of gutta-percha used in different cables is as follows^[5].

Name of Cable.
Red Sea		.267 × 10¹²	to .362 × 10¹²
Malta-Alexandria	1	.23 × 10¹²
Persian Gulf	1	.80 × 10¹²
Second Atlantic	3	.42 × 10¹²
Hooper's Persian Gulf Core	74	.7  × 10¹²
Gutta-percha at 24°C	3	.53 × 10¹²

368.] The following table, calculated from the experiments of M. Buff, described in Art. 271, shews the resistance of a cubic metre of glass in Ohms at different temperatures.

Temperature.	⁠Resistance.
200°C	227000
250°	13900
300°	1480
350°	1035
400°	735

369.] Mr. C. F. Varley^[6] has recently investigated the conditions of the current through rarefied gases, and finds that the electromotive force $E$ is equal to a constant $E_{0}$ together with a part depending on the current according to Ohm's Law, thus

$E=E_{0}+RC.$

For instance, the electromotive force required to cause the current to begin in a certain tube was that of 323 Daniell's cells, but an electromotive force of 304 cells was just sufficient to maintain the current. The intensity of the current, as measured by the galvanometer, was proportional to the number of cells above 304. Thus for 305 cells the deflexion was 2, for 306 it was 4, for 307 it was 6, and so on up to 380, or 304 + 76 for which the deflexion was 150, or 76 × 1.97.

From these experiments it appears that there is a kind of polarization of the electrodes, the electromotive force of which is equal to that of 304 Daniell's cells, and that up to this electromotive force the battery is occupied in establishing this state of polarization. When the maximum polarization is established, the excess of electromotive force above that of 304 cells is devoted to maintaining the current according to Ohm's Law.

The law of the current in a rarefied gas is therefore very similar to the law of the current through an electrolyte in which we have to take account of the polarization of the electrodes.

In connexion with this subject we should study Thomson's results, described in Art. 57, in which the electromotive force required to produce a spark in air was found to be proportional not to the distance, but to the distance together with a constant quantity. The electromotive force corresponding to this constant quantity may be regarded as the intensity of polarization of the electrodes.

370.] MM. Wiedemann and Rühlmann have recently^[7] investigated the passage of electricity through gases. The electric current was produced by Holtz's machine, and the discharge took place between spherical electrodes within a metallic vessel containing rarefied gas. The discharge was in general discontinuous, and the interval of time between successive discharges was measured by means of a mirror revolving along with the axis of Holtz's machine. The images of the series of discharges were observed by means of a heliometer with a divided object-glass, which was adjusted till one image of each discharge coincided with the other image of the next discharge. By this method very consistent results were obtained. It was found that the quantity of electricity in each discharge is independent of the strength of the current and of the material of the electrodes, and that it depends on the nature and density of the gas, and on the distance and form of the electrodes.

These researches confirm the statement of Faraday^[8] that the electric tension (see Art. 48) required to cause a disruptive discharge to begin at the electrified surface of a conductor is a little less when the electrification is negative than when it is positive, but that when a discharge does take place, much more electricity passes at each discharge when it begins at a positive surface. They also tend to support the hypothesis stated in Art. 57, that the stratum of gas condensed on the surface of the electrode plays an important part in the phenomenon, and they indicate that this condensation is greatest at the positive electrode.

1. Proc. R. S., April 27, 1871.
2. Phil. Mag., May, 1865.
3. Berlin Monatsbericht, July, 1868.
4. Pogg., Ann. cxxxviii, p. 286, Oct. 1869.
5. Jenkin's Cantor Lectures.
6. Proc. R. S., Jan. 12, 1871.
7. Berichte der Königl. Sächs. Gesellschaft, Oct. 20, 1871.
8. Exp. Res., 1501.