Chapter I: Conduction in Solids

It is found convenient to express the resistivity of metals in two different ways: (1) We may state the resistivity of one cubic centimetre of the material in microhms or absolute units taken between opposed faces. This is called the _volume-resistivity_; (2) we may express the resistivity by stating the resistance in ohms offered by a wire of the material in question of uniform cross-section one metre in length, and one gramme in weight. This numerical measure of the resistivity is called the _mass-resistivity_. The mass-resistivity of a body is connected with its volume-resistivity and the density of the material in the following manner:--The mass-resistivity, expressed in microhms per metre-gramme, divided by 10 times the density is numerically equal to the volume-resistivity per centimetre-cube in absolute C.G.S. units. The mass-resistivity per metre-gramme can always be obtained by measuring the resistance and the mass of any wire of uniform cross-section of which the length is known, and if the density of the substance is then measured, the volume-resistivity can be immediately calculated.

If R is the resistance in ohms of a wire of length l, uniform
cross-section s, and density d, then taking [rho] for the
volume-resistivity we have 10^9R = [rho]l/s; but lsd = M, where M is
the mass of the wire. Hence 10^9R = [rho]dl^2/M. If l = 100 and M = 1,
then R = [rho]'= resistivity in ohms per metre-gramme, and 10^9[rho]'
= 10,000d[rho], or [rho] = 10^5[rho]'/d, and [rho]' = 10,000MR/l^2.

The following rules, therefore, are useful in connexion with these
measurements. To obtain the mass-resistivity per metre-gramme of a
substance in the form of a uniform metallic wire:--Multiply together
10,000 times the mass in grammes and the total resistance in ohms, and
then divide by the square of the length in centimetres. Again, to
obtain the volume-resistivity in C.G.S. units per centimetre-cube, the
rule is to multiply the mass-resistivity in ohms by 100,000 and divide
by the density. These rules, of course, apply only to wires of uniform
cross-section. In the following Tables I., II. and III. are given the
mass and volume resistivity of ordinary metals and certain alloys
expressed in terms of the international ohm or the absolute C.G.S.
unit of resistance, the values being calculated from the experiments
of A. Matthiessen (1831-1870) between 1860 and 1865, and from later
results obtained by J. A. Fleming and Sir James Dewar in 1893.

TABLE I.--_Electric Mass-Resistivity of Various Metals at 0 deg. C.,
or Resistance per Metre-gramme in International Ohms at 0 deg. C._
(Matthiessen.)

+---------------------+---------------------+-------------+
| | Resistance at 0 deg.| |
| | C. in International | Approximate |
| Metal. | Ohms of a Wire | Temperature |
| | 1 Metre long and | Coefficient |
| | Weighing | near 20 |
| | 1 Gramme. | deg. C. |
+---------------------+---------------------+-------------+
|Silver (annealed) | .1523 | 0.00377 |
|Silver (hard-drawn) | .1657 | .. |
|Copper (annealed) | .1421 | 0.00388 |
|Copper (hard-drawn) | .1449 (Matthiessen's Standard) |
|Gold (annealed) | .4025 | 0.00365 |
|Gold (hard-drawn) | .4094 | .. |
|Aluminium (annealed) | .0757 | .. |
|Zinc (pressed) | .4013 | .. |
|Platinum (annealed) | 1.9337 | .. |
|Iron (annealed) | .765 | .. |
|Nickel (annealed) | 1.058[1] | .. |
|Tin (pressed) | .9618 | 0.00365 |
|Lead (pressed) | 2.2268 | 0.00387 |
|Antimony (pressed) | 2.3787 | 0.00389 |
|Bismuth (pressed) | 12.8554[1] | 0.00354 |
|Mercury (liquid) | 12.885[2] | 0.00072 |
+---------------------+---------------------+-------------+

The data commonly used for calculating metallic resistivities were
obtained by A. Matthiessen, and his results are set out in the Table
II. which is taken from Cantor lectures given by Fleeming Jenkin in
1866 at or about the date when the researches were made. The figures
given by Jenkin have, however, been reduced to international ohms and
C.G.S. units by multiplying by ([pi]/4) X 0.9866 X 10^5 = 77,485.

Subsequently numerous determinations of the resistivity of various
pure metals were made by Fleming and Dewar, whose results are set out
in Table III.

TABLE II.--_Electric Volume-Resistivity of Various Metals at 0 deg.
C., or Resistance per Centimetre-cube in C.G.S. Units at 0 deg. C._

+---------------------+---------------------+
| | Volume-Resistivity. |
| Metal. | at 0 deg. C. in |
| | C.G.S. Units |
+---------------------+---------------------+
|Silver (annealed) | 1,502 |
|Silver (hard-drawn) | 1,629 |
|Copper (annealed) | 1,594 |
|Copper (hard-drawn) | 1,630[3] |
|Gold (annealed) | 2,052 |
|Gold (hard-drawn) | 2,090 |
|Aluminium (annealed) | 3,006 |
|Zinc (pressed) | 5,621 |
|Platinum (annealed) | 9,035 |
|Iron (annealed) | 10,568 |
|Nickel (annealed) | 12,429[4] |
|Tin (pressed) | 13,178 |
|Lead (pressed) | 19,580 |
|Antimony (pressed) | 35,418 |
|Bismuth (pressed) | 130,872 |
|Mercury (liquid) | 94,896[5] |
+---------------------+---------------------+

TABLE III.--_Electric Volume-Resistivity of Various Metals at 0 deg.
C., or Resistance per Centimetre-cube at 0 deg. C. in C.G.S. Units._
(Fleming and Dewar, _Phil. Mag._, September 1893.)

+--------------------------+------------------+------------------+
| | Resistance | Mean Temperature |
| Metal. | at 0 deg. C. per | Coefficient |
| | Centimetre-cube | between 0 deg. C.|
| | in C.G.S. Units. | and 100 deg. C. |
+--------------------------+------------------+------------------+
|Silver (electrolytic and | | |
| well annealed)[6] | 1,468 | 0.00400 |
|Copper (electrolytic and | | |
| well annealed)[6] | 1,561 | 0.00428 |
|Gold (annealed) | 2,197 | 0.00377 |
|Aluminium (annealed) | 2,665 | 0.00435 |
|Magnesium (pressed) | 4,355 | 0.00381 |
|Zinc | 5,751 | 0.00406 |
|Nickel (electrolytic)[6] | 6,935 | 0.00618 |
|Iron (annealed) | 9,065 | 0.00625 |
|Cadmium | 10,023 | 0.00419 |
|Palladium | 10,219 | 0.00354 |
|Platinum (annealed) | 10,917 | 0.003669 |
|Tin (pressed) | 13,048 | 0.00440 |
|Thallium (pressed) | 17,633 | 0.00398 |
|Lead (pressed) | 20,380 | 0.00411 |
|Bismuth (electrolytic)[7] | 110,000 | 0.00433 |
+--------------------------+------------------+------------------+

_Resistivity of Mercury._--The volume-resistivity of pure mercury is a
very important electric constant, and since 1880 many of the most
competent experimentalists have directed their attention to the
determination of its value. The experimental process has usually been
to fill a glass tube of known dimensions, having large cup-like
extensions at the ends, with pure mercury, and determine the absolute
resistance of this column of metal. For the practical details of this
method the following references may be consulted:--"The Specific
Resistance of Mercury," Lord Rayleigh and Mrs Sidgwick, _Phil.
Trans._, 1883, part i. p. 173, and R. T. Glazebrook, _Phil. Mag._,
1885, p. 20; "On the Specific Resistance of Mercury," R. T. Glazebrook
and T. C. Fitzpatrick, _Phil. Trans._, 1888, p. 179, or _Proc. Roy.
Soc._, 1888, p. 44, or _Electrician_, 1888, 21, p. 538; "Recent
Determinations of the Absolute Resistance of Mercury," R. T.
Glazebrook, _Electrician_, 1890, 25, pp. 543 and 588. Also see J. V.
Jones, "On the Determination of the Specific Resistance of Mercury in
Absolute Measure," _Phil. Trans._, 1891, A, p. 2. Table IV. gives the
values of the volume-resistivity of mercury as determined by various
observers, the constant being expressed (a) in terms of the resistance
in ohms of a column of mercury one millimetre in cross-section and 100
centimetres in length, taken at 0 deg. C.; and (b) in terms of the
length in centimetres of a column of mercury one square millimetre in
cross-section taken at 0 deg. C. The result of all the most careful
determinations has been to show that the resistivity of pure mercury
at 0 deg. C. is about 94,070 C.G.S. electromagnetic units of
resistance, and that a column of mercury 106.3 centimetres in length
having a cross-sectional area of one square millimetre would have a
resistance at 0 deg. C. of one international ohm. These values have
accordingly been accepted as the official and recognized values for
the specific resistance of mercury, and the definition of the ohm. The
table also states the methods which have been adopted by the different
observers for obtaining the absolute value of the resistance of a
known column of mercury, or of a resistance coil afterwards compared
with a known column of mercury. A column of figures is added showing
the value in fractions of an international ohm of the British
Association Unit (B.A.U.), formerly supposed to represent the true
ohm. The real value of the B.A.U. is now taken as .9866 of an
international ohm.

TABLE IV.--_Determinations of the Absolute Value of the
Volume-Resistivity of Mercury and the Mercury Equivalent of the
Ohm._

+-----------------+------+----------------------+--------+----------+---------+
| | | | |Value of |Value of |
| | | |Value of|100 Centi-|Ohm in |
| Observer. | Date.| Method. |B.A.U. |metres of |Centi- |
| | | |in Ohms.|Mercury |metres of|
| | | | |in Ohms. |Mercury. |
+-----------------+------+----------------------+--------+----------+---------+
|Lord Rayleigh | 1882 | Rotating coil | .98651 | .94133 | 106.31 |
|Lord Rayleigh | 1883 | Lorenz method | .98677 | .. | 106.27 |
|G. Wiedemann | 1884 | Rotation through 180 | .. | .. | 106.19 |
| | | deg. | | | |
|E. E. N. Mascart | 1884 | Induced current | .98611 | .94096 | 106.33 |
|H. A. Rowland | 1887 | Mean of several | .98644 | .94071 | 106.32 |
| | | methods | | | |
|F. Kohlrausch | 1887 | Damping of magnets | .98660 | .94061 | 106.32 |
|R. T. Glazebrook | 1882 | Induced currents | .98665 | .94074 | 106.29 |
| | 1888 | | | | |
|Wuilleumeier | 1890 | | .98686 | .94077 | 106.31 |
|Duncan and Wilkes| 1890 | Lorenz | .98634 | .94067 | 106.34 |
|J. V. Jones | 1891 | Lorenz | .. | .94067 | 106.31 |
| | | +--------+----------+---------+
| | | Mean value .98653 | | |
| | | | | |
|Streker | 1885 | An absolute determin-| | .94056 | 106.32 |
|Hutchinson | 1888 | ation of resistance| | .94074 | 106.30 |
|E. Salvioni | 1890 | was not made. The | | .94054 | 106.33 |
|E. Salvioni | .. | value .98656 has | | .94076 | 106.30 |
| | | been used | | | |
| | | | +----------+---------+
| | | Mean value .94076 | 106.31 |
| | | | |
|H. F. Weber | 1884 | Induced current | Absolute measure- | 105.37 |
|H. F. Weber | .. | Rotating coil | ments compared | 106.16 |
|A. Roiti | 1884 | Mean effect of | with German silver| 105.89 |
| | | induced current | wire coils issued | |
|F. Himstedt | 1885 | | by Siemens and | 105.98 |
| | | | Streker | |
|F. E. Dorn | 1889 | Damping of a magnet | | 106.24 |
|Wild | 1883 | Damping of a magnet | | 106.03 |
|L. V. Lorenz | 1885 | Lorenz method | | 105.93 |
+-----------------+------+----------------------+-------------------+---------+

For a critical discussion of the methods which have been adopted in
the absolute determination of the resistivity of mercury, and the
value of the British Association unit of resistance, the reader may be
referred to the _British Association Reports_ for 1890 and 1892
(_Report of Electrical Standards Committee_), and to the
_Electrician_, 25, p. 456, and 29, p. 462. A discussion of the
relative value of the results obtained between 1882 and 1890 was given
by R. T. Glazebrook in a paper presented to the British Association at
Leeds, 1890.

_Resistivity of Copper._--In connexion with electro-technical work the
determination of the conductivity or resistivity values of annealed
and hard-drawn copper wire at standard temperatures is a very
important matter. Matthiessen devoted considerable attention to this
subject between the years 1860 and 1864 (see _Phil. Trans._, 1860, p.
150), and since that time much additional work has been carried out.
Matthiessen's value, known as _Matthiessen's Standard_, for the
mass-resistivity of pure hard-drawn copper wire, is the resistance of
a wire of pure hard-drawn copper one metre long and weighing one
gramme, and this is equal to 0.14493 international ohms at 0 deg. C.
For many purposes it is more convenient to express temperature in
Fahrenheit degrees, and the recommendation of the 1899 committee on
copper conductors[8] is as follows:--"Matthiessen's standard for
hard-drawn conductivity commercial copper shall be considered to be
the resistance of a wire of pure hard-drawn copper one metre long,
weighing one gramme which at 60 deg. F. is 0.153858 international
ohms." Matthiessen also measured the mass-resistivity of annealed
copper, and found that its conductivity is greater than that of
hard-drawn copper by about 2.25% to 2.5% As annealed copper may vary
considerably in its state of annealing, and is always somewhat
hardened by bending and winding, it is found in practice that the
resistivity of commercial annealed copper is about 1-1/4% less than
that of hard-drawn copper. The standard now accepted for such copper,
on the recommendation of the 1899 Committee, is a wire of pure
annealed copper one metre long, weighing one gramme, whose resistance
at 0 deg. C. is 0.1421 international ohms, or at 60 deg. F., 0.150822
international ohms. The specific gravity of copper varies from about
8.89 to 8.95, and the standard value accepted for high conductivity
commercial copper is 8.912, corresponding to a weight of 555 lb per
cubic foot at 60 deg. F. Hence the volume-resistivity of pure annealed
copper at 0 deg. C. is 1.594 microhms per c.c., or 1594 C.G.S. units,
and that of pure hard-drawn copper at 0 deg. C. is 1.626 microhms per
c.c., or 1626 C.G.S. units. Since Matthiessen's researches, the most
careful scientific investigation on the conductivity of copper is that
of T. C. Fitzpatrick, carried out in 1890. (_Brit. Assoc. Report_,
1890, Appendix 3, p. 120.) Fitzpatrick confirmed Matthiessen's chief
result, and obtained values for the resistivity of hard-drawn copper
which, when corrected for temperature variation, are in entire
agreement with those of Matthiessen at the same temperature.

The volume resistivity of alloys is, generally speaking, much higher than that of pure metals. Table V. shows the volume resistivity at 0 deg. C. of a number of well-known alloys, with their chemical composition.

TABLE V.--_Volume-Resistivity of Alloys of known Composition at 0 deg.
C. in C.G.S. Units per Centimetre-cube. Mean Temperature Coefficients
taken at 15 deg. C._ (Fleming and Dewar.)

+-----------------------+-----------+-------------+-------------------+
| |Resistivity| Temperature |Composition in per |
| Alloys. | at 0 | Coefficient | cents. |
| | deg. C. |at 15 deg. C.| |
+-----------------------+-----------+-------------+-------------------+
|Platinum-silver | 31,582 | .000243 |Pt 33%, Ag 66% |
|Platinum-iridium | 30,896 | .000822 |Pt 80%, Ir 20% |
|Platinum-rhodium | 21,142 | .00143 |Pt 90%, Rd 10% |
|Gold-silver | 6,280 | .00124 |Au 90%, Ag 10% |
|Manganese-steel | 67,148 | .00127 |Mn 12%, Fe 78% |
|Nickel-steel | 29,452 | .00201 |Ni 4.35%, remaining|
| | | | percentage |
| | | | chiefly iron, but|
| | | | uncertain |
|German silver | 29,982 | .000273 |Cu5Zn3Ni2 |
|Platinoid[9] | 41,731 | .00031 | |
|Manganin | 46,678 | .0000 |Cu 84%, Mn 12%, |
| | | |Ni 4% |
|Aluminium-silver | 4,641 | .00238 |Al 94%, Ag 6% |
|Aluminium-copper | 2,904 | .00381 |Al 94%, Cu 6% |
|Copper-aluminium | 8,847 | .000897 |Cu 97%, Al 3% |
|Copper-nickel-aluminium| 14,912 | .000643 |Cu 87%, Ni 6.5%, |
| | | | Al 6.5% |
|Titanium-aluminium | 3,887 | .00290 | |
+-----------------------+-----------+-------------+-------------------+

Generally speaking, an alloy having high resistivity has poor
mechanical qualities, that is to say, its tensile strength and
ductility are small. It is possible to form alloys having a
resistivity as high as 100 microhms per cubic centimetre; but, on the
other hand, the value of an alloy for electro-technical purposes is
judged not merely by its resistivity, but also by the degree to which
its resistivity varies with temperature, and by its capability of
being easily drawn into fine wire of not very small tensile strength.
Some pure metals when alloyed with a small proportion of another metal
do not suffer much change in resistivity, but in other cases the
resultant alloy has a much higher resistivity. Thus an alloy of pure
copper with 3% of aluminium has a resistivity about 5-1/2 times that of
copper; but if pure aluminium is alloyed with 6% of copper, the
resistivity of the product is not more than 20% greater than that of
pure aluminium. The presence of a very small proportion of a
non-metallic element in a metallic mass, such as oxygen, sulphur or
phosphorus, has a very great effect in increasing the resistivity.
Certain metallic elements also have the same power; thus platinoid has
a resistivity 30% greater than German silver, though it differs from
it merely in containing a trace of tungsten.

The resistivity of non-metallic conductors is in all cases higher than that of any pure metal. The resistivity of carbon, for instance, in the forms of charcoal or carbonized organic material and graphite, varies from 600 to 6000 microhms per cubic centimetre, as shown in Table VI.:--

TABLE VI.--_Electric Volume-Resistivity in Microhms per
Centimetre-cube of Various Forms of Carbon at 15 deg. C._

+-----------------------------------------+---------------+
| Substance. | Resistivity. |
+-----------------------------------------+---------------+
| Arc lamp carbon rod | 8000 |
| Jablochkoff candle carbon | 4000 |
| Carre carbon | 3400 |
| Carbonized bamboo | 6000 |
| Carbonized parchmentized thread | 4000 to 5000 |
| Ordinary carbon filament from glow-lamp | |
| "treated" or flashed | 2400 to 2500 |
| Deposited or secondary carbon | 600 to 900 |
| Graphite | 400 to 500 |
+-----------------------------------------+---------------+

The resistivity of liquids is, generally speaking, much higher than that of any metals, metallic alloys or non-metallic conductors. Thus fused lead chloride, one of the best conducting liquids, has a resistivity in its fused condition of 0.376 ohm per centimetre-cube, or 376,000 microhms per centimetre-cube, whereas that of metallic alloys only in few cases exceeds 100 microhms per centimetre-cube. The resistivity of solutions of metallic salts also varies very largely with the proportion of the diluent or solvent, and in some instances, as in the aqueous solutions of mineral acids; there is a maximum conductivity corresponding to a certain dilution. The resistivity of many liquids, such as alcohol, ether, benzene and pure water, is so high, in other words, their conductivity is so small, that they are practically insulators, and the resistivity can only be appropriately expressed in megohms per centimetre-cube.

In Table VII. are given the names of a few of these badly-conducting liquids, with the values of their volume-resistivity in megohms per centimetre-cube:--

TABLE VII.--_Electric Volume-Resistivity of Various Badly-Conducting
Liquids in Megohms per Centimetre-cube._

+---------------------------+----------------+---------------------+
| | Resistivity | |
| Substance. | in Megohms | Observer. |
| | per c.c. | |
+---------------------------+----------------+---------------------+
| Ethyl alcohol | 0.5 | Pfeiffer. |
| Ethyl ether | 1.175 to 3.760 | W. Kohlrausch. |
| Benzene | 4.700 | |
| Absolutely pure water | 25.0 at 18 deg.| Value estimated |
| approximates probably to | C. | by F. Kohlrausch |
| | | and A. Heydweiler. |
| All very dilute aqueous | 1.00 at 18 deg.| From results by |
| salt solutions having a | C. | F. Kohlrausch |
| concentration of about | | and others. |
| 0.00001 of an equivalent | | |
| gramme molecule[10] per | | |
| litre approximate to | | |
+---------------------------+----------------+---------------------+

The resistivity of all those substances which are generally called dielectrics or insulators is also so high that it can only be appropriately expressed in millions of megohms per centimetre-cube, or in megohms per quadrant-cube, the quadrant being a cube the side of which is 10^9 cms. (see Table VIII.).

Effects of Heat.--Temperature affects the resistivity of these different classes of conductors in different ways. In all cases, so far as is yet known, the resistivity of a pure metal is increased if its temperature is raised, and decreased if the temperature is lowered, so that if it could be brought to the absolute zero of temperature (-273 deg. C.) its resistivity would be reduced to a very small fraction of its resistance at ordinary temperatures. With metallic alloys, however, rise of temperature does not always increase resistivity: it sometimes diminishes it, so that many alloys are known which have a maximum resistivity corresponding to a certain temperature, and at or near this point they vary very little in resistance with temperature. Such alloys have, therefore, a negative temperature-variation of resistance at and above fixed temperatures. Prominent amongst these metallic compounds are alloys of iron, manganese, nickel and copper, some of which were discovered by Edward Weston, in the United States. One well-known alloy of copper, manganese and nickel, now called manganin, which was brought to the notice of electricians by the careful investigations made at the Berlin Physikalisch-Technische Reichsanstalt, is characterized by having a zero temperature coefficient at or about a certain temperature in the neighbourhood of 15 deg. C. Hence within a certain range of temperature on either side of this critical value the resistivity of manganin is hardly affected at all by temperature. Similar alloys can be produced from copper and ferro-manganese. An alloy formed of 80% copper and 20% manganese in an annealed condition has a nearly zero temperature-variation of resistance between 20 deg. C. and 100 deg. C. In the case of non-metals the action of temperature is generally to diminish the resistivity as temperature rises, though this is not universally so. The interesting observation has been recorded by J. W. Howell, that "treated" carbon filaments and graphite are substances which have a minimum resistance corresponding to a certain temperature approaching red heat (_Electrician_, vol. xxxviii. p. 835). At and beyond this temperature increased heating appears to increase their resistivity; this phenomenon may, however, be accompanied by a molecular change and not be a true temperature variation. In the case of dielectric conductors and of electrolytes, the action of rising temperature is to reduce resistivity. Many of the so-called insulators, such as mica, ebonite, indiarubber, and the insulating oils, paraffin, &c., decrease in resistivity with great rapidity as the temperature rises. With guttapercha a rise in temperature from 0 deg. C. to 24 deg. C. is sufficient to reduce the resistivity of one-twentieth part of its value at 0 deg. C., and the resistivity of flint glass at 140 deg. C. is only one-hundredth of what it is at 60 deg. C.

TABLE VIII.--_Electric Volume-Resistivity of Dielectrics reckoned in
Millions of Megohms (Mega-megohms) per Centimetre-cube, and in Megohms
per Quadrant-cube, i.e. a Cube whose Side is 10^9 cms._

+------------------------+---------------------------+----------+
| | Resistivity. | |
| +------------+--------------+ Tempera- |
| Substance. |Mega-megohms| Megohms per | tura |
| | per c.c. |Quadrant-cube.| Cent. |
+------------------------+------------+--------------+----------+
|Bohemian glass | 61 | .061 | 60 deg. |
|Mica | 84 | .084 | 20 deg. |
|Gutta-percha | 450 | .45 | 24 deg. |
|Flint glass | 1,020 | 1.02 | 60 deg. |
|Glover's vulcanized | | | |
| indiarubber | 1,630 | 1.63 | 15 deg. |
|Siemens' ordinary pure | | | |
| vulcanized indiarubber | 2,280 | 2.28 | 15 deg. |
|Shellac | 9,000 | 9.0 | 28 deg. |
|Indiarubber | 10,900 | 10.9 | 24 deg. |
|Siemens' high-insulating| | | |
| fibrous material | 11,900 | 11.9 | 15 deg. |
|Siemens' special | | | |
| high-insulating | | | |
| indiarubber. | 16,170 | 16.17 | 15 deg. |
|Flint glass | 20,000 | 20.0 | 20 deg. |
|Ebonite | 28,000 | 28. | 46 deg. |
|Paraffin | 34,000 | 34. | 46 deg. |
+------------------------+------------+--------------+----------+

A definition may here be given of the meaning of the term _Temperature
Coefficient_. If, in the first place, we suppose that the resistivity
([rho]t) at any temperature (t) is a simple linear function of the
resistivity ([rho]0) at 0 deg. C., then we can write [rho]t = [rho]0(1
+ [alpha]t), or [alpha] = ([rho]t - [rho]0)/[rho]0t.

The quantity [alpha] is then called the temperature-coefficient, and
its reciprocal is the temperature at which the resistivity would
become zero. By an extension of this notion we can call the quantity
d[rho]/[rho]dt the temperature coefficient corresponding to any
temperature t at which the resistivity is [rho]. In all cases the
relation between the resistivity of a substance and the temperature is
best set out in the form of a curve called a temperature-resistance
curve. If a series of such curves are drawn for various pure metals,
temperature being taken as abscissa and resistance as ordinate, and if
the temperature range extends from the absolute zero of temperature
upwards, then it is found that these temperature-resistance lines are
curved lines having their convexity either upwards or downwards. In
other words, the second differential coefficient of resistance with
respect to temperature is either a positive or negative quantity. An
extensive series of observations concerning the form of the
resistivity curves for various pure metals over a range of temperature
extending from -200 deg. C. to +200 deg. C. was carried out in 1892
and 1893 by Fleming and Dewar (_Phil. Mag._ Oct. 1892 and Sept. 1893).
The resistance observations were taken with resistance coils
constructed with wires of various metals obtained in a state of great
chemical purity. The lengths and mean diameters of the wires were
carefully measured, and their resistance was then taken at certain
known temperatures obtained by immersing the coils in boiling aniline,
boiling water, melting ice, melting carbonic acid in ether, and
boiling liquid oxygen, the temperatures thus given being +184 deg..5
C., +100 deg. C., 0 deg. C., -78 deg..2 C. and -182 deg..5 C. The
resistivities of the various metals were then calculated and set out
in terms of the temperature. From these data a chart was prepared
showing the temperature-resistance curves of these metals throughout a
range of 400 degrees. The exact form of these curves through the
region of temperature lying between -200 deg. C. and -273 deg. C. is
not yet known. As shown on the chart, the curves evidently do not
converge to precisely the same point. It is, however, much less
probable that the resistance of any metal should vanish at a
temperature above the absolute zero than at the absolute zero itself,
and the precise path of these curves at their lower ends cannot be
delineated until means are found for fixing independently the
temperature of some regions in which the resistance of metallic wires
can be measured. Sir J. Dewar subsequently showed that for certain
pure metals it is clear that the resistance would not vanish at the
absolute zero but would be reduced to a finite but small value (see
"Electric Resistance Thermometry at the Temperature of Boiling
Hydrogen," _Proc. Roy. Soc._ 1904, 73, p. 244).

The resistivity curves of the magnetic metals are also remarkable for
the change of curvature they exhibit at the magnetic critical
temperature. Thus J. Hopkinson and D. K. Morris (_Phil. Mag._
September 1897, p. 213) observed the remarkable alteration that takes
place in the iron resistance temperature curve in the neighbourhood of
780 deg. C. At that temperature the direction of the curvature of the
curve changes so that it becomes convex upwards instead of convex
downwards, and in addition the value of the temperature coefficient
undergoes a great reduction. The mean temperature coefficient of iron
in the neighbourhood of 0 deg. C. is 0.0057; at 765 deg. C. it rises
to a maximum value 0.0204; but at 1000 deg. C. it falls again to a
lower value, 0.00244. A similar rise to a maximum value and subsequent
fall are also noted in the case of the specific heat of iron. The
changes in the curvature of the resistivity curves are undoubtedly
connected with the molecular changes that occur in the magnetic metals
at their critical temperatures.

A fact of considerable interest in connexion with resistivity is the
influence exerted by a strong magnetic field in the case of some
metals, notably bismuth. It was discovered by A. Righi and confirmed
by S. A. Leduc (_Journ. de Phys._ 1886, 5, p. 116, and 1887, 6, p.
189) that if a pure bismuth wire is placed in a magnetic field
transversely to the direction of the magnetic field, its resistance is
considerably increased. This increase is greatly affected by the
temperature of the metal (Dewar and Fleming, _Proc. Roy. Soc._ 1897,
60, p. 427). The temperature coefficient of pure copper is an
important constant, and its value as determined by Messrs Clark, Forde
and Taylor in terms of Fahrenheit temperature is

[rho]t = [rho]32 {1 + 0.0023708(t - 32) + 0.0000034548(t - 32)^2}.

_Time Effects._--In the case of dielectric conductors, commonly called insulators, such as indiarubber, guttapercha, glass and mica, the electric resistivity is not only a function of the temperature but also of the time during which the electromotive force employed to measure it is imposed. Thus if an indiarubber-covered cable is immersed in water and the resistance of the dielectric between the copper conductor and the water measured by ascertaining the current which can be caused to flow through it by an electromotive force, this current is found to vary very rapidly with the time during which the electromotive force is applied. Apart from the small initial effect due to the electrostatic capacity of the cable, the application of an electromotive force to the dielectric produces a current through it which rapidly falls in value, as if the electric resistance of the dielectric were increasing. The current, however, does not fall continuously but tends to a limiting value, and it appears that if the electromotive force is kept applied to the cable for a prolonged time, a small and nearly constant current will ultimately be found flowing through it. It is customary in electro-technical work to consider the resistivity of the dielectric as the value it has after the electromotive force has been applied for one minute, the standard temperature being 75 deg. F. This, however, is a purely conventional proceeding, and the number so obtained does not necessarily represent the true or ohmic resistance of the dielectric. If the electromotive force is increased, in the case of a large number of ordinary dielectrics the apparent resistance at the end of one minute's electrification decreases as the electromotive force increases.

_Practical Standards._--The practical measurement of resistivity involves many processes and instruments (see WHEATSTONE'S BRIDGE and OHMMETER). Broadly speaking, the processes are divided into _Comparison Methods_ and _Absolute Methods_. In the former a comparison is effected between the resistance of a material in a known form and some standard resistance. In the _Absolute Methods_ the resistivity is determined without reference to any other substance, but with reference only to the fundamental standards of length, mass and time. Immense labour has been expended in investigations concerned with the production of a standard of resistance and its evaluation in absolute measure. In some cases the absolute standard is constructed by filling a carefully-calibrated tube of glass with mercury, in order to realize in a material form the official definition of the ohm; in this manner most of the principal national physical laboratories have been provided with standard mercury ohms. (For a full description of the standard mercury ohm of the Berlin Physikalisch-Technische Reichsanstalt, see the _Electrician_, xxxvii. 569.) For practical purposes it is more convenient to employ a standard of resistance made of wire.

Opinion is not yet perfectly settled on the question whether a wire
made of any alloy can be considered to be a perfectly unalterable
standard of resistance, but experience has shown that a platinum
silver alloy (66% silver, 33% platinum), and also the alloy called
manganin, seem to possess the qualities of permanence essential for a
wire-resistance standard. A comparison made in 1892 and 1894 of all
the manganin wire copies of the ohm made at the Reichsanstalt in
Berlin, showed that these standards had remained constant for two
years to within one or two parts in 100,000. It appears, however, that
in order that manganin may remain constant in resistivity when used in
the manufacture of a resistance coil, it is necessary that the alloy
should be _aged_ by heating it to a temperature of 140 deg. C. for ten
hours; and to prevent subsequent changes in resistivity, solders
containing zinc must be avoided, and a silver solder containing 75% of
silver employed in soldering the manganin wire to its connexions.

The authorities of the Berlin Reichsanstalt have devoted considerable attention to the question of the best form for a wire standard of electric resistance. In that now adopted the resistance wire is carefully insulated and wound on a brass cylinder, being doubled on itself to annul inductance as much as possible. In the coil two wires are wound on in parallel, one being much finer than the other, and the final adjustment of the coil to an exact value is made by shortening the finer of the two. A standard of resistance for use in a laboratory now generally consists of a wire of manganin or platinum-silver carefully insulated and enclosed in a brass case. Thick copper rods are connected to the terminals of the wire in the interior of the case, and brought to the outside, being carefully insulated at the same time from one another and from the case. The coil so constructed can be placed under water or paraffin oil, the temperature of which can be exactly observed during the process of taking a resistance measurement. Equalization of the temperature of the surrounding medium is effected by the employment of a stirrer, worked by hand or by a small electric motor. The construction of a standard of electrical resistance consisting of mercury in a glass tube is an operation requiring considerable precautions, and only to be undertaken by those experienced in the matter. Opinions are divided on the question whether greater permanence in resistance can be secured by mercury-in-glass standards of resistance or by wire standards, but the latter are at least more portable and less fragile.

A full description of the construction of a standard wire-resistance
coil on the plan adopted by the Berlin Physikalisch-Technische
Reichsanstalt is given in the Report of the British Association
Committee on Electrical Standards, presented at the Edinburgh Meeting
in 1892. For the design and construction of standards of electric
resistances adapted for employment in the comparison and measurement
of very low or very high resistances, the reader may be referred to
standard treatises on electric measurements.

BIBLIOGRAPHY.--See also J. A. Fleming, _A Handbook for the Electrical
Laboratory and Testing Room_, vol. i. (London, 1901); _Reports of the
British Association Committee on Electrical Standards_, edited by
Fleeming Jenkin (London, 1873); A. Matthiessen and C. Vogt, "On the
Influence of Temperature on the Conducting Power of Alloys," _Phil.
Trans._, 1864, 154, p. 167, and _Phil. Mag._, 1865, 29, p. 363; A.
Matthiessen and M. Holtzmann, "On the Effect of the Presence of Metals
and Metalloids upon the Electric Conducting Power of Pure Copper,"
_Phil. Trans._, 1860, 150, p. 85; T. C. Fitzpatrick, "On the Specific
Resistance of Copper," _Brit. Assoc. Report_, 1890, p. 120, or
_Electrician_, 1890, 25, p. 608; R. Appleyard, _The Conductometer and
Electrical Conductivity_; Clark, Forde and Taylor, _Temperature
Coefficients of Copper_ (London, 1901). (J. A. F.)