# 1911 Encyclopædia Britannica/Units, Physical

UNITS, PHYSICAL. In order that our acquaintance with any part of nature may become exact we must have not merely a qualitative but a quantitative knowledge of facts. Hence the moment that any branch of science begins to develop to any extent, attempts are made to measure and evaluate the quantities and effects found to exist. To do this we have to select for each measurable magnitude a unit or standard of reference (Latin, unitas, unity), by comparison with which amounts of other like quantities may be numerically defined. There is nothing to prevent us from selecting these fundamental quantities, in terms of which other like quantities are to be expressed, in a perfectly arbitrary and independent manner, and as a matter of fact this is what is generally done in the early stages of every science. We may, for instance, .choose a certain length, a certain volume, a certain mass, a certain force or power as our units of length, volume, mass, force or power, which have no simple or direct relation to each other. Similarly we may select for more special measurements any arbitrary electric current, electromotive force, or resistance, and call them our units. The progress of knowledge, however, is greatly assisted if all the measurable quantities are brought into relation with each other by so selecting the units that they are related in the most simple manner, each to the other and to one common set of measurable magnitudes called the fundamental quantities.

The progress of this co-ordination of units has been greatly aided by the discovery that forms of physical energy can be converted into one another, and that the conversion is by definite rule and amount (see Energy). Thus the mechanical energy associated with moving masses can be converted into heat, hence heat can be measured in mechanical energy units. The amount of heat required to raise one gramme of water through 1° C. in the neighbourhood of 10° C. is equal to forty-two million ergs, the erg being the kinetic energy or energy of motion associated with a mass of 2 grammes when moving uniformly, without rotation, with a velocity of 1 cm. per second. This number is commonly called the “mechanical equivalent of heat,” but would be more exactly described as the “mechanical equivalent of the specific heat of water at 10° C.” Again, the fact that the maintenance of an electric current requires energy, and that when produced its energy can be wholly utilized in heating a mass of water, enables us to make a similar statement about the energy required to maintain a current of one ampere through a resistance of one ohm for one second, and to define it by its equivalent in the energy of a moving mass. Physical units have therefore been selected with the object of establishing simple relations between each of them and the fundamental mechanical units. Measurements based on such relations are called absolute measurements. The science of dynamics, as far as that part of it is concerned which deals with the motion and energy of material substances, starts from certain primary definitions concerning the measurable quantities involved. In constructing a system of physical units, the first thing to consider is the manner in which we shall connect the various items. What, for instance, shall be the unit of force, and how shall it be determined by simple reference to the units of mass, length and time?

The modern absolute system of physical measurement is founded upon dynamical notions, and originated with C. F. Gauss. We are for the most part concerned in studying motions in nature; and even when we find bodies at rest in equilibrium it is because the causes of motion are balanced rather than absent. Moreover, the postulate which lies at the base of all present-day study of physics is that in the ultimate issue we must seek for a mechanical explanation of the facts of nature if we are to reach any explanation intelligible to the human mind. Accordingly the root of all science is the knowledge of the laws of motion, and the enunciation of these laws by Newton laid the foundation of a more exact knowledge of nature than had been possible before. Our fundamental scientific notions are those of length, time, and mass. No metaphysical discussion has been able to resolve these ideas into anything simpler or to derive them from each other. Hence in selecting units for physical measurements we have first to choose units for the above three quantities.

Fundamental Units.—Two systems of fundamental units are in common use: the British system, having the yard and pound as the standard units of length and mass, frequently termed the “foot-pound-second” (F.P.S.) system; and the “centimetre-gramme-second” system (C.G.S.), having the centimetre and gramme as standard units of length and mass, termed the “metric” system. The fundamental unit of time is the same in both systems, namely, the “mean solar second,” 86,400 of which make 1 solar day (see Time). Since these systems and the corresponding standards, together with their factors of conversion, are treated in detail in the article Weights and Measures, we need only deal here with such units as receive special scientific use, i.e. other than in ordinary commercial practice. The choice of a unit in which to express any quantity is determined by the magnitude and proportional error of the measurement. In astronomy, where immense distances have to be very frequently expressed, a common unit is the mean radius of the earth's orbit, the “astronomical unit” of length, i.e. 92,900,000 miles. But while this unit serves well for the region of our solar system, its use involves unwieldy numerical coefficients when stellar distances are to be expressed. Astronomers have therefore adopted a unit of length termed the “light year,” which is the distance traversed by light in a year; this unit is 63,000 times the mean radius of the earth's orbit. The relative merits of these units as terms in which astronomical distances may be expressed is exhibited by the values of the distance of the star α. Centauri from our earth, namely, 25,000,000,000,000 miles = 275,000 astronomical units = 4·35 light years.

As another example of a physical unit chosen as a matter of convenience, we may refer to the magnitudes of the wave-lengths of light. These quantities are extremely small, and admit of correct determination to about one part in ten-thousand, and range, in the visible spectrum, from about 6 to 4 ten-millionths of a metre. Since their values are determined to four significant figures, it is desirable to choose a unit which represents the value as an integer number; the unit is therefore a ten-thousand millionth of a metre, termed a “tenth metre,” since it is 10−10 metres. Sometimes the, thousand-millionth of a metre, the “micro millimetre,” denoted by μμ, serves as a unit for wave lengths. Another relatively minute unit is the “micron,” denoted by μ, and equal to one-millionth of a metre; it is especially used by bacteriologists.

Units in Mechanics.—The quantities to be measured in mechanics (q.v.) are velocity and acceleration, dependent on the units of length and time only, momentum, force, energy or work and power, dependent on the three fundamental' units. The unit of velocity in the British system is 1 foot, 1 yard, or 1 mile per second; or the time to which the distance is referred may be expressed in hours, days, &c., the choice depending upon the actual magnitude of the velocity or on custom. Thus the muzzle velocity of a rifle or cannon shot is expressed in feet per second, whereas the speed of a train is usually expressed in miles per hour. Similarly, the unit on the metric system isr metre, or any decimal multiple thereof, per second, per hour, &c. Since acceleration is the rate of increase of velocity per unit time, it is obvious that the unit of acceleration depends solely upon the units chosen to express unit velocity; thus if the unit of velocity be one foot per second, the unit of acceleration is one foot per second per second, if one metre per second the unit is one metre per second per second, and similarly for other units of velocity. Momentum is defined as the product of mass into velocity; unit momentum is therefore the momentum of unit mass into unit velocity; in the British system the unit of mass may be the pound, ton, &c., and the unit of velocity any of those mentioned above; and in the metric system, the gramme, kilogramme, &c., may be the unit of mass, while the metre per second, or any other metric unit of velocity, is the remaining term of the product.

Force, being measured .by the change of momentum in unit time, is expressed in terms of the same units in which unit momentum is defined. The common British unit is the “poundal,” the force which in one second retards or accelerates the velocity of a mass of one pound by one foot per second. The metric (and scientific) unit, named the “dyne,” is derived from the centimetre, gramme, and second. The poundal and dyne are related as follows:—1 poundal= 13,825·5 dynes.

A common unit of force, especially among engineers, is the “weight of one pound,” by which is meant the force equivalent to the gravitational attraction of the earth on a mass of one pound. This unit obviously depends on gravity; and since this varies with the latitude and height of the place of observation (see Earth, Figure of), the “force of one pound” of the engineer is not constant. Roughly, it equals 32·17 poundals or 980 dynes. The most frequent uses of this engineer's unit are to be found in the expressions for pressure, especially in the boilers and cylinders of steam engines, and in structures, such as bridges, foundations of buildings, &c. The expression takes the form: pounds per square foot or inch, meaning a force equivalent to so many pounds' weight distributed over a square foot or inch, as the case may be. Other units of pressure (and therefore special units of force) are the “atmosphere” (abbreviated “atmo”), the force exerted on unit area by the column of air vertically above it; the “millimetre or centimetre of mercury,” the usual scientific units, the force exerted on unit area by a column of mercury one millimetre or centimetre high; and the “foot of water,” the column being one foot of Water. All these units admit of ready conversion:—1 atmo= 760 mm. mercury= 32 feet of water= 1,013,600 dynes.

Energy of work is measured by force acting over a distance. The scientific unit is the “erg,” which is the energy expended when a force of one dyne acts over one centimetre. This unit is too small for measuring the quantity of energy associated, for instance, with engines; for such purposes a unit ten-million times as great, termed the “joule,” is used. The British absolute unit is the “poundal-foot.” As we noticed in the case of units of force, common-life experience has led to the introduction of units dependent on gravitation, and therefore not invariable: the common British practical unit of this class is the “foot pound”; in the metric system its congener is the “kilogramme metre.”

Power is the rate at which force does work; it is therefore expressed by “units of energy per second.” The metric unit in use is the “watt,” being the rate equal to one joule per second. Larger units in practical use are: “kilowatt, equal to 1000 watts; the corresponding energy unit being the kilowatt-second, and 3600 kilowatt-seconds or 1 kilowatt-hour called a “Board of Trade unit” or a “kelvin.” This last is a unit of energy, not power. In British engineering practice the common unit of power is the “horse-power” (HP ), which equals 550 foot-pounds performed per second, or 33,000 foot-pounds per minute; its equivalent in the metric system is about 746 watts, the ratio varying, however, with gravity.

Units of Heat.—In studying the phenomena of heat, two measurable quantities immediately present themselves:—(1) temperature or thermal potential, and (2) quantity of heat. Three arbitrary scales are in use for measuring temperature (see Thermometry), and each of these scales affords units suitable for the expression of temperature. On the Centigrade scale the unit, termed a “Centigrade degree,” is one-hundredth of the interval between the temperature of water boiling under normal barometric pressure (760 mm. of mercury) and that of melting ice; the “Fahrenheit degree” is one-hundred and-eightieth, and the “Réaumur degree” is one-eightieth of the same difference. In addition to these scales there is the “thermo-dynamic scale,” which, being based on dynamical reasoning, admits of correlation with the fundamental units. This subject is discussed in the articles Thermodynamics and Thermometry.

Empirical units of “quantity of heat” readily suggest themselves as the amount of heat necessary to heat a unit mass of any substance through unit temperature. In the metric system the unit, termed a “calorie,” is the quantity of heat required to raise a gramme of water through one degree Centigrade. This quantity, however, is not constant, since the specific heat of water varies with temperature (see Calorimetry). In defining the calorie, therefore, the particular temperatures must be specified; consequently there are several calories particularized by special designations:—(1) conventional or common gramme calorie, the heat required to raise 1 gramme of water between 150° C. and 17° C. through 1° C.; (2) “mean or average gramme calorie,” one-hundredth of the total heat required to raise the temperature of 1 gramme of water from 0° C. to 100° C.; (3) “zero gramme calorie,” the heat required to raise 1 gramme of water from 0° C. to 1° C. These units are thus related:—1 common calorie= 1·987 mean calories=0·992 zero calories. A unit in common use in thermo-chemistry is the major calorie, which refers to one kilogramme of water and 1° C. In the British system the common unit, termed the “British Thermal Unit” (B.Th.U.), is the amount of heat required to raise one pound of water through one degree Fahrenheit.

A correlation of these units of quantity of heat with the fundamental units of mass, length and time attended the recognition of the fact that heat was a form of energy; and their quantitative relationships followed from the experimental determinations of the so-called “mechanical equivalent of heat,” i.e. the amount of mechanical energy, expressed in ergs, joules, or foot-pounds, equivalent to a certain quantity of heat (cf. Calorimetry). These results show that a gram-calorie is equivalent to about 4·2 joules, and a British thermal unit to 780 foot-pounds.

Electrical Units.—The next most important units are the electrical units. We are principally concerned in electrical work with three quantities called respectively, electric current, electromotive force, and resistance. These are related to one another by Ohm’s law, which states that the electric current in a circuit is directly as the electromotive force and inversely as the resistance, when the current is unvarying and the temperature of the circuit constant. Hence if we choose units for two of these quantities, the above law defines the unit for the third. Much discussion has taken place over this question. The choice is decided by the nature of the quantities themselves. Since resistance is a permanent quality of a substance, it is possible to select a certain piece of wire or tube full of mercury, and declare that its resistance shall be the unit of resistance, and if the substance is permanent we shall possess an unalterable standard or unit of resistance. For these reasons the practical unit of resistance, now called the international ohm, has been selected as one of the above three electrical units.

It has now been decided that the second unit shall be the unit of electric current. As an electric current is not a thing, but a process, the unit current can only be reproduced when desired. There are two available methods for creating a standard or unit electric current. If an unvarying current is passed through a neutral solution of silver nitrate it decomposes or electrolysis it and deposits silver upon the negative pole or cathode of the electrolytic cell. According to Faraday’s law and all subsequent experience, the same current deposits in the same time the same mass of silver. Hence we may define the unit current by the mass of silver it can liberate per second. Again, an electric current in one circuit exerts mechanical force upon a magnetic pole or a current in another circuit suitably placed, and we may measure the force and define by it a unit electric current. Both these methods have been used. Thirdly, the unit of electromotive force may be defined as equal to the difference of potential between the ends of the unit of resistance when the unit of current flows in it.

Apart, however, from the relation of these electrical units to each other, it has been found to be of great importance to establish a simple relation between the latter and the absolute mechanical units. Thus an electric current which is passed through a conductor dissipates its energy as Absolute electrical units. heat, and hence creates a certain quantity of heat per unit of time. Having chosen our units of energy and related unit of quantity of heat, we must so choose the unit of current that when passed through the unit of resistance it shall dissipate 1 unit of energy in 1 unit of time.

A further consideration has weight in selecting the size of the units, namely, that they must be of convenient magnitude for the ordinary measurements. The founders of the modern system of practical electrical units were a committee appointed by the British Association in British Association units 1861, at the suggestion of Lord Kelvin, which made its first report in 1862 at Cambridge (see B. A. Report). The five subsequent reports containing the results of the committee’s work, together with a large amount of most valuable matter on the subject of electric units, were collected in a volume edited by Prof. Fleeming Jenkin in 1873, entitled Reports of the Committee on Electrical Standards. This committee has continued to sit and report annually to the British Association since that date. In their second report in 1863 (see B.A. Report, Newcastle-on-Tyne) the committee recommended the adoption of the absolute system of electric and magnetic units on the basis originally proposed by Gauss and Weber, namely, that these units should be derived from the fundamental dynamical units, but assuming the units of length, mass and time to be the metre, gramme and second instead of the millimetre milligramme and second as proposed by Weber. Considerable differences of opinion existed as to the choice of the fundamental units, but ultimately a suggestion of Lord Kelvin’s was adopted to select the centimetre, gramme, and second, and to construct a system of electrical units (called the C.G.S. system) derived from the above fundamental units. On this system the unit of force is the dyne and the unit of work the erg. The dyne is the uniform force which when acting on a mass of 1 gramme for 1 second gives it a velocity of 1 centimetre per second. The erg is the work done by 1 dyne when acting through a distance of 1 centimetre in its own direction. The electric and magnetic units were then derived, as previously suggested by Weber, in the following manner: If we consider two very small spheres placed with centres 1 centimetre apart in air and charged with equal quantities of electricity, then if the force between these bodies is 1 dyne each sphere is said to be charged with 1 unit of electric quantity on the electrostatic system. Again, if we consider two isolated magnetic poles of equal strength and consider them placed 1 centimetre apart in air, then if the force between them is 1 dyne these poles are said to have a strength of 1 unit on the electromagnetic system. Unfortunately the Committee did not take into account the fact that in the first case the force between the electric charges depends upon and varies inversely as the dielectric constant of the medium in which the experiment is made, and in the second case it depends upon the magnetic permeability of the medium in which the magnetic poles exist. To put it in other words, they assume that the dielectric constant of the cir cum ambient medium was unity in the first case, and that the permeability was also unity in the second case.

The result of this choice was that two systems of measurement were created, one depending upon the unit of electric quantity so chosen, called the electrostatic system, and the other depending upon the unit magnetic pole defined as above, called the electromagnetic system of C.G.S. units. Moreover, it was found that in neither of these systems were the units of very convenient magnitude. Hence, finally, the committee adopted a third system of units called the practical system, in which convenient decimal multiples or fractions of the electromagnetic units were selected and named for use. This system, moreover, is not only consistent with itself, but may be considered to be derived from a system of dynamical units in which the unit of length is the earth quadrant or 10 million metres, the unit of mass is 10−11 of a gramme and the unit of time is 1 second. The units on this system have received names derived from those of eminent discoverers. Moreover, there is a certain relation between the size of the units for the same quantity on the electrostatic (E.S.) system and that on the electromagnetic (E.M.) system, which depends upon the velocity of light in the medium in which the measurements are supposed to be made. Thus on the E.S. system the unit of electric quantity is a point charge which at a distance of 1 cm. acts on another equal charge with a force of 1 dyne. The E.S. unit of electric current is a current such that 1 E.S. unit of quantity flows per second across each section of the circuit. On the E.M. system we start with the definition that the unit magnetic pole is one which acts on another equal pole at a distance of 1 cm. with a force of 1 dyne. The unit of current on the E.M. system is a current such that if flowing in circular circuit of r cm. radius each unit of length of it will act on a unit magnetic pole at the centre with a force of 1 dyne. This E.M. unit of current is much larger than the E.S. unit defined as above. It is v times greater, where v=3×1010 is the velocity of light in air expressed in cms. per second. The reason for this can only be understood by considering the dimensions of the quantities with which we are concerned. If L, M, T denote length, mass, time, and we adopt certain sized units of each, then we may measure any derived quantity, such as velocity, acceleration, or force in terms of the derived dynamical units as already explained. Suppose, however, we alter the size of our selected units of L, M or T, we have to consider how this alters the corresponding units of velocity, acceleration, force, &c. To do this we have to consider their dimensions. If the unit of velocity is the unit of length passed over per unit of time, then it is obvious that it varies directly as the unit of length, and inversely as the unit of time. Hence we may say that the dimensions of velocity are L/T or LT−1; similarly the dimensions of acceleration are L/T2 or LT−2, and the dimensions of a force are MLT−2.

For a fuller explanation see above (Units, Dimensions of), or Everett’s Illustrations of the C.G.S. System of Units.

Accordingly on the electrostatic system the unit of electric quantity is such that f=q2/Kd2, where q is the quantity of the two equal charges, d their distance, f the mechanical force or stress between them, and K the dielectric Electrostatic and electro-magnetic units. constant of the dielectric in which they are immersed. Hence since f is of the dimensions MLT−2, q2 must be of the dimensions of KML3T−2, and q of the dimensions M12L32T−1K12. The dimensions of K, the dielectric constant, are unknown. Hence, in accordance with the suggestion of Sir A. Rücker (Phil. Mag., February 1889), we must treat it as a fundamental quantity. The dimensions of an electric current on the electrostatic system are therefore those of an electric quantity divided by a time, since by current we mean the quantity of electricity conveyed per second. Accordingly current on the E.S. system has the dimensions M12L32T−1K12.

We may obtain the dimensions of an electric current on the magnetic system by observing that if two circuits traversed by the same or equal currents are placed at a distance from each other, the mechanical force or stress between two elements of the circuit, in accordance with Ampere’s law (see Electro-kinetics), varies as the square of the current C, the product of the elements of length ds, ds′ of the circuits, inversely as the square of their distance d, and directly as the permeability, μ, of the medium in which they are immersed. Hence C2ds dsμ/d2 must be of the dimensions of a force or of the dimensions MLT−2. Now, ds and ds′ are lengths, and d is a length, hence the dimensions of electric current on the E.M. system must be M12L12T−1μ12. Accordingly the dimensions of current on the E.S. system are M12L32T−2K12, and on the E.M. system the are M12L12T−1μ12, where μ and K, the permeability and dielectric constant of the medium, are of unknown dimensions, and therefore treated as fundamental quantities.

The ratio of the dimensions of an electric current on the two systems (E.S. and E.M.) is therefore LT−1K12μ12. This ratio must be a mere numeric of no dimensions, and therefore the dimensions of √ must be those of the reciprocal of a velocity. We do not know what the dimensions of, μ and K are separately, but we do know, therefore, that their product has the dimensions of the reciprocal of the square of a velocity.

Again, we may arrive at two dimensional expressions for electromotive force or difference of potential. Electrostatic difference of potential between two places is measured by the mechanical work required to move a small conductor charged with a unit electric charge from one place to the other against the electric force. Hence if V stands for the difference of potential between the two places, and Q for the charge on the small conductor, the product QV must be of the dimensions of the work or energy, or of the force×length, or of ML2T−2. But Q on the electrostatic system of measurement is of the dimensions M12L32T−1K12; the potential difference V must be, therefore, of the dimensions M12L12T−1K12. Again, since by Ohms law and Joule’s law electromotive force multiplied by a current is equal to the power expended on a circuit, the dimensions of electromotive force, or, what is the same thing, of potential difference, in the electromagnetic system of measurement must be those of power divided by a current. Since mechanical power means rate of doing work, the dimensions of power must be ML2T−3. We have already seen that on the electromagnetic system the dimensions of a current are M12L12T−1μ12; therefore the dimensions of electromotive force or potential on the electromagnetic system must be M12L32T−2μ12. Here again we find that the ratio of the dimensions on the electrostatic system to the dimensions on the electromagnetic system is L−1TK12μ12.

In the same manner we may recover from fundamental facts and relations the dimensions of every electric and magnetic quantity on the two systems, starting in one case from electrostatic phenomena and in the other case from electromagnetic or magnetic. The electrostatic dimensional expression will always involve K, and the electromagnetic dimensional expression will always Involve μ, and in every case the dimensions in terms of K are to those in terms of μ for the same quantity in the ratio of a power of LT−1K12μ12. This therefore confirms the view that whatever may be the true dimensions in terms of fundamental units of μ and K, their product is the inverse square of a velocity.

Table I. gives the dimensions of all the principal electric and magnetic quantities on the electrostatic and electromagnetic systems.

It will be seen that in every case the ratio of the dimensions on the two systems is a power of LT−1K12μ12, or of a velocity multiplied by the square root of) the product K and μ; In other words, it is the product of a velocity multiplied by the geometric mean of K and μ. This quantity 1/√ must therefore be of the dimensions of a velocity, and the questions arise, What is the absolute value of this velocity? and, How is it to be determined? The answer is, that the value of the velocity in concrete numbers maybe obtained by measuring the magnitude of any electric quantity in two ways, one making use only of electrostatic phenomena, and the other only of electromagnetic. To take one instance:—It is easy to show that the electrostatic capacity of a sphere suspended in air or in vacuo at a great distance from other conductors is given by a number equal to its radius in centimetres. Suppose such a sphere to be charged and discharged rapidly with electricity from any source, such as a battery. It would take electricity from the source at a certain rate, and would in fact act like a resistance in permitting the passage through it or by it of a certain quantity of electricity per unit of time. If K is the capacity and n is the number of discharges per second, then nK is a quantity of the dimensions of an electric conductivity, or of the reciprocal of a resistance. If a conductor, of which the electrostatic capacity can be calculated, and which has associated with it a commutator that charges and discharges it n times per second, is arranged in one branch of a Wheatstone's Bridge, it can be treated and measured as if it were a resistance, and its equivalent resistance calculated in terms of the resistance of all the other branches of the bridge (see Phil. Mag., 1885, 20, 258).

 Quantity. Symbol. Dimensionson the Electro-static SystemE.S. Dimensionson the Electro-magneticSystem E.M. Ratio of E.S. to E.M. Magnetic permeability (μ) L−2 T2 K−1 ⁠μ L−2 T2 K−1 μ−1 Magnetic force of field (H) L12  M12 T−2 K12 L−12 M12 T−1 μ− 12 L T−1 K12 μ12 Magnetic flux density or induction (B) L32  M12 K−12 L−12 M12 T−1 μ12 L−1T  K−12 μ−12 Total magnetic flux (Z) L12  M12 K−12 L32 M12 T−1 μ12 L−1T  K−12 μ−12 Magnetization (I) L−32 M12 K−12 L−12 M12 T−1 μ12 L−1T  K−12 μ12 Magnetic pole strength (m) L12  M12 K−12 L32 M12 T−1 μ12 L−1T  K−12 μ−12 Magnetic moment (M) L32  M12 K−12 L52 M12 T−1 μ12 L−1T  K−12 μ−12 Magnetic potential or magnetomotive force (M.M.F.) L32  M12 T−2 K12 L12 M12 T−1 μ−12 L⁠T−1 K12  μ12 Specific inductive capacity (K) K L−2 T2 μ−1 L2 T−2 K  μ Electric force (e) L−12 M12 T−1 K−12 L12 M12 T−2 μ12 L−1 T K− 12 μ− 12 Electric displacement (D) L−12 M12 T−1 K12 L−32 M12 μ−12 L T−1 K12 μ12 Electric quantity (Q) L32  M12 T−1 K12 L12 M12 μ−12 L⁠T−1 K12 μ12 Electric current (A) L32  M12 T−2 K12 L12 M12 T−1 μ−12 L⁠T−1 K12 μ12 Electric potential ${\Big \}}$ ⁠(V) ${\Big \}}$ L12  M12 T−1 K−12 L32 M12 T−2 μ12 L−1 T K−12  μ−12 Electromotive force ⁠(E.M.F.) Electric resistance (R) L−1 T  K−1 L  T−1 μ L−2 T2  K−1  μ−1 Electric capacity (C) L⁠K L−1 T2 μ−1 L2  T−2 K⁠μ Self inductance ${\Big \}}$ (L) ${\Big \}}$ L−1 T2 K−1 L  μ L−2 T2  K−1  μ−1 Mutual inductance ⁠(M)

Accordingly, we have two methods of measuring the capacity of a conductor. One, the electrostatic method, depends only on the measurement of a length, which in the case of a sphere in free space is its radius; the other, the electromagnetic method, determines the capacity in terms of the quotient of a time by a resistance. The ratio of the electrostatic to the electromagnetic value of the same capacity is therefore of the dimensions of a velocity multiplied by a resistance in electromagnetic value, or of the dimensions of a velocity squared. This particular experimental measurement has been carried out carefully by many observers, and the result has been always to show that the velocity v which expresses the ratio is very nearly equal to 30 thousand million centimetres per second; v=nearly 3× 1010. The value of this important constant can be determined by experiments made to measure electric quantity, potential, resistance or capacity, both in electrostatic and in electromagnetic measure. For details of the various methods employed, the reader must be referred to standard treatises on Electricity and Magnetism, where full particulars will be found (see Maxwell, Treatise on Electricity and Magnetism, vol. ii. ch. xix. 2nd ed.; also Mascart and Joubert, Treatise on Electricity and Magnetism, vol. ii. ch. viii., Eng. trans. by Atkinson).

Table II. gives a list of some of these determinations of v, with references to the original papers.

It will be seen that all the most recent values, especially those in which a comparison of capacity has been made, approximate to 3 × 1010 centimetres per second, a value which is closely in accord with the latest and best determinations of the velocity of light.

We have in the next place to consider the question of practical electric units and the determination and construction of concrete standards. The committee of the British Association charged with the duty of arrangingPractical units a system of absolute and magnetic units settled also on a system of practical units of convenient magnitude, and gave names to them as follows:–

 109 absolute electromagnetic units of resistance = 1 ohm 108 ⁠ 〃 ⁠ 〃 units of electromotive force = 1 volt 110th of an ⁠〃⁠ 〃 unit of current = 1 ampere 110th of an ⁠〃⁠ 〃 unit of quantity = 1 coulomb 10−9 ⁠〃⁠ 〃 units of capacity = 1 farad 10−15 ⁠〃⁠ 〃 units of capacity = 1 microfarad

Since the date when the preceding terms were adopted, other multiples of absolute C.G.S. units have received practical names, thus:—

107 ergs or absolute C. G. S. units of energy = 1 joule
107 ergs per second or C.G.S. units of power = 1 watt
109 absolute units of inductance = 1 henry
108 absolute units of magnetic flux = 1 weber
1 absolute unit of magneto motive force = 1 gauss

An Electrical Congress was held in Chicago, U.S.A. in August 1893, to consider the subject of international practical electrical units, and the result of a conference between scientific representatives of Great Britain, the United States, France, Germany, Italy, Mexico, Austria, Switzerland, Sweden and British North America, after deliberation for six days, was a unanimous agreement to recommend the following resolutions as the definition of practical international units. These resolutions and definitions were confirmed at other conferences, and at the last one held in London in October 1908 were finally adopted. It was agreed to take:—

“As a unit of resistance, the International Ohm, which is based upon the ohm equal to 109 units of resistance of the C.G.S. system of electromagnetic units, and is represented by the resistance offered to an unvarying electric current by a column of mercury at the temperature of melting ice 14·4521 grammes in mass, of a constant cross-sectional area and of the length of 106·3 cm.

“As a unit of current, the International Ampere, which is one-tenth of the unit of current of the C.G.S. system of electromagnetic units, and which is represented sufficiently well for practical use by the unvarying current which, when passed through a solution of nitrate of silver in water, deposits silver at the rate of 0·00111800 of a gramme per second.

“As a unit of electromotive force, the International Volt, which is the electromotive force that, steadily applied to a conductor whose resistance is one international ohm, will produce a current of one international ampere. It is represented sufficiently well for practical purposes by 1000010184 of the E.M.F. of a normal or saturated cadmium Weston cell at 20° C., prepared in the manner described in a certain specification.

“As a unit of quantity, the International Coulomb, which is the quantity of electricity transferred by a current of one international ampere in one second.

“As the unit of capacity, the International Farad, which is the capacity of a condenser charged to a potential of one international volt by one international coulomb of electricity.

“As a unit of work, the Joule, which is equal to 107 units of work in the C.G.S. System, and which is represented sufficiently well for practical use by the energy expended in one second by an international ampere in an international ohm.

“As a unit of power, the Watt, which is equal to 107 units of power in the C.G.S. System, and which is represented sufficiently well for practical use by the work done at the rate of second.

“As the unit of inductance, the Henry, which is the induction in a circuit when an electromotive force induced in this circuit is one international volt, while the inducing current varies at the rate of one ampere per second.”

 Date. Name. Reference. ElectricQuantityMeasured. v inCentimetresper Second. 1856 W. Weber and R. Kohlrausch Electrodynamische Massbestimmungen and Pogg. Ann. xcix., August 10, 1856 Quantity 3·107 × 1010 18671868 ${\Big \}}$ Lord Kelvin⁠and W. F. King Report of British Assoc., 1869. p. 434; and Reports on Electrical Standards, F. Jenkin, p. 186 Potential 2·81  × 1010 1868 J. Clerk Maxwell Phil. Trans. Roy. Soc., 1868, p. 643 Potential 2·84  × 1010 1872 Lord Kelvin and Dugald M‘Kichan Phil. Trans. Roy. Soc., 1873, p. 409 Potential 2·89  × 1010 1878 W. E. Ayrton and J. Perry Journ. Soc. Tel. Eng. vol. viii. p. 126 Capacity 2·94  × 1010 1880 Lord Kelvin and Shida Phil. Mag., 1880, vol. 10, x. p. 431 Potential 2·995 × 1010 1881 A. G. Stoletow Soc. Franc. de Phys., 1881 Capacity 2·99  × 1010 1882 F. Exner Wien. Ber., 1882 Potential 2·92  × 1010 1883 Sir J. J. Thomson Phil. Trans. Roy. Soc., 1883, p. 707 Capacity 2·963 × 1010 1884 I. Klemencic Journ. Soc. Tel. Eng., 1887, p. 162 Capacity 3·019 × 1010 1888 F. Himstedt Electrician, March 23, 1888, vol. xx. p. 530 Capacity 3·007 × 1010 1888 Lord Kelvin, Ayrton and Perry British Association, Bath; and Electrician, Sept. 28, 1888 Potential 2·92  × 1010 1888 H. Fison Electrician, vol. xxi. p. 215; and Proc. Phys.  Soc. Lond., June 9, 1888 Capacity 2·965 × 1010 1889 Lord Kelvin Proc. Roy. Inst., 1889 Potential 3·004 × 1010 1889 H. A. Rowland Phil. Mag. 1889 Quantity 2·981 × 1010 1889 E. B. Rosa Phil. Mag., 1889 Capacity 3·000 × 1010 1890 Sir J. J. Thomson and G. F. C. Searle Phil. Trans., 1890 Capacity 2·995 × 1010 1891 M. E. Maltby Wied. Ann. 1897 Alternating Currents 3·015 × 1010

In connexion with the numerical values in the above definitions much work has been done. The electrochemical equivalent of silver or the weight in grammes deposited per second by 1 C.G.S. electromagnetic unit of current has been the subject of much research. The following determinations of it have been given by various observers:—

 Name. Value. Reference. E. E. N. Mascart 0·011156 Journ. de physique, 1884, (2), 3, 283. F. and W. Kohlrausch 0·011183 Wied. Ann., 1886, 27, 1. Lord Rayleigh and Mrs Sedgwick 0·011179 Phil. Trans. Roy. Soc., 1884, 2, 411. J. S. H. Pellat and A. Potier 0·011192 Journ. de Phys., 1890, (2), 9, 381. Karl Kahle 0·011183 Wied, Ann., 1899, 67, 1. G. W. Patterson and K. E. Guthe 0·011192 Physical Review, 1898, 7, 251. J. S. H. Pellat and S. A. Leduc 0·011195 Comptes rendus, 1903, 136, 1649.

Although some observers have urged that the 0·01119 is nearer to the true value than 0·01118, the preponderance of the evidence seems in favour of this latter number and hence the value per ampere-second is taken as 0·0011800 gramme. The exact value of the electromotive force of a Clark cell has also been the subject of much research. Two forms of cell are in use, the simple tubular form and the H-form introduced by Lord Rayleigh. The Berlin Reichsanstalt has issued a specification for a particular H-form of Clark cell, and its E.M.F. at 15° C. is taken as 1·4328 international volts. The E.M.F. of the cell set up in accordance with the British Board of Trade specification is taken as 1·434 international volts at 15° C. The detailed specifications are given in Fleming’s Handbook for the Electrical Laboratory and Testing Room (1901), vol. i. chap. 1; in the same book will be found copious references to the scientific literature of the Clark cell. One objection to the Clark cell as a concrete standard of electromotive force is its variation with temperature and with slight impurities in the mercurous sulphate used in its construction. The Clark cell is a voltaic cell made with mercury, mercurous sulphate, zinc sulphate, and zinc as elements, and its E.M.F. decreases 0·08% per degree Centigrade with rise of temperature. In 1891 Mr Weston proposed to employ cadmium and cadmium sulphate in place of zinc and zinc sulphate and found that the temperature coefficient for the cadmium cell might be made as low as 0·004 % per degree Centigrade. Its E.M.F. is, however, 1·0184 international volts at 20° C. For details of construction and the literature of the subject see Fleming’s Handbook for the Electrical Laboratory, vol. i. chap. 1.

In the British Board of Trade laboratory the ampere and the volt are not recovered by immediate reference to the electrochemical equivalent of silver or the Clark cell, but by means of instruments called a standard ampere balance and a standard 100-volt electrostatic voltmeter. In the standard ampere balance the current is determined by weighing the attraction between two coils traversed by the current, and the ampere is defined to be the current which causes a certain attraction between the coils of this standard form of ampere balance. The form of ampere balance in use at the British Board of Trade electrical standards office is described in Fleming’s Handbook for the Electrical Laboratory, vol. i., and that constructed for the British National Physical Laboratory in the report of the Committee on Electrical Standards (Brit. Assoc. Rep., 1905). This latter instrument will recover the ampere within one-thousandth part. For a further description of it and for full discussion of the present position of knowledge respecting the values of the international practical units the reader is referred to a paper by Dr F. A. Wolff read before the International Electrical Congress at St Louis Exhibition, U.S.A., in 1904, and the subsequent dispassion (see Journ. Inst. Elec. Eng. Lond., 1904–5, 34, 190, and 35, 3.

The construction of the international ohm or practical unit of resistance involves a knowledge of the specific resistance of mercury. Numerous determinations of this constant have been made. The results are expressed either in terms of the length in cm. of the column of pure mercury of 1 sq. mm. in section which at 0° C. has a resistance of 109 C.G.S. electromagnetic units, or else in terms of the weight of mercury in grammes for a column of constant cross sectional area and length of 100·3 cm. The latter method was adopted at the British Association Meeting at Edinburgh in 1892, but there is some uncertainty as to the value of the density of mercury at 0° C. which was then adopted. Hence it was proposed by Professor J. Viriamu Jones that the re determination of the ohm should be made when required by means of the Lorentz method (see J. V. Jones, “The Absolute Measurement of Electrical Resistance,” Proc. Roy. Inst. vol. 14, part iii. p. 601). For the length of the mercury column defining the ohm as above, Lord Rayleigh in 1882 found the value 106·27 cm., and R. T. Glazebrook in the same year the value 106·28 cm. by a different method, while another determination by Lord Rayleigh and Mrs Sedgwick in 1883 gave 106·22 cm. Viriamu Jones in 1891 gave the value 106·30 cm., and one by W. E. Ayrton in 1897 by the same method obtained the value 106·27 to 106·28 cm. Hence the specific resistance of mercury cannot be said to be known to 1 part in 10,000, and the absolute value of the ohm in centimetres per second is uncertain to at least that amount. (See also J. Viriamu Jones, “On a Determination of the International Ohm in Absolute Measure,” Brit. Assoc. Report, 1894.)

The above-described practical system based on the C.G.S. double system of theoretical units labours under several very great disadvantages. The practical system is derived from and connected with an abnormally large unit of length (the earth quadrant) and an absurdly small unit of mass. Also in consequence of the manner in Rational system of electrical units. which the unit electric quantity and magnetic pole strength are defined, a coefficient, 4π, makes its appearance in many practical equations. For example, on the present system the magnetic force H in the interior of a long spiral wire of N turns per centimetre of length when a current of A amperes circulates in the wire is 4π AN/10. Again, the electric displacement or induction D through a unit of area is connected with the electric force E and the dielectric constant K by the equation- D=KE/4π. In numerous electric and magnetic equations the constant 4π makes its appearance where it is apparently meaningless. A system of units in which this constant is put into its right place by appropriate definitions is called a rational system of electric units. Several physicists have proposed such systems. Amongst others that of Professor G. Giorgi especially deserves mention.Giorgi’s system of electrical units. We have seen that in expressing the dimensions of electric and magnetic qualities we cannot do so simply by reference to the units of length, mass and time, but must introduce a fourth fundamental quantity. This we may take to be the dielectric constant of the ether or its magnetic permeability, and thus we obtain two systems of measurement. Professor Giorgi proposes that the four fundamental quantities shall be the units of length, mass, time and electrical resistance, and takes as the concrete units or standards the metre, kilogramme, second and ohm. Now this proposal not only has the advantage that the theoretical units are identical with the actual practical concrete units, but it is also a rational system. Moreover, the present practical units are unaltered; the ampere, volt, coulomb, weber, joule and watt remain the actual as well as theoretical units of current, electromotive force, quantity, magnetic flux, work and power. But the unit of magnetic force becomes the ampere-turn per metre, and the unit of electric force the volt per metre; thus the magnetic units are measured in terms of electric units. The numerical value of the permeability of ether or air becomes 4π×10−7 and the dielectric constant of the ether or air becomes 1/4π×9×109 their product is therefore 1/(3×108)2, which is the reciprocal of the square of the velocity of light in metres per second.

For a discussion of the Giorgi proposals, see a paper by Professor M. Ascoli, read before the International Electrical Congress at St Louis, 1904 (Journ. Inst. Elect. Eng. Lond., 1904, 34, 176).

It can hardly be said that the present system of electrical units is entirely satisfactory in all respects. Great difficulty would of course be experienced in again altering the accepted practical concrete units, but if at any future time a reformation should be possible, it would be desirable to bear in mind the recommendations made by Oliver Heaviside with regard to their rationalization. The British Association Committee defined the strength of a magnetic pole by reference to the mechanical stress between it and another equal pole: hence the British Association unit magnetic pole is a pole which at a distance of one centimetre attracts or repels another equal pole with a force of one dyne. This, we have seen, is an imperfect definition, because it omits all reference to the permeability of the medium in which the experiment takes place; but it is also unsatisfactory as a starting-point for a system of units for another reason. The important quantity in connexion with polar magnets is not a mechanical stress between the free poles of different magnets, but the magnetic flux emanating from, or associating with, them. From a technical point of view this latter quality is far more important than the mechanical stress between the magnetic poles, because we mostly employ magnets to create induced electromotive force, and the quantity we are then mostly concerned with is the magnetic flux proceeding from the poles. Hence the most natural definition of a unit magnet pole is that pole from which proceeds a total magnetic flux of one unit. The definition of one unit of magnetic iiux must then be that flux which, when inserted into or withdrawn from a conducting circuit of one turn having unit area and unit conductivity, creates in it a flow or circulation of one unit of electric quantity. T he definition of a unit magnetic pole ought, therefore, to have been approached from the definition of a unit of electric quantity.

On the C.G.S. or British Association system, if a magnetic filament has a pole strength m—that is to say, if it has a magnetization I, and a section s, such that Is equals m—then it can be shown that the total flux emanating from the pole is $4\pi m$ . The factor $4\pi$ , in consequence of this definition, makes its appearance in many practically important expressions. For instance, in the well-known magnetic equation connecting the vector values of magnetization ${\text{I}}$ , magnetic force ${\text{H}}$ and magnetic flux density ${\text{B}}$ , where we have the equation

${\text{B}}={\text{H}}+4\pi {\text{I}}$ ,

the appearance of the quantity $4\pi$ disguises the real physical meaning of the equation.

The true remedy for this difficulty has been suggested by Heaviside to be the substitution of rational for irrational formulae and definitions. He proposes to restate the definition of a unit magnetic pole in such a manner as to remove this constant $4\pi$ Heaviside’s rational system. rational from the most frequently employed equations. His start system mg-point IS a new definition according to which a unit magnetic pole is said to have a strength of in units if it attracts or repels another equal pole placed at a distance of d centimetres with a force of m24πd2 dynes. It follows from this definition that a rational unit magnetic pole is weaker or smaller than the irrational or British Association unit pole in the- ratio of 1/√ to 1, or 0.28205 to 1. The magnetic force due to a rational pole of strength m at a distance of d centimetres being m/4πd2 units, if we suppose a magnetic filament having a pole of strength m in rational units to have a smaller sphere of radius r described round its pole, the magnetic force on the surface of this sphere is m/4πr2 units, and this is therefore also the numerical value of the flux density. Hence the total magnetic flux through the surface of the sphere is

$4\pi r^{2}\times m/4\pi r^{2}$ units $=m$ units;

and therefore the number which denotes the total magnetic flux coming out of the pole of strength in in rational units is also in.

The Heaviside system thus gives us an obvious and natural definition of a unit magnetic pole, namely, that it is a pole through which proceeds the unit of magnetic flux. It follows, therefore, that if the intensity of magnetization of the magnetic filament is I and the section is s, the total flux traversing the centre of the magnet is Is units; and that if the filament is an endless or poleless iron filament magnetized uniformly by a resultant external magnetic force H, the flux density will be expressed in rational units by the equation B =I+H. The physical meaning of this equation is that the Hux per square centimetre in the iron is simply obtained by adding together the flux per square centimetre, if the iron is supposed to be removed, and the magnetization of the iron at that place. On the rational system, sinceihe unit pole strength has been decreased in the ratio of I to 1/√, or of 3.5441 to 1, when compared with the magnitude of the present irrational unit pole, and since the unit of magnetic flux is the total flux proceed in from a magnetic pole, it follows that Heaviside's unit of magnetic flux is larger than the C.G.S. unit of magnetic flux in the ratio of 3.5441 to 1.