Page:EB1911 - Volume 27.djvu/766

Rh 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, 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 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 MundefinedLundefinedT−1Kundefined. 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 MundefinedLundefinedT−1Kundefined.

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 ), 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,, of the medium in which they are immersed. Hence C2ds ds′/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 MundefinedLundefinedT−1−. Accordingly the dimensions of current on the E.S. system are MundefinedLundefinedT−2Kundefined, and on the E.M. system the are MundefinedLundefinedT−1−, where 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−1Kundefinedundefined. This ratio must be a mere numeric of no dimensions, and therefore the dimensions of √$\overline$ must be those of the reciprocal of a velocity. We do not know what the dimensions of, 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 MundefinedLundefinedT−1Kundefined; the potential difference V must be, therefore, of the dimensions MundefinedLundefinedT−1K−. 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 MundefinedLundefinedT−1−; therefore the dimensions of electromotive force or potential on the electromagnetic system must be MundefinedLundefinedT−2undefined. Here again we find that the ratio of the dimensions on the electrostatic system to the dimensions on the electromagnetic system is L−1TK−−.

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 , and in every case the dimensions in terms of K are to those in terms of for the same quantity in the ratio of a power of LT−1Kundefinedundefined. This therefore confirms the view that whatever may be the true dimensions in terms of fundamental units of 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−1Kundefinedundefined, or of a velocity multiplied by the square root of) the product K and ; In other words, it is the product of a velocity multiplied by the geometric mean of K and . This quantity 1/√$\overline$ 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