(c) Mutual Potential of a Magnet-pole and a Circuit. - If a magnet-pole of strength m were brought up to P, m times as much work will be done as if the magnet-pole had been of unit strength, and the work would be just as great whether the pole m were brought up to the circuit, or the circuit up to the pole. Hence, the mutual potential will be But, as in Art. 349, we may regard mo as representing the number of lines of force of the pole which are intercepted by and pass through the circuit, and we may write N for that number, and say V = - CN, or the mutual potential of a magnet-pole and a circuit is equal to the strength of the current multiplied by the number of the magnet-pole's lines of force that are intercepted by the circuit, taken with reversed sign. (d) As in the case of the magnetic shell, so with the circuit, the value of the potential changes by 4 C from a point on one side of the circuit to a point just on the other side; that is to say, being 2 #С on one side and +2 ′′C on the other side work equal to 4 C must be done in carrying a unit-pole from one side to the other round the outside of the circuit. The work done in thus threading the circuit along a path looped S times round it would be 4 SC. S Seose 351. (e) Mutual Potential of two Circuits. Two closed circuits will have a mutual potential, depending on the strengths of their respective currents, on their distance apart, and on their form and position. If their currents be respectively C and C', and if the distance between two elements ds and ds' of the circuits be called r, and e the angle between the elements, it can be shown that their mutual potential is =-CC' ds ds'. This expression represents the work that would have to be done to bring up either of the circuits from an infinite distance to its present position near the other, and is a negative quantity if they attract one another. Now, suppose the strength of current in each circuit to be unity; their mutual potential will in that case be ds ds', a quantity which depends purely upon the geometrical form and position of the circuits, and for which we may substitute the single symbol M, which we will call the coefficient of mutual potential"; we may now write the mutual potential of the two circuits when the currents are C and C' as == CC'M. es Scose But we have seen in the case of a single circuit that we may represent the potential between a circuit and a unit-pole as the product of the strength of the current C into the number N of the magnet-pole's lines of force intercepted by the circuit. Hence the symbol M must represent the number of each other's lines of force mutually intercepted by both circuits, if each carried unit current. If we call the two circuits A and B, then, when each carries unit current, A intercepts M lines of force belonging to B, and B intercepts M lines of force belonging to A. Now suppose both currents to run in the same (clock-wise) direction; the front or S-seeking face of one circuit will be opposite to the back or N-seeking face of the other circuit, and they will attract one another, and will actually do work as they approach one another, or (as the negative sign shows) negative work will be done in bringing up one to the other. When they have attracted one another up as much as possible the circuits will coincide in direction and position as nearly as can ever be. Their potential energy will have run down to its lowest minimum, their mutual potential being a negative maximum, and their coefficient of mutual potential M, having its greatest possible value. Two circuits, then, are urged so that their coefficient of mutual potential M shall have the greatest possible value. This justifies Maxwell's Rule (Art. 204), because M represents the number of lines of force mutually intercepted by both circuits. And since in this position each circuit induces as many lines of magnetic force as possible through the other, the coefficient of mutual potential M is also called the coefficient of mutual induction (Art. 454). LESSON XXVII. -The Electromagnetic System of Units 352. Magnetic Units. All magnetic quantities, strength of poles, intensity of magnetization, etc., are expressed in terms of special units derived from the fundamental units of length, mass, and time, explained in the Note on Fundamental and Derived Units (Art. 280). Most of the following units have been directly explained in the preceding Lesson, or in Lesson XI.; the others follow from them. Unit Magnet-pole. - The unit magnetic pole is one of such a strength, that when placed at a distance of 1 centimetre (in air) from a similar pole of equal strength, repels it with a force of 1 dyne (Art. 141). Magnetic Potential. - Magnetic potential being measured by work done in moving a unit magnetic pole against the magnetic forces, the unit of magnetic potential will be measured by the unit of work done on unit-pole. Unit Difference of Magnetic Potential. - Unit difference of magnetic potential exists between two points when it requires the expenditure of one erg of work to bring a (N-seeking) unit magnetic pole from one point to the other against the magnetic forces. Magnetomotive-force, or magnetizing power, is measured in same units as difference of magnetic potential. Intensity of Magnetic Field is measured by the force it exerts upon a unit magnetic pole: hence, Unit Intensity of Field is that intensity of field which acts Magnetic Flux, or total induction of magnetic lines, is 353. Electromagnetic Units. - The preceding magnetic units give rise to the following set of electrical units, in which the strength of currents, etc., are expressed in magnetic measure. They are sometimes called "absolute C.G.S." units. The relation of this "electromagnetic set of units to the "electrostatic" set of units of Art. 283 is explained in Art. 359. Unit Strength of Current.-A current has unit strength when one centimetre length of its circuit bent into an arc of one centimetre radius (so as to be always one centimetre away from the magnet-pole) exerts a force of one dyne on a unit magnet-pole placed at the centre (Art. 207). Unit of Difference of Potential (or of Electromotive-force). -Potential is work done on a unit of electricity; hence unit difference of potential exists between two points when it requires the expenditure of one erg of work to bring a unit of electricity from one point to the other against the electric force. Also, unit electromotive-force is generated by cutting one magnetic line per second. Unit of Resistance. - A conductor possesses unit resistance when unit difference of potential between its ends causes a current of unit strength to flow through it. Unit of Quantity of Electricity is that quantity which is conveyed by unit current in one second. Unit of Capacity. - Unit capacity requires unit quantity to charge it to unit potential. Unit of Induction. - Unit induction is such that unit electromotive-force is induced by the variation of the current at the rate of one unit of current per second. 66 354. Practical Units and Standards.*- Several of the above "absolute" units in the C.G.S. system would be inconveniently large and others inconveniently small for practical use. The following are therefore chosen as practical units:Resistance. The Ohm, = 109 absolute units of resistance (and theoretically the resistance represented by the velocity of one earth-quadrant per second, see Art. 357), but actually represented by the resistance of a uniform column of mercury 106-3 centimetres long and 14:4521 grammes in mass, at 0° C. Such a column of mercury is represented by a standard" ohm (see Appendix B). Current. The Ampere (formerly called the "weber"), = 10-1 absolute units; practically represented by the current which deposits silver at the rate of 0·001118 gramme per second (see Appendix B). Electromotive-force. - The Volt, 108 absolute units, is that E.M.F. which applied to 1 ohm will produce in it a current of 1 ampere; being of the E.M.F. of a Clark standard cell at 15° C. (See Appendix C.) Quantity. The Coulomb, = 10-1 absolute units of quantity; being the quantity of electricity conveyed by 1 ampere in one second. Capacity. The Farad, == 10-9 (or one one-thousandmillionth) of absolute unit of capacity; being the capacity of a condenser such as to be changed to a potential of 1 volt by 1 coulomb. The microfarad or millionth part of 1 farad 10-15 absolute units. Work. The Joule, = 107 absolute units of work (ergs), is represented by energy expended in one second by 1 ampere in 1 ohm. Power. The Watt, = 107 absolute units of power (ergs per second), is power of a current of 1 ampere flowing * The word "unit expresses our conception in the abstract of a unit quantity, such as those defined in the preceding Articles. A "standard" is the concrete thing with which we compare quantities to be measured, such as a centimetre scale or a standard cell. under a pressure of 1 volt. It is equal to 1 joule per second, and is approximately of one horse-power. Induction. - The Henry, = 109 absolute units of induction, is the induction in a circuit when the electromotiveforce induced in this circuit is 1 volt, while the inducing current varies at the rate of 1'ampere per second. Seeing, however, that quantities a million times as great as some of these, and a million times as small as some, have to be measured by electricians, the prefixes mega- and micro- are sometimes used to signify respectively one million" and "onemillionth part." Thus a megohm is a resistance of one million ohms, a microfarad a capacity of roads of a farad, etc. The prefix kilo- is used for "one thousand," and milli- for "onethousandth part"; thus a kilowatt is 1000 watts, and milliampere is the thousandth part of 1 ampere. 66 The "practical" system may be regarded as a system of units derived not from the fundamental units of centimetre, gramme, and second, but from a system in which, while the unit of time remains the second, the units of length and mass are respectively the earth-quadrant and 10-11 gramme. 355. Use of Index Notation. - Seeing that electricians have to deal with quantities requiring in some cases very large numbers, and in other cases very small numbers, to express them, a system of index notation is adopted, in order to obviate the use of long rows of ciphers. In this system the significant figures only of a quantity are put down, the ciphers at the end, or (in the case of a long decimal) at the beginning, being indicated by an index written above. Accordingly, we may write a thousand (= 10 × 10 × 10) as 108, and the quantity 42,000 may be written 42 × 103. The British National Debt of £770,000,000 may be written £77 x 107. Fractional quantities will have negative indices when written as exponents. Thus 1 (= 0·01) 110 ÷ 10 = 10-2. And so the decimal 0.00028 will be written 28 × 10-5 (being = 28 x .00001). The convenience of this method will be seen by an example or two on electricity. The electrostatic capacity of the earth is 630,000,000 times that of a sphere of one centimetre radius, 63 × 107 (electrostatic) units. The resistance of selenium is about 40,000,000,000, or 4 × 1010 times as great as that of copper; that of air is about 1026, or = 100,000,000,000,000,000,000,000,000 times as great. The velocity of light is about 30,000,000,000 centimetres per second, or 3 × 1010. 356. Dimensions of Magnetic and Electromagnetic Units. |