volts E, and equate these two expressions: E = .000010384 x H x 424, 9.81 which reduces to E=HX.043. For the work (in kilogram Rule 61, page 78. meters) done by a current (volt-coulombs) we have the general expression: = 9.81 or 9.81 (1) Making W 1 (i. e. one kilogram-meter) and transforming, we have, as the coulombs corresponding to 1 kilogram-meter: One coulomb of electricity liberates a weight (in grams) of an element equal to the product of the following: .000010384 × equivalent of element in question X number of equivalents valency of the element. Therefore, the coulombs corresponding to one kilogram-meter, liberates this weight multiplied by or, indicating weight by G, G = 9.81 E .000010384 X equiv. X number equiv. 9.81 but .000010384 × 9.81 = .000101867. valency F (3) (4) Rule 73, page 98-99. The voltage of an armature of a definite number of turns of wire and a fixed speed, varies with the lines included within its longitudinal area, as such lines are cut in every revolution. These lines vary with its area, and the latter varies with the square of its linear dimensions. To maintain a constant voltage if the size is changed, the number of turns must be varied inversely as the square of the linear dimensions. This ensures the cutting of the same number of lines of force per revolution. If, therefore, its size is reduced from x to the turns of wire must be changed from x to x2. The relative diameters of the two sizes of wire is found by dividing a similar linear dimension by the relative size of the wire. But + 22 = 1 = diameter of x3 the wire for maintenance of a constant voltage with change of size. But The capacity of a wire varies with the square of its diameter and ()' — 1. = Therefore the amperage, if a constant voltage is maintained, will vary inversely as the sixth power of the linear dimensions of an armature. CHAPTER XIV. NOTATION IN POWERS OF TEN. THIS adjunct to calculations has become almost indispensable in working with units of the C. G. S. system. It consists in using some power of 10 as a multiplier which may be called the factor. The number multiplied may be called the characteristic. The following are the general principles. The power of 10 is shown by an exponent which indicates the number of ciphers in the multiplier. Thus 102 indicates 100; 108 indicates 1000 and so on. The exponent, if positive, denotes an integral number, as shown in the preceding paragraph. The exponent, if negative, denotes the reciprocal of the indicated power of 10. Thus 10-2 indicates 1o; 103 indicates Too and so on. The compound numbers based on these are reduced by multiplication or division to simple expressions. Thus: 3.14 × 107 3.14 × 10,000,000 31,400,000. 3.14 × 107 = or 3.14 = Re gard must be paid to the decimal point as is done here. To add two or more expressions in this notation if the exponents of the factors are alike in all respects, add the characteristics and preserve the same factor. Thus: (51 × 10o) + (54 × 106 )= 105 × 10o. (9.1 × 10-9) + (8.7 × 10-9) = 17.8 × 10-o. To subtract one such expression from another, subtract the characteristics and preserve the same factor. Thus: (54 × 10o) — (51 × 10°) = 3 × 10o. If the factors have different exponents of the same sign the factor or factors of larger exponent must be reduced to the smaller exponent, by factoring. The characteristic of the expression thus treated is multiplied by the odd factor. This gives a new expression whose characteristic is added. to the other, and the factor of smaller exponent is preserved for both. Thus: (5 × 107) + (5 × 109) = (5 × 107) + (5 × 100 × = 505 × 107. 107) The same applies to subtraction. Thus: (5 × 109) - (5 × 107) = (5 × 100 × 107) (5 x 107) 495 × 107. = If the factors differ in sign, it is generally best to leave the addition or subtraction to be simply ex pressed. However by following the above rule it can be done. Thus: Add 5 X 10-2 and 5 × 103. 5 X 108 = 5 × 105 × 102: (5 × 105 × 10-2) + (5 × 10-2) = 500005 × 10%. This may be reduced to a = 5000.05. fraction 500005 100 To multiply add the exponents of the factors, for the new factor, and multiply the characteristics for a new characteristic. The exponents must be added algebraically: that is, if of different signs the numerically smaller one is subtracted from the other one, its sign is given the new exponent. Thus: = (25 × 10) X (9 x 108) 225 × 1014. (9 × 108) × (98 × 10-2) = 882 × 10o. To divide, subtract (algebraically) the exponent of the divisor from that of the dividend for the exponent of the new factor, and divide the characteristics one by the other for the new characteristic. Algebraic subtraction is effected by changing the sign of the subtrahend, subtracting the numerically smaller number from the larger, and giving the result the sign of the larger number. (Thus to subtract 7 from 5 proceed thus: 5 — 7 = −2.) Thus: (25 × 10°) (5 x 10°) = 5 x 10-* (28 × 10-3) + (5 × 103) = 5.6 × 10-1. |