Absolute standards of inductance . ndition for equal radii). ] Absolute Standards of Inductance. iii To do this it was necessary to employ the following expansions/ ^ =—^n—2 = —i—25 if—¥ — ^ k is the complementary )= logj + ij^ogj-^) , i3 ,L 4 2 2 \ , i35 J, 4 2 2 2 \ +2-4^6^« V°?;?-i:2-M-pj4-E!i5!z! /s/wi J_ A J_ J_\ + ? • • +^^-r2-^4) , i 35 J. 4 2 2 I \ , Ils; ,s/i 4 A A A- _l\ ^2^4^6^8 V^^/.^~~~~+ . . Puttins: c—— the formula derived is^ 2a ^ira ^ ^[ 4 64 256 (128) , j_2_ V I 31 A 247 ^a I 163,695 s + 1 4 ^^ i536+(i28rx84 • • ^Gray: Absolute Meas. in E
Absolute standards of inductance . ndition for equal radii). ] Absolute Standards of Inductance. iii To do this it was necessary to employ the following expansions/ ^ =—^n—2 = —i—25 if—¥ — ^ k is the complementary )= logj + ij^ogj-^) , i3 ,L 4 2 2 \ , i35 J, 4 2 2 2 \ +2-4^6^« V°?;?-i:2-M-pj4-E!i5!z! /s/wi J_ A J_ J_\ + ? • • +^^-r2-^4) , i 35 J. 4 2 2 I \ , Ils; ,s/i 4 A A A- _l\ ^2^4^6^8 V^^/.^~~~~+ . . Puttins: c—— the formula derived is^ 2a ^ira ^ ^[ 4 64 256 (128) , j_2_ V I 31 A 247 ^a I 163,695 s + 1 4 ^^ i536+(i28rx84 • • ^Gray: Absolute Meas. in Elec. and Mag., Vol. II, Part I, page —No. I—06 8 (12) 112 Bulletin of the Bureau of Standards. [Vol. 2, No. I. The last three terms in each of the parentheses are the newterms, and form a continuation of Maxwells formula, equation (6),for the case y — o. Hence, this formula gives the mutual induct-ance between two circles of the same radius, when placed at a dis-tance apart equal to w \ \ dxW\ \ w 1 i! / // / 1 ; // // / / / / / Fig. 11. Consider a current sheet, the current in it being unity and the length of the circular cylindrical sheet being b. dxThen in a length dx there will be a current — and in another dxelement dx a current of strength —7-. The mutual inductance between these two circular elements is then dM^M,,/^.^-^ where M^, ^ is the mutual inductance between two circular fila-ments at points whose coordinates are x and x and hence whosedistance apart is ix — x). The total self inductance of the currentsheet is u_ u_ j.^ dx dx \ Absolute Standards of Inductance. 113 Using the following formulae of integration, which may readily bededuced by integration by parts, we obtain formula (13). b_ b_ I dx j {x — xY log {x —x)dx — 2b- (;2+i)(;2+2) log b 2(2;^+3) ^« + 2 (;^+i)X;2+2)where ;2 is integral, even and positive, and +- +- n+2 )(;^+2; I ^^ L — ^iraii /^Y, 8a 43i\ 102410 I3IO72 35 if ;^ is even, if n
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