Miscellaneous papers . hevalues of /, /2 thence found when substituted in yjr merelygive the inductions of the first and second order, which wehave already calculated on p. 68. Here we only consider the 80 INDUCTION IN ROTATING SPHERES II angle through which the lines of flow are turned. Eetainingonly lowest powers we find ta-i-*--!=*/_* I- /x k \2»+l 2ti+3 so that the angle in question is 8 2ir(of E p~ 1~ V~\2n+l~ 2n+Z, Thus all the layers appear rotated: the rotation is least atthe surface of the sphere and increases continuously we imagine a plane section taken through the equato
Miscellaneous papers . hevalues of /, /2 thence found when substituted in yjr merelygive the inductions of the first and second order, which wehave already calculated on p. 68. Here we only consider the 80 INDUCTION IN ROTATING SPHERES II angle through which the lines of flow are turned. Eetainingonly lowest powers we find ta-i-*--!=*/_* I- /x k \2»+l 2ti+3 so that the angle in question is 8 2ir(of E p~ 1~ V~\2n+l~ 2n+Z, Thus all the layers appear rotated: the rotation is least atthe surface of the sphere and increases continuously we imagine a plane section taken through the equator ofthe sphere and join corresponding points of the differentlayers we get a system of congruent curves, which is verysuitable for representing the state of the sphere. The equa-tion of one of these curves clearly is ^ y = «tan- p= Jx2 + y2, V or very approximately 2ttco ( E2 x2 2n+l 2n + 3 In Fig. 8 these curves are drawn for a copper sphere forwhich E = 50 mm., n = 1, when it makes 1, 2, 3, 4 revolutionsper Fig. 8. Large 3. Thirdly, let us assume that //, is so large that for q(BX) ofrotatTon. and %(sX) we ma7 Put their approximate values. Further,assume that the ratio r/K is neither very nearly = 1 nor verynearly = 0. The former case has been considered already;the latter requires special consideration. Substituting theapproximate values in the exact formula we find /—- 2n+l S71 €**-»> + €-* l + 92V 1 - 3+TXS-s)_e-A(S-s) 2»+l/R\»+1 _v(R_p) e II INDUCTION IN ROTATING SPHERES 81 Since S is not nearly equal to s, and both are very great,the second term in the denominator vanishes compared withthe first, and we get ft + ft V^l = ^QyV^-Xe^ + e-^). The second term in the bracket vanishes in comparisonwith the first except when p = r; if then we are content withan approximate knowledge of the current at the inner surfacewe may write ft + ftV^T = ^p(JJ Since s or r has disappeared from this equation, we mayassume that it holds also for a solid sphere. In fact
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Keywords: ., bookcentury1800, bookdecade1890, booksubjectphysics, bookyear1896
