Rainich G. Y.'s A New Kind of Representation of Surfaces PDF

By Rainich G. Y.

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Extra resources for A New Kind of Representation of Surfaces

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28): V = f R) E = 4;€O ~r Outside (r - 100 r3 qp 2 qp R2= qR2 ~= JE2dT. f r2 . j Inside (r 2 {R r4 (r 411"dr)+ Jo 00 1 )1 R + R6 "5 2 (R3 ) R r5 ( r < R) E = 4;€OWrr. f { is VE. da + Jv E2dT} , where V is large enough to enclose all the charge, but otherwise arbitrary.

But in this case there are no natural boundary conditions, and no persuasive symmetry conditions, to fix the answer. " is simply ill-posed: it does not give us sufficient information to determine the answer. 51 Compare Newton's law of universal gravitation to Coulomb's law: F = _Gmlm2 ~r; A F =- 1 qlq2 471"100-;:2 A r. Evidently 4;<0 -t G and q -t m. The gravitational energy of a sphere (translating Prob. 28 X 1041J. 87 X 107 years.! 52 First eliminate z, using the formula for the ellipsoid: a(x,y)=_m Q 1 47rab yI C2(X2 f a4) + C2(y2 fb4) + 1 - (x2 f a2) - (y2fb2) .

Dipole. z rL 411"fO /Ldq = )"dx -- x 1 Z2~ ( x 411"fOJo x Z L A x+ ) ] I ~ - - ---1-).. Z 411"fO () ] I 1-).. 411"fO L xdx I (X2+z2)3/2 - - ---1-).. 1 0 - [ 411"fO Z ~. ,jz2 + L2 For z » L you expect it to look like a point char ge q = )"L: E -+ 4_ 1 ¥-z. It checks, for with z » L the x 1I"fO Z 1 ~kz. 4 From Ex. 1, with L -+ ~ and z -+ vz2 + (~)2 (distance from center of edge to F), field of one edge is: ~ El = . - : z2+ a4 z. 5 ~ "Horizontal" components cancel, leaving: E = 4;fO {I cos ()} z. Here, 1-2= r2 + z2, cos() = ~ (both constants), while dl = 27rr.

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A New Kind of Representation of Surfaces by Rainich G. Y.

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