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hw5_solutions - TIlE STATIC ELECTRIC FIELD 101 where the...

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TIlE STATIC ELECTRIC FIELD 101 where the last two terms are the so-called Exponential Integrals for which there is no closed form solution but which are well tabulated t . (d) The result can be shown by simple substitution into the cylindrical coordinate version of Poisson's equation, namely 1 8 2 C1> ~CI> r 2 8cjJ2 + 8z 2 ""--" ~ No variation in </J No variation in z 2 I 8 ( 8C1» V CI>=-- r- + r8r 8r -p(r) =-- Note that differentiation of the Exponential Integral terms simply yield the integrands. The electron charge density in a hydrogen atom. (a) Since there is spherical symmetry, we consider a spherical Gaussian surface S with radius r and apply Gauss's law as where where we used the integral which can easily be shown using integration by parts twice. As a result, the electric field due to the electron cloud only can be found as E 1 4qe [a 3 -2r/a (ar 2 a 2 r a 3 )] - --e -+ +- r 41T1:or2 a 3 4 2 2 4 - 41T~:r2 { 1- e- 2r / a [2(~r + 2 (~) + I] } Note that the constant a in the above expression is called the Bohr radius given by a ~ 0.529 x 10- 10 m. The electric potential can be found by integrating the electric field. We find i r qe -2r/a[1 1] qe 1 CI>(r) = - Er(r)dr = -- e - + - - --- 00 41TfO a r 41TfO r . See Chapter 5 of M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, .. aphs, and mathematical Tables, National Bureau of Standards, Tenth Printing, 1972.
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102 niB STATIC ELECTRIC FIELD (b) Using superposition principle, the total electric field is given by where ETc is the field due to the electron cloud only (found in part (a» and E1'p is the field due to the proton charge -qe located at the origin given by The electric potential can be found by integrating the electric field. We find 4·22. Spherical sheD of charge. (a) The total charge in the spherical shell region specified by a S r S b is given by Q = [ p( r')dv' = [21f r [b p( r')r,2 sin fJ' dr' dfJ' drp' iv' io io ia b K lb =(21T)(2) 2r,2dr, = 41TK dr' = 41TK(b - a) l a r' a (b) Due to spherical symmetry, the electric field has the form E = t E1'(r).
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