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Extraneous chagres are uniformly distributed with space density rho gt 0 overa ball of radiusR made of uniformistropis dielectric with permittivity epsilon. Find : (a) magnitudeof the electric field strength as a function of distance r fromthe centreof the ball, drawteh approixmateplots E(r) adn varphi (r ), (b) the space and surface densitiesof the boundcharges. |
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Answer» SOLUTION :`di v vec(D) = (1)/(r^(2)) (del)/(del r) D_(r ) = rho` `r^(2) D_(r ) = rho (r^(3))/(3) + A D, = (1)/(3) rho r + (A)/(r^(2)), r LT R` `A = 0` as `D_(r ) = oo` at `r = 0`, THUS, `E_( r) = (rho r)/(3 epsilon epsilon_(0))` For`r gt R, D_( r) = (B)/(r^(2))` By continuity of `D`, at`r = R , B =(rho R^(3))/(3)` so, `E_(r ) = (rho R^(3))/(3 epsilon_(0) r^(2)) , r gt R` `varphi = (rho R^(3))/(3 epsilon r), r gt R` and `varphi = (rho r^(2))/(6 epsilon epsilon_(0)) + C, r lt R` `C = + (rho R^2)/(3 epsilon_(0)) + (rho R^(2))/(6 epsilon epsilon_(0))`, by continuity of `varphi`. See answer sheet for graphs of `E ( r)` and `varphi (r )` (b) `rho' = d ivvec(P) = (1)/(r^(2)) (del)/(del r) {(r^(3))/(3) rho (1 - (1)/(epsilon))} = - (rho (epsilon - 1))/(epsilon)` `sigma' = P_(1r) - P_(2r) = P_(1r) = (1)/(3) rho R(1 - (1)/(epsilon))` |
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