InterviewSolution
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InterviewSolution
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A nonumiform but spherically symmetric distribution of charge has charge density ` prop` given as follow - ` prop =prop_0 (1 - r //R)` for prop le R lt brgt ` prop =0 `for `r le R` where ` prop_0 = 3 Q // pi R^3` is a constant ( a) Show that the total charge contained in the charge distrubution is `Q`. (b) Show that , for the region defined by ` r le R`, the electric field is identical to that produced by a point charge Q. Obtain an expression for the electric field in the region ` r le R`. (d) Compare your results in part (b) and (c ) `r=R`. |
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Answer» Correct Answer - (c ) `(kQ r//R^2) (4 -3 r//R)` (a) For total charge `q = int rho d v rArr q = int rho_0 ( 1 - r/R ) ( 4 pi r^2 dr ) = rho_0^x` ` 4 pi int_0^R (r^2 - r^3/R) dr` `q =rho_0 4 pi [R^3 /3 - R^4 /(4R) ] = rho 4 pi [R^3 /(12)]` …(1) put `rho _0 = (3Q)/(piR^3)` in equation (1) `q = (3 Q)/(pi R^3) xx ( pi R^3)/3 = Q` (b) From gauss theorem for `r le R` `r le R oint vec E.s vec S = (sumq)/(varepsilon_0) E(4 pi r^2) = (sumq)/(varepsilon_0)` Since `sin q = Q` for `r le R` so `E = 1/( 4 pi in_0) Q/r^2` (c) `E = (sumq)/(4 pi varepsilon_0 r^2)` ...(1) `sum q = int rho d V = int_0^r rho 4 pi r der = rho_0 xx 4 pi int_0^r (r^2-r^3)/(R) dr` ` = (3Q)/(piR^2) xx 4 pi [ r^2 /3 - r^2 /(4R)]` ` q = ( 12Q)/R^2 (r^3 /3 - r^4 /(4R))` ...(2) Sub. (2) in (1) ` E = ( 12 Q)/( 4 pi varepsilon_0r^2 xx R^3) ( r^3 /3 - R^4 /(4 R) ) = (KQr)/R^3 ( 4 - (3r)/R)` (d) In (b) `r =R rArr E = (KQ)/R^2` in ( c) `r = R rArr E = (KQ)/R^2`. |
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