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A solid insulating sphere of radius R ha...

A solid insulating sphere of radius R has a nonuniform charge density that varies with r according to the expression `rho = Ar^2`, where A is a constant and `rltR` is measured from the centre of the sphere. Show that (a) the magnitude of the electric field outside `(rgtR)` the sphere is `E = AR^5//5epsilon_0r^2` and (b) the magnitude of the electric field inside `(rltR)` the sphere is `E = Ar^3//5epsilon_0`.

Text Solution

Verified by Experts

`intEdA = E(4pir^2) = q_("in")/epsilon_0`
(a) For `r gtR, q_("in") = int_(0)^® Ar^2 (4pir^2)dr = 4pi (AR^5)/5`
So `E = (AR^5)/(5epsilon_0r^2)`
(b) For rltR, `q_("in") = int_(0)^(R) Ar^2(4pir^2)dr = (4piAr^5)/5`
So `E = (Ar^3)/(5epsilon_0)`.
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