A uniform istropic dielectric si shaped as a spherical layer with radii `a` and `b`. Draw the approximate plots of the electric field strength `E` and the potential `varphi` vs the distance `r` from the centre fo the layer if the dielectric has a certain positive extraneous charge distributed uniformly,
(a) over the internal surface of the layer, (b) over the volume of the layer.

Explanation

(a) Positive extraneous charge is distributed uniformly over the internal surface layer. Let `sigma_(0)` be the surface density of the charge.
Clearlty `E = 0`, for `r lt a`
For `a lt r`
`epsilon_(0) E xx 4pi r^(2) = 4pi a^(2) sigma_(0)` by Gauss theorem.
or, `E = (sigma_(0))/(epsilon_(0)) ((a)/(r ))^(2) , a lt r lt b`
For `r gt b`, similarly
`E = (sigma_(0))/(epsilon_(0))((a)/(r ))^(2), r gt b`
Now, `E = - (del varphi)/(del r)`,
So by intergation from intensity where `varphi (oo) = 0`,
`varphi = (sigma_(0) a^(2))/(epsilon_(0) r) r gt b`

`a lt r lt b varphi = (sigma_(0) a^(2))/(epsilon epsilon r) + B, B` is a constant
or by continuity, `varphi = (sigma_(0) a^(2))/(epsilon_(0) epsilon) ((1)/(r ) - (1)/(b)) + (sigma_(0) a^(2))/(epsilon_(0) b)`
Positive extreneous charge si distributed uniformly over the internal volume of the dielectric
Let `rho_(0)` = volume density of the charge in the dielectric, for `a lt r lt b`.
`E = 0, r lt a`
`epsilon_(0) epsilon 4pi r^(2) E = (4pi)/(3) (r^(3) - a^(3)) rho_(0) , (a lt r lt b)`
or, `E = (rho_(0))/(3epsilon_(0)) (r - (a^(3))/(r^(2)))`
`E = (4pi)/(3) (b^(3) - a^(2)) rho_(0)//epsilon_(0) 4pi r^(2), r gt b`
or, `E = ((b^(2) - a^(2)) rho_(0))/(3epsilon_(0) r^(2))` for `r gt b`
By intergration,
`varphi = ((b^(2) - a^(3))rho_(0))/(3epsilon_(0) r)` for `r gt b`
or, `varphi = B = (rho_(0))/(3epsilon_(0) epsilon) ((r^(2))/(2) + (a^(2))/(r )), a lt r lt b`
By continuity
`(b^(3) - a^(3))/(3epsilon_(0) b) rho_(0) = B - (rho_(0))/(3 epsilon_(0) epsilon) ((b^(2))/(2) + (a^(3))/(b))`
or, `B = (rho_(0))/(3epsilon_(0) epsilon) {(epsilon (b^(2) - a^(2)))/(b) + ((b^(2))/(2) + (a^(3))/(b))}`
Finally `varphi = B - (rho_(0))/(3 epsilon_(0) epsilon) ((a^(2))/(2) + a^(2)) = B - (rho_(0) a^(2))/(2 epsilon_(0) epsilon), r lt a`
On the basic of obtained expressions `E(r )` and `(varphi) (r )` can be plotted as shown in the answer-sheet.