A space is filled up with a charge with volume desnsity `rho = rho_(0) e^(-alpha r^(3))`, where `rho_(0)` and `alpha` are positive constansts, `r` is the distance from the centre of this system. Find the magnitude of the electric field strength vector as a function of `r`. Investigate the obtained expresssion for the small and large values of `r, i.e.,` at `alpha r^(3) lt lt 1` and `alpha r^(3) gt gt 1`.

Explanation

Let us consider a spherical layer of radius `r` and thickness `dr`, having its centre coinciding with the centre of the system. Then using Gauss's theorem for the surface
`E_(r ) 4pi r^(2) = (q_("inclosed"))/(epsilon_(0)) = int_(0)^(r ) (rho d V)/(epsilon_(0))`
`= (1)/(epsilon_(0)) int_(0)^(r ) rho_(0) e^(-a r^(3)) 4pi r^(2) dr`.
After integration
`E_(r ) 4pi r^(2) = (rho 4pi)/(3 epsilon_(0) alpha) [1 - e^(-alpha r^(3))]`
or, `E_(r ) = (rho_(0))/(3 epsilon_(0) alpha r^(2)) [1- e^(-alpha r^(3))]`
Now, when `alpha r^(3) lt lt 1, E_(r )~~(rho_(0))/(3epsilon_(0))`
And when `alpha r^(3) gt gt 1, E_(r ) = (rho_(0))/(3 epsilon_(0) r^(2))`