Question

A spherically symmertic charge distribution, comprised of filled sphere at the center surround by a shell. The center sphere has radius, a, and charge distribution: rho = (rho1/a^2)r^2. The outer shell has inner radius, b, and outer radius, c and a uniform charge density, rho2. What is the electric field, E, everywhere?

Answer 1

given
radius of central sphere = a
charge distribution in central sphere, rho = (rho1/a^2)r^2
inner radius of outer sheell = b
outer radius = c
charge density in shell = rho2

so, from gauss' law
for r < a

E*4*pi*r^2 = q/epsilon
dq = rho*4*pi*r^2*dr = rho1*4*pi*r^4*dr/a^2
integrating from r = 0 to r = r
q = rho1*4*pi*r^5/5a^2

hence
E*4*pi*r^2 = rho1*4*pi*r^5/5a^2*epsilon
E = rho1*r^3/5a^2*epsilon

for a < r < b
from gauss law
E*4*pi*r^2 = rho1*4*pi*a^5/5a^2*epsilon
E = rho1*a^3/5r^2*epsilon

for b < r < c
from gauss law
E*4*pi*r^2 = rho1*4*pi*a^5/5a^2*epsilon + rho2*4*pi*(r^3 - b^3)/3*epsilon
E = rho1*a^3/5r^2*epsilon + rho2*(r - b^3/r^2)/3*epsilon

for r > c
from gauss law
E*4*pi*r^2 = rho1*4*pi*a^5/5a^2*epsilon + rho2*4*pi*(c^3 - b^3)/3*epsilon
E = rho1*a^3/5r^2*epsilon + rho2*(c^3 - b^3)/3*r^2*epsilon