The volume of a sphere of radius a is charged with a constant volume charge density of p. Find the field of this spherical charge @ a distance d>>a.
Let S be the sphere of radius d with center at center of the ball, and E be the electric dield at some point of S. Due to symmetricity of the ball and sphere the absolute value of E is the same at all points of S. Moreover the vector E is parallel to normal vector to S. Hence the flux Phi through S is equal to
Phi = integral_over_S (E dA) = E * Area_of_S = E * 4 pi * d^2.
On the other hand, due to Gauss' law
Phi is equal to the full charge Q inside S divided by eps_0:
Phi = Q/eps_0.
The ball is uniformly charged, so
Q = p * Volume_of_a_ball = p * 4/3 * pi * a^3.
Phi = E * 4 pi * d^2 = p * 4/3 * pi * a^3 / eps_0
E = (p * 4/3 * pi * a^3) / (4 pi * d^2 * eps_0 ) = a^3 * p / (3 * eps_0 * d^2)