Question

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.

Answer 1

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.

Hence

Phi = E * 4 pi * d^2 = p * 4/3 * pi * a^3 /
eps_0

Therefore

E = (p * 4/3 * pi * a^3) / (4 pi * d^2 * eps_0
)
= a^3 * p / (3 * eps_0 * d^2)


Answer: E = a^3 * p / (3 *
eps_0 * d^2)