Example M2.
A cloud of charge occupies the part of the sphere
$$
x^2 + y^2 + z^2 \le a^2
$$
in which $z\ge\sqrt{x^2+y^2}$. The charge density at $(x,y,z)$ is
$$
\rho_V(x,y,z) = 8\beta z\qquad\hbox{C/m${}^3$},
$$
for some constant $\beta$.
Find
-
the total charge in the cloud;
-
an integral expression for the electric field ${\bf E}(x,y,z)$,
valid for any point $(x,y,z)$ outside the cloud;
-
exact values for two components of the vector ${\bf E}(0,0,z)$,
assuming $z>a$; and
-
an iterated double integral expression for the vector component
not found in part (c).