Question

A charge of −33 µC is distributed uniformly throughout a spherical volume of radius 19.0 cm. Determine the electric field (in N/C) due to this charge at the following distances from the center of the sphere. (Enter the radial component of the electric field.)
(a)
8.0 cm
E =( ? N/C) r̂

(b)
12.0 cm
E =( ? N/C) r̂

(c)
37.0 cm
E =( ? N/C) r̂

Answer 1

(a) is a trick question seeing if you are into Gauss' law:
inside a sphere:
The electric field E times the area of the sphere equals the charge INSIDE the sphere / eo
so
4 pi r^2 E = q / eo
but what is q INSIDE r = 8.0 ?
well the whole charge Q = -33 µC
and the whole volume of the sphere = (4/3) pi (19*10^-2)^2
so
charge / unit volume = -33 µC / [ (4/3) pi (19*10^-2)^3 ]
so our charge Qin =
{ -33 µC / [ (4/3) pi (19*10^-2)^3 ] }*[ (4/3) pi (8 *10^-2)^3 ]
= -33 µC * { 8^3 /19^3}
Now you can do the problem at radius = 8 cm
4 pi r^2 E = Qin / eo
4 pi (8*10^-2)^2 E = -33 µC * { 8^3 /19^3} / eo
by the way
eo is about 8.85*10^-12 C^2 /(N m^2)

part b is the same with new radius, 12 not 8
in fact the charge inside your sphere goes up with r^3 but the area goes up with r^2 so in there the field is proportional to r
12/8 = 1.5
so answer to b = 1.5 * answer to a

part c use the whole charge -33 uC but the radius = 0.37 meters