A nonconducting spherical shell, with an inner radius of 4 cm and an outer radius of 6 cm, has charge spread non uniformly through its volume between its inner and outer surfaces. The volume charge density ρ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell ρ = b/r where r is the distance in meters from the center of the shell and b = 3 μ C/m2. What is the net charge in the shell?

In other words a infinitesimal segment dV caries the charge

dQ = ρ dV

Let dV be a spherical shell between between r and (r + dr):

dV = (4π/3) · ((r + dr) ² - r³)

= (4π/3) · (r³ + 3·r²·dr + 3·r· (dr) ² + / dr) ³ - r³)

= (4π/3) · (3·r²·dr + 3·r· (dr) ² + / dr) ³)

drop higher order terms

= 4·π·r²·dr

To get total charge integrate over the whole volume of your object, i. e.

from ri to ra:

Q = ∫ dQ = ∫ ρ dV

= ∫ri→ra { (b/r) ·4·π·r² } dr

= ∫ri→ra { 4·π·b·r } dr

= 2·π·b· (ra² - ri²)

With given parameters:

Q = 2·π · 3µC/m²· ((6cm) ² - (4cm) ²)

= 2·π · 3*10⁻⁶C/m²· ((6*10⁻²m) ² - (4*10⁻²m) ²)

= 3.77*10⁻⁸C

= 37.7nC