Final answer:
To calculate the volume charge density of a solid sphere with a radius r, one would integrate the product of the volume of an infinitesimal spherical shell and the charge density, which varies with the radius. The charge dq in the shell is given by dq = ar' × 4πr'^2 dr', where ar' is the varying charge density.
Step-by-step explanation:
The question pertains to the determination of the volume charge density within a solid sphere of insulating material with radius r. In physics, especially in the study of electromagnetism, volume charge density is a measure of how much electric charge is distributed within a given volume.
To find the charge in an infinitesimally thin spherical shell within the solid sphere, one would multiply the volume of the shell by the charge density at that radius. If the charge density is given by ρ(r) = ar', where a is a constant and r' is the radial distance from the center of the sphere to a point within it, then the charge dq in the shell can be expressed as dq = ar' × 4πr'^2 dr', where 4πr'^2 is the surface area of the shell and dr' is the shell's thickness.
This approach involves the concept of a Gaussian surface, which is crucial in applying Gauss's Law to find the resulting electric field inside and outside the sphere due to the non-uniform charge distribution.