Final answer:
The question relates to finding the potential difference between a solid cylindrical conductor and a concentric cylindrical shell with varying charge density. Integration of the charge density is used to calculate the electric field, which is then integrated to find the potential difference.
Step-by-step explanation:
Understanding the Potential Difference Between Cylindrical Conductors
The question concerns the calculation of the potential difference between a solid cylindrical conductor and a concentric cylindrical shell with charges of equal magnitude but opposite sign. Given that the volume charge density within the insulating material is proportional to the radial position, we must integrate to find the total charge enclosed.
In a scenario where we have a charge density that varies with the radius as ρ(r)=αr, integration is necessary to determine the electric field and hence the potential difference. For a cylindrical geometry, the charge within an infinitesimal cylindrical shell of radius r' and thickness dr' is given by dq = αr * 2πr'l dr', where l is the length of the cylinder. The electric field can be found by using Gauss's Law, and the potential difference can be obtained by integrating the electric field over the distance between the two conductors.