Question

A sphere of radius r has total charge q. The volume charge density (c/m3) within the sphere is rho(r) = c/r², where c is a constant to be determined. What is the volume charge density within the sphere?

Answer 1

The volume charge density ρ(r) within a sphere of radius R and total charge q is found by integrating the given density function ρ(r) = c/r² over the sphere's volume. The constant c is determined to be q/(4πR), resulting in a density function ρ(r) = q/(4πRr²).

Step-by-step explanation:

To find the volume charge density within a sphere where charge q is distributed with a density ρ(r) = c/r², we must determine the constant c. We can do this by integrating the charge density over the volume of the sphere to get the total charge q. The volume of an infinitesimal spherical shell at a radius r' and thickness dr' is given by 4πr'² dr'. Thus, the charge dq in that shell is the product of the volume and the charge density at r', resulting in dq = (c/r'²)4πr'² dr' = c4π dr'.

To obtain the total charge q, we integrate dq from 0 to R (the radius of the sphere), which gives us q = ∫ dq = c4π∫ dr' from 0 to R = c4πR. Solving for c, we find c = q/(4πR), and thus the volume charge density function ρ(r) becomes ρ(r) = q/(4πRr²).