Potential energy of a continous charge distribution

[asdf60]

How exactly does one find the potential energy of a charge distribution? More precisely, how does one get over the 1/r term in the integral goes crazy near r=0? Purcell says it is possible, but I'm not seeing how for an continuous distribution this is possible.

Consider for a line of length L with linear charge density p. Let's start just by finding the potential at on end of the line. It should be integral from 0 to L of (p*dx / x). Needless to say, the integral doesn't exist. What am I doing wrong?

 

[pseudovector]

The line has to have finite cross-section, otherwise the integral blows up. Moreover, you can only assign a potential to objects of finite charge and dimensions.

 

[asdf60]

what about for something like a conducting volume, where the charge is distributed over the surface (and hence density is in terms of area not volume)?

 

[pseudovector]

Again, the surface has to have finite thickness. If you have some charge distribution spread over some region in space, then that region must have finite dimensions, or else you'll get places with infinite charge density (charge to volume ratio), and the integral over such 'singularities' will blow up.