Potential energy of a continous charge distribution

Click For Summary

Homework Help Overview

The discussion revolves around calculating the potential energy of a continuous charge distribution, specifically focusing on the challenges posed by the 1/r term in the integral as it approaches r=0. The original poster seeks clarification on how to handle this issue, particularly for a line charge with a linear charge density.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to evaluate the potential at one end of a line charge by integrating the linear charge density, questioning the validity of the integral due to its divergence at r=0.
  • Some participants suggest that the line must have a finite cross-section to avoid divergence in the integral.
  • Questions arise regarding the treatment of charge distributions, such as those on a conducting surface, and how their dimensionality affects the potential calculation.

Discussion Status

The discussion is exploring the implications of finite dimensions on charge distributions and the resulting potential calculations. Participants are examining different scenarios and questioning the assumptions about charge density and dimensionality, but no consensus has been reached regarding a specific method or resolution.

Contextual Notes

Participants emphasize the necessity of finite dimensions for charge distributions to prevent infinite charge density and divergence in integrals. The discussion highlights the constraints imposed by the nature of continuous charge distributions and their geometric properties.

asdf60
Messages
81
Reaction score
0
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?
 
Physics news on Phys.org
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.
 
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)?
 
Last edited:
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.
 

Similar threads

Electrostatic Potential Energy of a Sphere/Shell of Charge
  • · Replies 2 ·
Replies
2
Views
4K
The Energy of a Continuous Charge Distribution (Griffiths EM Sect. 2.4.3 3rd ed)
  • · Replies 2 ·
Replies
2
Views
3K
Why is the electric force the negative of the position derivative of EPE?
  • · Replies 5 ·
Replies
5
Views
2K
A particle with spin 1/2 in a potential well
  • · Replies 5 ·
Replies
5
Views
2K
What Is the Dipole Moment of a Paired Charged Ring System?
  • · Replies 1 ·
Replies
1
Views
2K
Why Does Gauss' Law Yield Different Results for Spherical Charge Distributions?
  • · Replies 9 ·
Replies
9
Views
2K
Finite dual disk capacitor: estimating charge distribution
  • · Replies 1 ·
Replies
1
Views
2K
How Do You Calculate the Electrostatic Energy of a Hollow Conducting Sphere?
  • · Replies 5 ·
Replies
5
Views
2K
Computing dipole moment from charge density
  • · Replies 7 ·
Replies
7
Views
2K
Electric potential due to shell containing a charge at an offset outside
  • · Replies 5 ·
Replies
5
Views
873