I Is Leibniz integral rule allowed in this potential improper integral?

  • Thread starter Beelzedad
  • Start date
18
3
Electric potential at a point inside the charge distribution is:

##\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta}

\dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'##

where:

##\delta## is a small volume around point ##\mathbf{r}=\mathbf{r'}##
##\mathbf{r}## is coordinates of field point
##\mathbf{r'}## is coordinates of source point
##\rho (\mathbf{r'})## is the density of charge distribution

Taking the gradient of potential:

##\displaystyle \nabla \psi (\mathbf{r}) =\nabla\ \left[ \lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV' \right] =\lim\limits_{\delta \to 0} \int_{V'-\delta} \rho (\mathbf{r'})\ \nabla \left( \dfrac{1}{|\mathbf{r}-\mathbf{r'}|} \right) dV'##

In the last step, we have applied Leibniz integral rule (basic form).

The validity of this technique for improper integrals is discussed below:

The following passage from the book "Foundations of Potential Theory page 151" says the technique is not valid. But it says the equation ##\mathbf{E}=-\nabla \psi## still holds at points inside source regions ##V'##. It also gives a "little" proof of the argument.


243911

243912

243913

243914

243915


(continued below)
 
18
3
Proof:

243918

243919


Unfortunately I am not well versed in potential theory to understand this little proof that the book offers. Can anybody explain the proof in a way in which a Physics graduating student can understand?
 

Want to reply to this thread?

"Is Leibniz integral rule allowed in this potential improper integral?" You must log in or register to reply here.

Related Threads for: Is Leibniz integral rule allowed in this potential improper integral?

  • Posted
Replies
1
Views
4K
Replies
5
Views
2K
  • Posted
Replies
7
Views
3K
Replies
1
Views
13K
  • Posted
Replies
18
Views
3K
Replies
3
Views
703
Replies
1
Views
1K
  • Posted
Replies
1
Views
751

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top