• Support PF! Buy your school textbooks, materials and every day products Here!

Divergence in spherical coordinates

  • Thread starter wifi
  • Start date
  • #1
115
1
Problem:

For the vector function [tex]\vec{F}(\vec{r})=\frac{r\hat{r}}{(r^2+{\epsilon}^2)^{3/2}}[/tex]

a. Calculate the divergence of ##\vec{F}(\vec{r})##, and sketch a plot of the divergence as a function ##r##, for ##\epsilon##<<1, ##\epsilon##≈1 , and ##\epsilon##>>1.

b. Calculate the flux of ##\vec{F}## outward through a sphere of radius R centered at the origin, and verify that it is equal to the integral of the divergence inside the sphere.

c. Show that the flux is ##4 \pi## (independent of R) in the limit ##\epsilon \rightarrow 0##.

Solution:

a. Calculating the divergence is simply a matter of plugging the function correctly into the spherical coordinates divergence formula. I get [tex]F_r=\frac{r}{(r^2+{\epsilon}^2)^{3/2}} \Rightarrow \frac{\partial F}{\partial r}=3r^2(r^2+{\epsilon}^2)^{-3/2}-3r^4(r^2+{\epsilon}^2)^{-5/2} [/tex]

Therefore,

[tex]\nabla \cdot \vec{F}= 3(r^2+{\epsilon}^2)^{-3/2}-3r^2(r^2+{\epsilon}^2)^{-5/2} [/tex]

So for the different values of ##\epsilon## do I just substitute?
 

Answers and Replies

  • #2
115
1
Also can someone point me in the right direction for parts b. and c.? Thanks.
 
  • #3
115
1
Anyone?
 
  • #4
ehild
Homework Helper
15,427
1,827
Check your ∂F/∂r.

Anyway, the end result is correct.

As for b), what is flux?

ehild
 
Last edited:

Related Threads on Divergence in spherical coordinates

  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
3
Views
1K
Replies
7
Views
9K
Replies
6
Views
654
Replies
2
Views
12K
Replies
17
Views
7K
Replies
5
Views
10K
  • Last Post
Replies
7
Views
12K
Replies
3
Views
28K
Replies
4
Views
2K
Top