# Divergence in spherical coordinates

Problem:

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

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 $$F_r=\frac{r}{(r^2+{\epsilon}^2)^{3/2}} \Rightarrow \frac{\partial F}{\partial r}=3r^2(r^2+{\epsilon}^2)^{-3/2}-3r^4(r^2+{\epsilon}^2)^{-5/2}$$

Therefore,

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

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

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Also can someone point me in the right direction for parts b. and c.? Thanks.

Anyone?

ehild
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