- #1
mjordan2nd
- 177
- 1
Homework Statement
This is a three part problem. My first task is to calculate the divergence of [tex]\vec{r}/r^{a}[/tex]. Next, I am to calculate its curl. Then I'm supposed to find the charge density that would produce the field
[tex]\vec{E}=\frac{q\vec{r}}{4\pi\epsilon_{0}r^{a}}[/tex]
The Attempt at a Solution
I calculated the curl by first calculating the surface integral through a sphere of r/r^a:
[tex]\oint\frac{\vec{r}}{r^{a}}r^{2}d\phi sin\theta d\theta \hat{r}=\frac{4\pi\left|\vec{r}\right|}{r^{a-2}}[/tex]
By the divergence theorem [tex]\frac{4\pi\left|\vec{r}\right|}{r^{a-2}}=\int_{v}\nabla\bullet\frac{\vec{r}}{r^{a}}d\tau=\int_{v}\frac{4\pi\left|\vec{r}\right|}{r^{a-2}}\delta^{3}(\vec{r})d\tau[/tex]
Which implies that the divergence of r/r^a is [tex]\frac{4\pi\delta^{3}(\vec{r})}{r^{a-3}}[/tex].
This would eventually give me the charge density as [tex]\frac{q\delta^{3}(r)}{r^{a-3}}[/tex]
Does this look correct? Any help would be appreciated. Thanks in advance.