- #1
skrat
- 748
- 8
Homework Statement
Find ##\alpha ## and ##p## so that ##\nabla \times \vec{A}=0## and ##\nabla \cdot \vec{A}=0##, where in ##\vec{A}=r^{-p}[\vec{n}(\vec{n}\vec{r})-\alpha n^2\vec{r}]## vector ##\vec{n}## is constant.
Homework Equations
The Attempt at a Solution
##\nabla \times \vec{A}=0##
##\nabla \times \vec{A}=\nabla \times [r^{-p}\vec{n}(\vec{n}\vec{r})-r^{-p}\alpha n^2\vec{r}]=##
##=r^{-p}(\vec{r}\vec{n})\nabla \times \vec{n}+\nabla(r^{-p}(\vec{r}\vec{n}))\times \vec{n}-r^{-p}\alpha n^2\nabla \times \vec{r}-\nabla(r^{-p}\alpha n^2)\times \vec{r}=##
##=0+[(\vec{r}\vec{n})\nabla r^{-p}+\nabla(\vec{r}\vec{n})r^{-p}]\times \vec{n}-0-[r^{-p}\alpha n^2\nabla \times \vec{r}+\alpha n^2\nabla r^{-p}\times \vec{r}]=##
##=(\vec{r}\vec{n})(-p)r^{-p-2}\vec{r}\times \vec{n}##
##\nabla \cdot \vec{A}=0##
##\nabla \cdot \vec{A}=\nabla [r^{-p}\vec{n}(\vec{n}\vec{r})-r^{-p}\alpha n^2\vec{r}]=##
##=\nabla (r^{-p}(\vec{n}\vec{r}))\vec{n}-r^{-p}\alpha n^2\nabla \vec{r}-\nabla(r^{-p}\alpha n^2)\vec{r}=##
##=(r^{-p}\vec{n}-pr^{-p-2}(\vec{r}\vec{n})\vec{r})\vec{n}-3\alpha n^2 r^{-p}+p\alpha n^2r^{-p-1}##
I know something is wrong, I just don't know what and where :(