divergence and curl of vector field defined by \vec A = f(r)vec r

FourierX

1. The problem statement, all variables and given/known data
A vector field is defined by A=f(r)r
a) show that f(r) = constant/r^3 if [tex]\nabla[/tex]. A = 0
b) show that [tex]\nabla[/tex]. A is always equal to zero


2. Relevant equations
divergence and curl relations


3. The attempt at a solution
I tried using spherical co-ordinates to solve this. But I am not sure if i am totally right.

G01

I would also use spherical coordinates on this problem.

I cannot say if you are right since I haven't seen any of your work. I'll need to see some work if you want more specific advice.

olgranpappy

Certainly parts (a) and (b) can not be consistent as you have written them... i suppose part (b) should be a curl not a divergence?

FourierX

here is what i've done and where i got stuck:

[tex]\nabla[/tex]A = [tex]\frac{1}{r^{2}}[/tex] [tex]\frac{\partial}{\partial r} (r^{2}f(r))[/tex]

on simplifying this i got:

f(r) = [tex]\frac{-1}{2} r f'(r)[/tex]

then i integrated with a hope to get an expression for f(r) but did not end up with what i needed i.e. f(r) = constant/ r^{3}

nicksauce

Well the way you interpreted it, you have [tex]\mathbf{A}=f(r)\hat{r}[/tex], but the way the question is stated it is [tex]\mathbf{A}=f(r)\mathbf{r}=[/tex], or [tex]\mathbf{A}=f(r)r\hat{r}[/tex]. This extra factor of r should give the answer required.