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

[FourierX]

Homework Statement

A vector field is defined by A=f(r)r
a) show that f(r) = constant/r^3 if \nabla. A = 0
b) show that \nabla. A is always equal to zero

Homework Equations

divergence and curl relations

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]

Homework Statement

A vector field is defined by A=f(r)r
a) show that f(r) = constant/r^3 if \nabla. A = 0
b) show that \nabla. A is always equal to zero

Homework Equations

divergence and curl relations

The Attempt at a Solution

I tried using spherical co-ordinates to solve this. But I am not sure if i am totally right.

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:

\nablaA = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2}f(r))

on simplifying this i got:

f(r) = \frac{-1}{2} r f'(r)

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 \mathbf{A}=f(r)\hat{r}, but the way the question is stated it is \mathbf{A}=f(r)\mathbf{r}=, or \mathbf{A}=f(r)r\hat{r}. This extra factor of r should give the answer required.