Homework Help: Vector potential with current density

1. Dec 14, 2012

Faust90

1. The problem statement, all variables and given/known data
Hey,
I got the current density $\vec{j}=\frac{Q}{4\pi R^2}\delta(r-R)\vec{\omega}\times\vec{r}$ and now I should calculate the vector potential:
\vec{A}(\vec{r})=\frac{1}{4\pi}\int\frac{j(\vec{r})}{|r-r'|}.

2. Relevant equations
3. The attempt at a solution
here my attempt till now:
http://phymat.de/physics.png [Broken]
I'm really not sure how to go on now. Is this right what I wrote there?

Last edited by a moderator: May 6, 2017
2. Dec 14, 2012

TSny

In the second equation of the notes, should the vector r in the numerator have a prime?

[EDIT: Also, do $\theta$ and $\phi$ refer to the unprimed position or the primed position?]

Last edited: Dec 14, 2012
3. Dec 14, 2012

Faust90

Hi,

Edit: I should only calculate the potential A(r) on the z-axis (for r=z e_z). But I dont know how this can be helpful.

Last edited: Dec 14, 2012
4. Dec 14, 2012

TSny

Ahh. If $\vec{\omega}$ is also along the z-axis, then life is looking Wunderbar!

5. Dec 15, 2012

Faust90

Hi, thanks,
Yes Omega is also along the z-Axis, but I don't know how this could help me. Didn't I have to solve the integral first?

6. Dec 15, 2012

TSny

I don't think so. You can get the answer from symmetry considerations. Note that the current $\vec{j}(r') d^3r'$ in a small volume element makes an infinitesitmal contribution to the vector potential at $r$ of amount $d\vec{A}(r)$, and $d\vec{A}(r)$ has the same direction as $\vec{j}(r')$.

Do you see that each nonzero value of $\vec{j}(r') d^3r'$ is parallel to the xy plane? So, the total value of $\vec{A}(r)$ must be parallel to the xy plane at any observation point $r$.

For this problem the observation point is at some z on the z-axis. Note that the current distribution $\vec{j}(r')$ is axially symmetric about the z axis. So $\vec{A}(z)$ would have to point perpendicular to the z-axis and yet be rotationally invariant about the z-axis. There is only one possibility for the value of $\vec{A}$ at z.