Finding the magnetic field of a loop at far distances

[1v1Dota2RightMeow]

Homework Statement

Loop of current $I$ sitting in the xy plane. Current goes in counter clockwise direction as seen from positive z axis. Find:

a) the magnetic dipole moment
b) the approximate magnetic field at points far from the origin
c) show that, for points on the z axis, your answer is consistent with the exact field (Ex. 5.6), when z R.

Homework Equations

$\vec{m} = I \int d\vec{A}$
$\vec{A}_{dip} (\vec{r}) = \frac{\mu_0 \vec{m} \times \hat r}{4 \pi r^2}$
$\vec{B}=\nabla \times \vec{A}$

The Attempt at a Solution

a) I got that this is $I\pi R^2 \hat z$
b) I got $\vec{A}_{dip} (\vec{r}) = \frac{\mu_0 I R^2}{4 r^2}(\hat z \times \hat r)$, but I don't know how to interpret $(\hat z \times \hat r)$. I tried taking the vector product by treating them as cylindrical coordinates and using the conversion to cartesian, which resulted in $\hat \phi$. So it says that the vector potential curls around the z axis, which doesn't make sense...

Then I know I have to do $\vec{B}=\nabla \times \vec{A}$, but that's contingent on the previous part being correct.

c) ?

Any tips?

 

[Charles Link]

Hello again. Suggest you google "magnetic dipole". The Wikipedia has a good explanation. One item here that might create some confusion: In some SI units, I have seen the authors write $B=\mu_o H +\mu_o M$ instead of $B=\mu_o H +M$. In any case, try reading Wikipedia. I think the answer is often computed in spherical coordinates rather than cylindrical, but it appears Wikipedia's answers are independent of coordinate system. $\\$ editing... Also, there is a formula for the curl of a cross-product of two vectors. It can often be found on the cover or appendix of an E&M textbook. $\\$ editing... additional item... A magnetic dipole can also be modeled as a "+" pole at one end and a "-" pole at the other and the inverse square law can be used for the $H$ from each pole. The calculations using the pole model are a little simpler than Biot-Savart type integrals around the loop. I believe about a month or two ago, I verified the Wikipedia formulas using the pole method when someone else posted a question about the magnetic field of a magnetic dipole. $\\$ Here is a "link" to that discussion: https://www.physicsforums.com/threads/equation-of-magnetic-field-produced-by-a-solenoid.888895/