Curl of Z-unit vector in spherical coordinates

[SalcinNossle]

Homework Statement

There is a sphere of magnetic material in a uniform magnetic field $\vec H_0=H_0\vec a_z$,
and after some calculations I got the magnetic moment vector $\vec M_0=M_0\vec a_z$, where $M_0$ is something that isn't important to my question. I am unsure if this magnetic moment vector is correct, as I am running into trouble when I try to figure out the equivalent volume current on the sphere:

$\vec J_{mv}=\nabla\times\vec M_0$,

Is it possible for me to take the curl of a Z-vector, in spherical coordinate system?

Edit: I made a mistake in the title, I meant to take curl of Z, not cross with unit normal of surface of the sphere. Sorry!

 

[TSny]

Hello. Are you asking if it is possible to take the curl of the unit vector $\hat{a}_z$ in spherical coordinates?

If so, the answer is yes. You can do it as an academic exercise, but that would not be the easiest way to get the answer. Note that $\hat{a}_z$ is a constant vector - it is the same vector at all points of space. So, you are taking the curl of a constant vector and the result should be immediate. Or, you can easily see what the result is by taking the curl in Cartesian coordinates rather than spherical coordinates.

If you do want to take the curl in spherical coordinates, then you would want to express $\hat{a}_z$ in terms of the spherical coordinate unit vectors $\hat{a}_r$, $\hat{a}_{\theta}$, and (if necessary) $\hat{a}_{\phi}$.

 

[SalcinNossle]

Thank you! I did take the curl in spherical coordinates for practice, and I got zero! I think it' correct.

 

[TSny]

OK. That's good.

 

[paranormal]

I would like to clarify that the Z-unit vector and the unit normal vector of the surface of a sphere are two different things. The Z-unit vector is a unit vector in the direction of the Z-axis, while the unit normal vector of the surface of a sphere is a vector perpendicular to the surface at a specific point.

To answer your question, yes, it is possible to take the curl of a Z-vector in a spherical coordinate system. In fact, the curl operation is defined in any coordinate system. However, the result may be different depending on the coordinate system used.

In your case, it seems like you are trying to calculate the equivalent volume current on the sphere using the magnetic moment vector. This can be done by taking the curl of the magnetic moment vector, as you have correctly stated. However, please note that the magnetic moment vector should be defined at every point on the sphere, not just at a single point.

In summary, it is possible to take the curl of a Z-vector in a spherical coordinate system, but it may not be relevant for your specific problem. I would suggest double-checking your calculations and making sure that the magnetic moment vector is defined correctly at every point on the sphere.