What is the divergence of a unit vector not in the r direction?

[SiggyYo]

Hi guys,

I've run across a problem. In finding the potential energy between two electrical quadrupoles, I've come across the expression for the energy as follows:

U_{Q}=\frac{3Q_{0}}{4r^{4}}\left[(\hat{k}\cdot \nabla)(5(\hat{k}\cdot \hat{r})^3-2(\hat{k}\cdot \hat{r})^2-(\hat{k}\cdot \hat{r}))\right],

where \hat{k} is the orientation of the quadrupoles, and \hat{r} is the direction between the quadrupoles.

If I let \hat{r} be in the \hat{z}-direction, I get

U_{Q}=\frac{3Q_{0}}{4r^{4}}\left[(\hat{k}\cdot \nabla)(5(\cos{\theta})^3-2(\cos{\theta})^2-(\cos{\theta}))\right].

My problem now is, that I don't know what to do about the divergence of the \hat{k}-vector. I would like to do the differentiation in cartesian coordinates, but have them translated into spherical polar coordinates. I know, that the result should probably involve a \frac{1}{r}-factor, but I can't seem to do it right. I've tried to rewrite \hat{k} in polar coordinates and tried using the chain rule on the derivative, but I get 3 as an answer. So I don't know if the initial expression is wrong, or I just don't know how to take the derivative. Can anyone please help?

Thanks,

 

[chiro]

Hey SiggyYo.

Have you tried representing a transformation between spherical and cartesian?

(Example: for (r,theta) -> (x,y) we have y/x = arctan(theta) and x^2 + y^2 = r^2 which can be used to get (x,y)).

 

[SiggyYo]

Thank you chiro for the quick response.

I am afraid I don't know what you mean. Wouldn't I just obtain the usual
x=r\sin{\theta}\cos{\phi}
y=r\sin{\theta}\sin{\phi}
z=r\cos{\theta}?

Also, I want \hat{k} to be a unit vector, which gives me r=1. How do I take this into account, when trying to get a result with a factor of \frac{1}{r}? I am really lost on this one :P

 

[chiro]

If k is a unit vector, then I don't think you will have any extra terms.

I'm not really sure what you are doing or trying to say: you have a conversion from polar to R^3 and provided the formula is correct, you should be able to plug these definitions in.

Also is the r term in your equation related to some vector in polar or is it some other variable?