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

In summary, the conversation is about finding the potential energy between two electrical quadrupoles using an expression involving the orientation of the quadrupoles and the direction between them. The speaker is having trouble with the differentiation and is looking for help in translating the expression into spherical polar coordinates. They also mention a unit vector and how it affects the result. The other person suggests using a transformation between spherical and cartesian coordinates to solve the problem.
  • #1
SiggyYo
5
0
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:

[itex]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],[/itex]

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

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

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

My problem now is, that I don't know what to do about the divergence of the [itex]\hat{k}[/itex]-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 [itex]\frac{1}{r}[/itex]-factor, but I can't seem to do it right. I've tried to rewrite [itex]\hat{k}[/itex] 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,
 
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  • #2
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)).
 
  • #3
Thank you chiro for the quick response.

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

Also, I want [itex]\hat{k}[/itex] to be a unit vector, which gives me [itex]r=1[/itex]. How do I take this into account, when trying to get a result with a factor of [itex]\frac{1}{r}[/itex]? I am really lost on this one :P
 
  • #4
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?
 

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

1. What is the meaning of divergence in physics?

The divergence of a vector field is a measure of the net flow of a vector quantity out of a given region. It is a mathematical operation that describes the spreading or contracting of a vector field at a specific point.

2. How is divergence related to the r direction?

In physics, the r direction usually refers to the radial direction in spherical coordinates. The divergence of a unit vector not in the r direction indicates the rate at which the vector field is spreading or contracting in all directions except the radial direction.

3. Why is it important to consider the divergence of a unit vector not in the r direction?

The divergence of a vector field is a fundamental concept in fluid dynamics, electromagnetism, and other areas of physics. It helps us understand the flow of fluids and the behavior of electric and magnetic fields. Considering the divergence of a unit vector not in the r direction allows us to analyze the behavior of a vector field in all directions, not just in the radial direction.

4. How is the divergence of a unit vector not in the r direction calculated?

The divergence of a unit vector not in the r direction can be calculated using the divergence operator, which is a vector calculus operator. It involves taking the dot product of the vector field with the gradient operator. The resulting scalar value represents the net flow of the vector field out of a given region.

5. What are some real-world applications of understanding the divergence of a unit vector not in the r direction?

Understanding the divergence of a vector field has many practical applications in various fields of physics and engineering. For example, in fluid dynamics, it helps us analyze the flow of fluids in pipes and channels. In electromagnetism, it is essential for understanding the behavior of electric and magnetic fields. It also has applications in weather forecasting, aerodynamics, and fluid mechanics.

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