# Divergence of a/r^2

1. Sep 15, 2014

### oddjobmj

1. The problem statement, all variables and given/known data
Calculate the divergence and curl of $\vec{E}$=α$\frac{\vec{r}}{r^2}$

2. Relevant equations

Div($\vec{E}$)=$\vec{∇}$°$\vec{E}$
Div($\vec{E}$)=$\vec{∇}$x$\vec{E}$

Table of coordinate conversions, div, and curl:
http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

3. The attempt at a solution
My confusion is stemming from the spherical coordinates. I believe I have the divergence down because the result is a scalar so the direction doesn't matter (phi & theta) so it's just a matter of the magnitude of the field at any given point which is dependent only on the distance from the origin. Or can I not make this assumption? I'm not sure how to bring phi and theta into the mix when dealing with the curl.

For the divergence I know that $\vec{r}$=r$\hat{r}$ so $\vec{E}$ simplifies to:

α$\frac{\hat{r}}{r}$

Using the table linked above to find the form of the divergence in spherical coordinates I believe I can ignore the theta and phi 'contributions' because divergence will depend only on r.

div(E)=$\frac{α}{r^2}$$\frac{∂(r)}{∂r}$=$\frac{α}{r^2}$

I think the curl should be zero by observation but I don't know how to show this. Can I do the same as with divergence and ignore the theta and phi contributions because this is simply a function of r?

Thank you!

2. Sep 15, 2014

### ZetaOfThree

Just use the formula in the table for the curl in spherical coordinates. Then just make sure you do the derivatives correctly. Easy.

3. Sep 15, 2014

### oddjobmj

I'm sorry for not being more clear. Referencing the table in specific I don't know what A$\theta$ and A$\phi$ are. That's what is confusing me.

4. Sep 15, 2014

### ZetaOfThree

Those are the components of $\vec{A}$ in the $\hat{\theta}$ and $\hat{\phi}$ directions. In your case, $\vec{A}=\vec{E}=a\frac{\vec{r}}{r^2}$.

5. Sep 15, 2014

### oddjobmj

edit: sorry, accidental double post

6. Sep 15, 2014

### oddjobmj

Right, I understand that A is a placeholder for E in the table. It is the components of E in those directions that I don't know how to find.

7. Sep 15, 2014

### nrqed

In general, $\vec{E} = E_r \hat{r} + E_\theta \hat{\theta} + E_\phi \hat{\phi}$

8. Sep 15, 2014

### oddjobmj

Right, I'm trying to find out what Er$\hat{r}$ and so on are functionally (starting from the value given for $\vec{E}$ in the problem). I can't take the partial derivative of the general form for E in the direction of theta without breaking E down into the relevant components first. I am given an actual value for E which is a function of r.

9. Sep 15, 2014

### ZetaOfThree

You should read up on spherical coordinates, it will be very helpful. The $\theta$ and $\phi$ components of $\vec{E}$ are zero because it only points in the radial direction and $E_r=\frac{a}{r}$.

10. Sep 15, 2014

### nrqed

You are told that $\vec{E}= a \frac{\vec{r}}{r^2}$. By comparing with the general form, what do you conclude about $E_\theta$ and $E_\phi$ ?

11. Sep 15, 2014

### oddjobmj

Which is why I assumed by observation that the curl would be zero. I just wanted to verify that I didn't have to consider a general representation of phi and theta, to be complete, as they vary for any arbitrary r. For some reason I was thinking back to r in cartesian coordinates and projecting it onto some equivalent to x, y, and z for theta and phi. Sorry for the confusion!

Thank you

12. Sep 15, 2014

### ZetaOfThree

This is wrong. Divergence absolutely depends of the direction of $\vec{E}$.

You really shouldn't intuit your way through these exercises, you need to do them nrqed's way to find the components of $\vec{E}$ and then plug them into your formulas from the table.

Spherical coordinates is an orthogonal coordinate system, meaning that $\hat{r}$, $\hat{\theta}$, and $\hat{\phi}$ are perpendicular, so their projections onto each other are zero.

13. Sep 15, 2014

### oddjobmj

This seems to contradict your earlier statement. In this case E only depends on r. I considered Er which is effectively the function they gave me for E and used the partial derivative equation from the table and the only non-zero component was the one dependent on r. In other words, the partial with respect to theta is zero because only r shows up in the equation which is not a function of theta and thus E is a constant with respect to theta (and phi). The partial of a constant is zero. The calculated divergence in this case is listed above in my original post (non-zero). Do you disagree with the result?

I didn't -just- intuit my way through the problem but I did try to think about the problem and compare that to my results to make sure that I understand what is going on and that the results make sense.

Last edited: Sep 15, 2014