Finding flux from electric field

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1. Oct 30, 2015

Titan97

1. The problem statement, all variables and given/known data
If $\vec{E}=k\frac{x\hat i +y\hat j}{x^2+y^2}$, find flux through a sphere of radius R centered at origin.

2. Relevant equations
$\int E.da=\int(\nabla\cdot E)\cdot da$

3. The attempt at a solution
I was able to solve this problem without finding divergence of electric field.
If $\vec{r}= {x\hat i +y\hat j}$.
Then, $$E=k\frac{\vec{r}}{r^2}=k\frac{\hat r}{r}$$
This is like the electric field due to an infinite line charge which is given by $$E=\frac{\lambda}{2\pi\epsilon_0 r}$$
So the field specified in question can be assumed to be from an infinitely long charged wire.
I can easily find the flux using Gauss' law by finding the charged enclosed by the spherical surface.

But when I tried finding the divergence of E, it came out to be zero.
I used the formula $$E=\frac{\partial E}{\partial x}+\frac{\partial E}{\partial y}$$

2. Oct 30, 2015

TSny

The divergence of E is indeed zero at all points other than points on the z axis (i.e., on the line charge itself). However, you can see that $\partial E/ \partial x$ and $\partial E/ \partial y$ are undefined on the z axis.

This is similar to the example of the E field of a single point charge where the divergence is zero everywhere except at the location of the point charge (where the divergence is undefined). To deal with the divergence at the location of the charge, you can use a dirac delta function.

See http://www.physicspages.com/2011/11/14/dirac-delta-function-in-three-dimensions/

3. Oct 30, 2015

Titan97

Divergence is zero in this case at all points (except at x-axis). Does that mean flux is zero through a spherical surface?

4. Oct 30, 2015

TSny

If the divergence is zero at every point inside a spherical surface, then the net flux through the surface would be zero. For this problem of a line charge along z axis and spherical surface centered at the origin, there are points inside the spherical surface where the divergence is not zero (namely, at points on the z axis where the divergence is undefined). The net flux through the surface is not zero, as you have shown using Gauss' law.

5. Oct 31, 2015

Titan97

I tried finding divergence in cylindrical coordinates. It worked perfectly. Why?

6. Oct 31, 2015

TSny

In cylindrical coordinates the divergence of $\vec{E}$ for this problem is $$\frac{1}{r} \frac{\partial (r E_r)}{\partial r}$$ The partial derivative is zero everywhere, but the $\frac{1}{r}$ factor is undefined on the z axis. So, the divergence does not have a definite value on the z axis.

7. Oct 31, 2015

Titan97

Thats not what I got.
$r=\sqrt{x^2+y^2}$
$x=r\cos\theta$
$y=r\sin\theta$
$\hat i=(\cos\theta) \hat r-(\sin\theta) \hat \theta$
$\hat j=(\sin\theta) \hat r+(\cos\theta) \hat \theta$
$$E=a\frac{x\hat i+y\hat j}{x^2+y^2}=a\frac{(r\cos\theta)(\cos\theta \hat r-\sin\theta \hat \theta)+(r\sin\theta)(\sin\theta \hat r+\cos\theta \hat \theta)}{r^2}$$
$$E=a\frac{r\hat r}{r^2}=a\frac{\hat r}{r}$$
Hence, $$\nabla\cdot E=-a\frac{1}{r^2}$$
So, divergence is non zero in cylindrical coordinates. But should divergence change if the coordinate system changes? The vector field is still the same.

8. Oct 31, 2015

TSny

9. Oct 31, 2015

Titan97

Why is it $\frac{1}{r}\frac{\partial rA}{\partial r}$?

10. Oct 31, 2015

TSny

It is due to the fact that the unit vectors in cylindrical coordinates change direction as you move around in space. Thus, when taking derivatives of a vector you need to take derivatives of the unit vectors as well as the coefficients of the unit vectors. The derivation of the form of the divergence is found in many texts.

11. Nov 5, 2015

Titan97

@TSny , $$E=a\frac{\hat r}{r}$$
$$\nabla\cdot E=a\nabla\cdot\frac{\hat r}{r}$$
This gives
$$\nabla\cdot E=ak\delta(r)$$
How do I find $k$?

12. Nov 5, 2015

TSny

To express $\nabla\cdot\frac{\hat r}{r}$ in terms of $\delta(r)$ where $r$ is the radial coordinate in cylindrical coordinates, examine the integral $\int \nabla\cdot\frac{\hat r}{r} dV$ over a cylindrical volume centered on the z-axis. Use the divergence theorem to get the answer by converting the integral to a surface integral. Then try to express $\nabla\cdot\frac{\hat r}{r}$ in terms of $\delta(r)$ such that when you do the volume integral you get the same result as the surface integral. You will see that you must take $\nabla\cdot\frac{\hat r}{r}$ proportional to $\delta(r)/r$ rather than just proportional to $\delta(r)$.

13. Nov 5, 2015

Titan97

$$\int\int\int \nabla\cdot\frac{\hat r}{r}dV=\int\int\frac{\hat r}{r}\cdot dS$$
Let the cylinder have centre at origin and axis along z-axis.There are three surfaces.
Since field is along radial direction, flux through the plane surfaces will be zero. There will only be flux through the curved surface.
$$dS=rd\theta dz \hat r$$
$$\int\int\frac{\hat r}{r}\cdot dS=\int_{-z}^z\int_0^{2\pi}d\theta dz=4\pi z$$

14. Nov 5, 2015

TSny

OK.

Now evaluate $\int\int\int \nabla\cdot\frac{\hat r}{r}dV$ for the same volume and force it to equal $4\pi z$ by deducing the necessary relation between $\nabla\cdot\frac{\hat r}{r}$ and $\delta(r)$.

15. Nov 6, 2015

Titan97

$\int\int\int \nabla\cdot\frac{\hat{r}}{r}dV=0$ for all $r\neq 0$
$\int\int\int \nabla\cdot\frac{\hat{r}}{r}dV=4\pi z$ for $r=0$

So $$\nabla\cdot\frac{\hat{r}}{r}=4\pi z\delta(r)$$

16. Nov 6, 2015

TSny

This isn't correct. In $\int\int\int (\nabla\cdot\frac{\hat{r}}{r})dV$ you need to write out the volume element $dV$ in cylindrical coordinates and integrate over the same cylinder that you used for evaluating the surface integral.

17. Nov 6, 2015

Titan97

How can I evaluate that integral? It will come out to be zero since the divergence is zero.

18. Nov 6, 2015

Titan97

How can I evaluate that integral? It will come out to be zero since the divergence is zero.

19. Nov 6, 2015

TSny

We are assuming that the divergence is proportional to a delta function $\delta(r)$ so that it will be zero everywhere except at $r = 0$, where it is singular. (It is good that it is singular at $r=0$ since we know that at the location of a line charge the volume charge density is singular.) So, you can try letting $\vec{\nabla} \cdot \frac{\hat{r}}{r} = k \delta(r)$ and then try to determine k such that the volume integral will equal $4 \pi z$.

You should find that you can't make it work. But, in the process you might see how to express $\vec{\nabla} \cdot \frac{\hat{r}}{r}$ in terms of $\delta(r)$ to make it work. In a previous post, I alluded to what you should find.

EDIT: Alternately, you can try expressing $\vec{\nabla} \cdot \frac{\hat{r}}{r}$ in terms of Cartesian coordinate delta functions. For example, suppose you try $\vec{\nabla} \cdot \frac{\hat{r}}{r} = k \delta(x) \delta(y)$. Can you find $k$ such that the volume integral of $\vec{\nabla} \cdot \frac{\hat{r}}{r}$ over your cylindrical volume equals $4 \pi z$?

Last edited: Nov 6, 2015
20. Nov 7, 2015

Titan97

δ(r) is infinite at r=0. But is $\nabla\cdot\frac{\hat r}{r}$ infinite at r=0?