# Finding the charge density,when Electric field intensity is given.

1. Aug 16, 2014

### Nero26

1. The problem statement, all variables and given/known data

Given, E=ar20/r2 (mV/m) in free space,find ρv at the point (3,-4,1)(cm).
where, ρv=Electric charge density per unit volume
E=Electric field intensity
2. Relevant equations

∇.(εE)=ρv

3. The attempt at a solution

ε∇.E= -2$\frac{20ε}{r^3}$
for point (3,-4,1)(cm) r=$\sqrt{26}$x10-2 m
∴ρv=-2.67nC/m3

But the answer given was -1.42nC/m3

I'm trying to know whether the approach was correct or not. I couldn't figure out any reason for the problem.

PS:The problem is adapted from "Fundamentals of Engineering Electromagnetics" by David K Cheng, pp90,Ex-3.4,2nd Edition

2. Aug 17, 2014

### TSny

3. Aug 17, 2014

### Nero26

I still can't get it.Here, ar is the unit vector along radius of sphere. E is independent of θ and ø so their partial derivatives of E will be zero.
∴ ε∇.E=-$\frac{40ε}{r^3}$,so where am I doing wrong?

4. Aug 17, 2014

### Orodruin

Staff Emeritus
The expression you are using for the divergence in spherical coordinates is not correct. It is not just the radial derivative of the radial component. Did you check the links provided?

5. Aug 17, 2014

### Nero26

Thanks all, Now I've got it.I was wrong in interpreting the question, ar is meant for cylindrical coordinates and aR for spherical coordinates in my book.

6. Aug 24, 2014

### rude man

Good thing, because I was about to point out that there is no unique solution if ar
had been the unit vector in spherical.

For example,
ρv = qδ(x)δ(y)δ(z), with q = 20/k, k = 9e9 SI, then ρv(3,-4,1 cm.) = 0.

An infinite number of alternative ρv would also exist, all satisfying ρv(3,-4,1) = 0.