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

[Nero26]

Homework Statement

Given, E=a_r20/r² (mV/m) in free space,find ρ_v at the point (3,-4,1)(cm).
where, ρ_v=Electric charge density per unit volume
E=Electric field intensity

Homework Equations

∇.(εE)=ρ_v

The Attempt at a Solution

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

Thanks for your help.

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

 

[TSny]

ε∇.E= -2$\frac{20ε}{r^3}$

You should review the divergence operator in spherical coordinates. See:

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

or

http://mathworld.wolfram.com/SphericalCoordinates.html

 

[Nero26]

You should review the divergence operator in spherical coordinates. See:

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

or

http://mathworld.wolfram.com/SphericalCoordinates.html

I still can't get it.Here, a_r 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?
Thank you for your response.

 

[Orodruin]

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?

 

[Nero26]

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?

You should review the divergence operator in spherical coordinates. See:

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

or

http://mathworld.wolfram.com/SphericalCoordinates.html

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

 

[rude man]

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

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

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

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