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## Homework Statement

Consider the following electric field:

[itex]

\vec{E}=\frac{\rho }{3\varepsilon _{0}}\vec{r}

[/itex] where [itex]r\leq R[/itex]

and [itex]

\vec{E}=\frac{\rho R^3 }{3\varepsilon _{0}r^2}\hat{e_{r}} [/itex]

where [itex] r>R

[/itex]

(a) calculate the divergence of the electric field in the two regions

(b) calculate the electric flux through a sphere of radius r<R and show it is equal to:

[itex]\int_{0}^{r}r^2 dr[/itex] [itex]\int_{0}^{\pi }\sin \theta d\theta [/itex][itex]\int_{0}^{2\pi }d\phi \frac{\rho }{\varepsilon _{0}}[/itex]

(c) calculate the electric flux through a sphere of radius r>R and show it's equal to:

[itex]\int_{0}^{R}r^2 dr[/itex] [itex]\int_{0}^{\pi }\sin \theta d\theta [/itex][itex]\int_{0}^{2\pi }d\phi \frac{\rho }{\varepsilon _{0}}[/itex]

## Homework Equations

Divergence in spherical coordinates: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

## The Attempt at a Solution

So far I have only been able to do part a.

Both regions of the electric field only have [itex] E_{r}[/itex] components which simplifies the dot product.

So the div for reg. 1 is

[itex] \frac{1}{r^2}\frac{3\rho r^2}{3\varepsilon _{0}} = \frac{\rho }{\varepsilon _{0}}[/itex]

and for reg 2 the divergence is 0 since [itex]\frac{\partial }{\partial r} \frac{r^2\rho R^3}{3r^2\varepsilon _{0}}=0[/itex]

The flux of an E in a region would be [itex]\triangledown \cdot d\vec{A}[/itex]. I can see in the volume integrals I am supposed to get that the terms in the integrands are the elements of the general differential surface area element of a spherical region. I am just stuck as to how to proceed and use the divergence theorem to show parts b and c.

Any help and assistance will be greatly appreciated!