Is My Approach to Deriving Quantities for a Charged Sphere Correct?

[Sekonda]

Hey,

I have a series of questions on a basic charged sphere and deriving quantities such as the infinitesimal charge, the total charge and the electric field. The question is part (b) in the image below:

So I found dQ' by equating it to the surface area of a shell at a distance r' multiplied by the corresponding charge density to attain:

dQ'=\frac{4\pi \rho _{0}r'^6dr'}{R^4}

Is this right?

And then for the next part I just integrated over r' for some r'<R to attain:

Q=\frac{4\pi\rho _{0}r&#039;^7}{7R^4}

and then the last part I wish to query is my electric field magnitude, which I attained from equating the product of the electric field and area of some shell at distance r' to the charge divided by the permitivitty of free space to attain:

E=\frac{\rho _{0}r&#039;^5}{7R^4\epsilon _{0}}

Is this right?

Thanks guys!
Any feedback appreciated,
SK

 

[Sekonda]

Oh and provided I've done the above correct, how would I go about answering the next part? It's below:

Thanks for any help!
SK