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:

$$Q=\frac{4\pi\rho _{0}r'^7}{7R^4}$$

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:

$$E=\frac{\rho _{0}r'^5}{7R^4\epsilon _{0}}$$

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