Finding the potential of a charged, solid sphere using the charge density

[LionelHutz]

Homework Statement

Solid ball of charge with radius R and volume charge density ρ(r) = ρ₀r², centred at the origin.

I have already found the electric field for r<R and r>R and also the potential at the origin by using the formula:
V = -∫E.dl

Now i want to find the potential at the origin using the charge density but am at a loss for the first step

Homework Equations

V = (1/4∏ε)∫(ρ/r)d$\tau$

The Attempt at a Solution

I don't need the full solution, just the first step would be much appreciated.

 

[Oxvillian]

Find the potential at the center due to a thin shell of charge and then integrate over all the shells in the ball?

 

[vela]

Just evaluate the integral. Or is there something specific about the integral you don't understand?

 

[LionelHutz]

Yes I'm unsure of what my limits are in this case and what d$\tau$ is equal to

 

[vela]

The volume element $d\tau$ will depend on which coordinate system you choose to use. To calculate the potential at a point, you integrate over all space, but really you only need to worry about where the charge density $\rho$ doesn't vanish because everywhere else won't contribute to the integral.

 

[LionelHutz]

So would the integral from 0 to R work if d$\tau$ = r²sinθdrdd$\phi$ ?

 

[LionelHutz]

Is V = ρ₀R⁴ / 4ε₀ the answer to this problem?

 

[LionelHutz]

V = $\frac{1}{4\pi\epsilon_{0}}$$∫^{2\pi}_{0}$d$\phi$$∫^{\pi}_{0}$sin$\theta$d$\theta$$∫^{R}_{0}$ $\frac{ρ_{0}r^{2}}{r}$ r$^{2}$dr

Solving this:

V= $\frac{ρ_{0}R^{4}}{4\epsilon_{0}}$
I'm quite happy with this answer, but if it is incorrect please let me know.
Thanks for the help

 

[vela]

I haven't worked it out, but your method looks fine. Does it match the answer you got using the previous method?