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

1. Aug 13, 2012

### LionelHutz

1. The problem statement, all variables and given/known data

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

2. Relevant equations

V = (1/4∏ε)∫(ρ/r)d$\tau$

3. The attempt at a solution

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

2. Aug 13, 2012

### Oxvillian

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

3. Aug 13, 2012

### vela

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

4. Aug 13, 2012

### LionelHutz

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

5. Aug 13, 2012

### vela

Staff Emeritus
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.

6. Aug 14, 2012

### LionelHutz

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

7. Aug 14, 2012

### LionelHutz

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

8. Aug 14, 2012

### 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

9. Aug 14, 2012

### vela

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

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