Electric field inside a solid sphere

[roam]

Homework Statement

We have a uniformly charged solid sphere whose radius is R and whose total charge is q. I'm trying to find the electric field inside a (r<R).

The correct answer must be:

$E=\frac{1}{4 \pi \epsilon_0} \frac{q}{R^3} r \hat{r}$

How did they get that answer?

The Attempt at a Solution

Since the sphere is not a shell, the E is not 0. And it should be found by considering a concentric Gaussian sphere with radius smaller than R. So by using Gauss's law for electric fields in the integral form we obtain:

$\oint E . da = E \oint da = E (4 \pi r^2) = \frac{q}{\epsilon_0}$

$\therefore E = \frac{q}{4 \pi \epsilon_0 r^2}$

For E outside the sphere we use r>R, and inside the sphere we use r<R. But why is my answer so different from the correct result?

Any help is greatly appreciated.

 

[TSny]

$\oint E . da = \frac{q}{\epsilon_0}$

In Gauss' law as stated above, what is the interpretation of q?

 

[AGNuke]

In the first answer, q is the charge spread out throughout the solid sphere, while in your attempt, q must be the charge enclosed by the gaussian surface.

 

[roam]

In Gauss' law as stated above, what is the interpretation of q?

Thank you very much for the hint! The q in Gauss's law is the amount of charge inside the Gaussian surface. The q in the question is the total amount of charge spread across a volume of 4/3 π R³.

I found the charge per unit volume by q dividing by the volume to get:

$q_{per \ unit \ volume}=\frac{q}{\frac{4}{3} \pi R^3}$

And to find the total charge

Then I multiplied this by 4/3πr² to find the total charge in the Gaussian surface of that volume. I got

$q=\frac{qr^3}{R^3}$

Substituting this back into Gauss's equation I got the right answer. I hope this was the correct thinking and approach.

 

[TSny]

Good work

 

[roam]

Thanks a lot for your help! :)