Electric Field Inside a Charged Insulating Sphere

[yango_17]

Homework Statement

A charged spherical insulating shell has inner radius a and outer radius b. The charge density on the shell is ρ.

What is the magnitude of the E-field at a distance r away from the center of the shell where r < a?

Homework Equations

Gauss' Law

The Attempt at a Solution

I read through the Gauss' law chapter in my textbook, and it stated that the internal electric field of such a charged insulator is zero. Therefore, I know the answer is 0 but I do not know why. Any explanation as to why this answer is correct would be appreciated. Thanks.

 

[axmls]

Well, Gauss's law states that: $$\oint \mathbf{E} \cdot \ d \mathbf{A} = \frac{Q_{enc}}{\epsilon_0}$$ What is the enclosed charge if you use a Gaussian surface inside the cavity?

 

[yango_17]

0. Is it because there is no charge in the cavity?

 

[axmls]

Correct. If you imagine your Gaussian surface to be a sphere inside the cavity centered at the center of the larger sphere, no charge is inside (so Q_enc = 0). All the charge is in the insulated sphere in this problem.

Now, the integral being equal to 0 is not a necessary condition to declare that the electric field is 0. For instance, $\int _{-1} ^1 x \ dx = 0$, so this is not enough to declare that there is no electric field inside.
Edit: changing the argument because I saw a flaw in it. Simply put: by symmetry, the electric field should cancel out everywhere inside the sphere.

If it didn't cancel out, which direction would get precedence? Why should any direction of a perfectly symmetrical sphere get the privilege of having the electric field not cancel out?