Use Gauss’s Law to prove that the electric field anywhere inside the hollow of a charged
spherical shell must be zero.
My attempt:
$$\int \mathbf{E}\cdot \mathbf{dA} = \frac{q_{net}}{e}$$
$$\int E \ dAcos\theta = \frac{q_{net}}{e}$$
$$E \int dA = \frac{q_{net}}{e}$$
$E\ 4\pi r^2 = \frac{q_{net}}{e}$ and since it is a hollow of a charged spherical shell the $q_{net}$ or $q_{in}$ is $0$ so: $E = 0$.
Is my reasoning on this problem correct? Essentially $E$ is $0$ because there is no charge enclosed.
Best Answer
You may have forgotten to consider the case where $\vec E \perp\vec A$. Then, also flux is zero. But, it is easy to tell using symmetry that then $\vec E$ would form closed loops which is not permissible. Hence, $E$ has to be zero units.
And yes, your reasoning is correct. You can show this for any (imaginary) shell inside your shell. Hence, Electric Field is zero everywhere.