Electric Fields inside Conductors & Gauss's Law

[pandabear27]

I am just starting to learn AP Physics C E&M, and I have some questions.

1. If I have a metal conductor sphere (neutral or charged), why is it that the electric field inside is always equal to 0?

2. Also, a question about Gauss's Law.
$$E \cdot A = \frac{Q_{enc}}{\varepsilon_0}$$
Since $Q_{enc}$ only accounts for the charges INSIDE of the Gaussian surface, when we solve for E, then does that only account for the electric field that is created by the enclosed charges, or also charges that could be outside of the Gaussian surface?

I would really appreciate any help. Thank you!!

 

[anuttarasammyak]

Hello.

2. all charges both inside and outside the surface contribute to $\mathbf{E}$.
$$\int_S \mathbf{E}\cdot d\mathbf{S}$$ is proportional to enclosed charges.

 

[pandabear27]

Got it, thank you!!

 

[Ibix]

1. If I have a metal conductor sphere (neutral or charged), why is it that the electric field inside is always equal to 0?

In a conductor, charges can move. What will happen to them if there is an E field inside the conductor?

2. Also, a question about Gauss's Law.
$$E \cdot A = \frac{Q_{enc}}{\varepsilon_0}$$

Gauss' Law doesn't say that - it says $$\oint \vec E\cdot d\vec A=\frac{Q_{\mathrm{enc}}}{\varepsilon_0}$$The integral is important because Gauss is making a statement about the flux through a closed surface, not just some random non-closed surface.

Notice that a field line from a charge inside the surface will cross the surface once, and every field line from the charge will have the same sign for $\vec E\cdot d\vec A$ where it crosses. Thus the charge's contribution to the integral must be non-zero.

However, a field line originating outside the surface will cross twice, once inward and once outward, meaning it will make two opposite-signed contributions to the integral. They don't necessarily cancel exactly, but once you do the integral all these contributions do net to zero. So it's only charges inside the surface that matter for calculating the total flux.