Question

According to Gauss's law, how is the electric flux through a closed surface related to the net charge enclosed by it, and what is the flux when no charge is enclosed?

Hint:

Flux depends only on the charge trapped inside the surface, not on charges outside it.

Answer 1

Step 1. Gauss's law connects the electric flux passing through any closed surface to the total charge enclosed inside that surface.
Step 2. The law is written as \(\Phi_E = \oint \vec{E} \cdot d\vec{A} = \dfrac{q_{enc}}{\epsilon_0}\), where \(q_{enc}\) is the net charge enclosed and \(\epsilon_0\) is the permittivity of free space.
Step 3. This means the flux does not depend on the shape or size of the closed surface, and it does not depend on charges lying outside the surface. Only the charge trapped inside matters.
Step 4. If the closed surface encloses zero net charge, that is \(q_{enc} = 0\), then substituting into the formula gives \(\Phi_E = 0\). Any field lines entering the surface must also leave it, so the net outward flux is zero.
Answer. The flux equals the enclosed charge divided by \(\epsilon_0\), that is \(\Phi_E = q_{enc}/\epsilon_0\), and it is zero when no net charge is enclosed.