The absence of an E-field inside a hollow charged conductor might be justified by a method different from that of Sec. 4.1. Consider the possible existence of clectric flux inside such a conductor which (a) begins and ends on the conductor, (b) begins on the conductor and ends in space, (c) closes on itself or (d) does something other than (a), (b) or \((\mathrm{c})\).

Electric flux is the measure of the total electric field passing through a closed surface, like the hollow charged conductor. Recall that the electric flux is given by the integral of the electric field over the surface area: \(\Phi = \int \vec{E} \cdot d\vec{A}\)

Gauss's Law states that the electric flux passing through any closed surface is equal to the total charge enclosed by the surface divided by the permittivity of free space. \(\Phi = \frac{Q}{\varepsilon_0}\) Let's apply this law to all four cases. (a) If the electric flux begins and ends on the conductor, it means the closed surface does not contain any charges within it. Since the surface encloses no charges, according to Gauss's Law, the electric field inside the hollow portion of the conductor is zero. (b) If the electric flux begins on the conductor and ends in space, this would create an electric field line that does not terminate or close on another charge. This is not possible, as electric field lines must end on a negative charge or go to infinity. So this case is invalid. (c) If the electric flux closes on itself, it means there would be a net charge inside the closed loop. However, we know that charges reside on the surface of a conductor, not inside its hollow portion. Therefore, this case cannot happen and the electric field inside the hollow conductor is zero. (d) For any other possibility different from (a), (b), or (c), it is highly probable that they also violate some basic principles of electric fields and charges. Since the cases (a), (b), and (c) have shown that electric filed inside a hollow conductor is zero, the possibility of any other case generating a nonzero electric field seems unlikely.

Considering all the possibilities of electric flux and using Gauss's Law, we have shown that there is no electric field present inside a hollow charged conductor, which is consistent with the conclusions from Section 4.1.