Gauss theorem states that ∫limits_sE·ds = 0 . It means that

The correct answer is Option (3) → the total flux through the closed surface is zero.

Gauss's law says that the surface integral of \( \vec{E} \) over a closed surface is the total electric flux through it, and equals \( q_{\mathrm{enc}}/\varepsilon_0 \). A zero value means the net charge enclosed is zero, so as much flux enters the surface as leaves it.

Option (1) E is zero on the surface. — Incorrect. A charge outside the surface produces a field on it, yet the total flux is zero; the field need not vanish for the integral to be zero.

Option (2) E is perpendicular to the surface. — Incorrect. A field perpendicular to the surface gives the largest flux through each element, not a zero total.

Option (3) the total flux through the closed surface is zero. — Correct. The integral itself is the total flux through the closed surface, so it is zero.

Option (4) the flux is only going out of the surface. — Incorrect. Flux that only goes out would make the integral positive, not zero.

NCERT (2026-27):
Gauss’s Law – “… We state Gauss’s law without proof: Electric flux through a closed surface S

    \( \displaystyle = \frac{q}{\varepsilon_0} \qquad (1.31) \)

q = total charge enclosed by S. The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface.”