Q4. Careful measurement of the field at the surface of a black box shows a net outward flux of 8.0 × 103 N m2/C. (a) What is the net charge inside the box? (b) If the net outward flux were zero, could you conclude that there were no charges inside? (NCERT Exercise 1.16, page 43)
(a) q = ε0ϕ = 8.854 × 10−12 × 8.0 × 103 ≈ 7.1 × 10−8 C. (b) No. Zero flux means zero net charge inside; the box could hold equal positive and negative charges (NCERT page 30).
Q5. A point charge causes a flux of −1.0 × 103 N m2/C through a spherical Gaussian surface of radius 10.0 cm centred on it. (a) What flux would pass if the radius were doubled? (b) What is the charge? (NCERT Exercise 1.19, page 43)
(a) The same, −1.0 × 103 N m2/C: the flux depends only on the enclosed charge. (b) q = ε0ϕ = 8.854 × 10−12 × (−1.0 × 103) ≈ −8.9 × 10−9 C (NCERT Eq. (1.31), page 30).
Q6. A uniformly charged conducting sphere of diameter 2.4 m has a surface charge density of 80.0 µC/m2. What is the total flux leaving its surface? (NCERT Exercise 1.21(b), page 44)
Its charge is Q = σ × 4πR2 ≈ 1.45 × 10−3 C (part (a), on the continuous-charge page), so ϕ = Q/ε0 = (1.45 × 10−3)/(8.854 × 10−12) ≈ 1.6 × 108 N m2/C (NCERT Eq. (1.31), page 30).
Q7. Assertion: the flux through a closed surface does not depend on charges outside it. Reason: the field on the surface does not depend on charges outside it. Are both true?
The assertion is true, the reason false: outside charges do change the field on the surface, but every line they send in also leaves, so they add nothing to the net flux (NCERT page 30).
Q8. A point charge q sits at one corner of a cube. Find the flux through each face of the cube.
Eight such cubes fit round the charge and together enclose it, so by symmetry each one gets q/(8ε0). The three faces that meet at the charge lie along its field lines, so their flux is zero. The other three share q/(8ε0) equally: q/(24ε0) each. (The cube itself cannot be a Gaussian surface, because the charge sits on it, but the face integrals here are understood with the singular corner removed and the removed patch shrunk to zero. The three faces through the corner have zero normal field away from that point. This is reasoning from NCERT Eq. (1.31), page 30.)