A large plane sheet carries a uniform negative surface charge density. To find its field, Gauss’s law is applied to a closed cylinder whose axis is normal to the sheet. The two flat faces of the cylinder are on opposite sides of the sheet, and each has area \(A\). The field at each face has magnitude \(E\). The total electric flux through the cylinder is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(-2EA\), inward through the two flat faces
By symmetry, the field of the sheet is normal to it on both sides. For a negative sheet the field points towards the sheet on each side. The curved surface is parallel to the field, so no flux passes through it. Flux leaving the closed surface is taken as positive.
Option (1) zero, because the flux entering one face leaves through the other — Incorrect. Flux in equal to flux out happens in a uniform field with no charge inside. Here the field enters through both faces, and the cylinder encloses negative charge.
Option (2) \( 2EA \), outward through the two flat faces — Incorrect. This is the result for a positive sheet, whose field points away from the sheet. For a negative sheet the field points into the cylinder at both faces.
Option (3) \( -EA \), inward through one flat face only — Incorrect. The faces are on opposite sides of the sheet, and the field enters through each of them. Both faces carry flux.
Option (4) \( -2EA \), inward through the two flat faces — Correct. At each flat face the field points into the cylinder, so each face has a flux \( -EA \). The total is \( -2EA \). This fits Gauss’s law, since the charge enclosed is negative.
NCERT (2026-27):
Field due to a uniformly charged infinite plane sheet – “… As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux.”
Field due to a uniformly charged infinite plane sheet – “E is directed away from the plate if σ is positive and toward the plate if σ is negative. …”