Question

Write the statement of Gauss’s law for electrostatics. Derive an expression for electric field due to an uniformly charged infinite non – conducting sheet at a point near to it. Draw suitable diagram.  

Answer 1

Gauss theorem states that the total flux through a closed surface is 1/ε₀ times the net charge

enclosed by the closed surface.

Mathematically, it can be expressed as

Consider a thin, infinite plane sheet of charge with uniform surface charge density σ. We wish to calculate its electric field at a point P at distance r from it.

By symmetry, electric field E points outwards normal to the sheet. Also, it must have same magnitude and opposite direction at two points P and P' equidistant from the sheet and on opposite sides. We choose cylindrical Gaussian surface of cross-sectional area A and length 2r with its axis perpendicular to the sheet.

As the lines of force are parallel to the curved surface of the cylinder, the flux through the curved surface is zero. The flux through the plane-end faces of the cylinder is

ϕ_E = EA + EA = 2 EA  (A = cross sectional area of plane-end faces)

Charge enclosed by the Gaussian surface,

Clearly, E is independent of r, the distance from the plane sheet.

(i)  If the sheet is positively charged (σ > 0), the field is directed away from it.

(ii)  If the sheet is negatively charged (σ > 0), the field is directed towards it. 

Variation of electric field (E) due to a uniformly charged infinite plane sheet with distance (r)