Question

(i) Use Gauss’s law to find the electric field due to a uniformly charged infinite plane sheet. What is the direction of field for positive and negative charge densities?

(ii) Find the ratio of the potential differences that must be applied across the parallel and series combination of two capacitors C₁ and C₂ with their capacitances in the ratio 1 : 2 so that the energy stored in the two cases becomes the same.

Solution

i) Electric field due to a uniformly charged infinite plane sheet:

Consider an infinite thin plane sheet of positive charge with a uniform charge density

straight sigma

on both sides of the sheet. Let a point be at a distance a from the sheet at which the electric field is required.
The gaussian cylinder is of area of cross section A.
Electric flux crossing the gaussian surface,

straight ϕ space equals space straight E space straight x space

Area of the cross section of the gaussian cylinder.
Here, electric lines of force are parallel to the curved surface area of the cylinder, the flux due to the electric field of the plane sheet of charge passes only through two circular sections of the cylinder.

straight ϕ space equals space straight E space straight x space 2 straight A space space bold space bold space bold. bold. bold. bold space bold left parenthesis bold i bold right parenthesis

According to Gauss's Theorem,

straight ϕ space equals space straight q over straight epsilon subscript straight o

Here, charge enclosed by the gaussian surface,

space space space space space space space straight q space equals space straight sigma space straight A therefore space space space straight ϕ space equals space σA over straight epsilon subscript straight o space space space space space bold space bold. bold. bold. bold space bold left parenthesis bold ii bold right parenthesis

From equations (i) and (ii), we get

straight E space straight x space 2 straight A space equals space σA over straight epsilon subscript straight o space space space space space space space space space straight E space equals space fraction numerator straight sigma over denominator 2 straight epsilon subscript straight o end fraction

The direction of electric field for positive charge is in the outward direction and perpendicular to the plane of infinite sheet.
Direction of electric field for negative charge is in the inward direction and perpendicular to the sheet.
ii) Given: Two capacitors are in the ratio of 1:2.
That is, C₂ = 2C₁When the capacitors are connected in parallel,
Total capacitance will be, C_P = C₁ + C₂ = 3 C₁
Energy stored in the capacitor,