(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.
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
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,
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.
According to Gauss's Theorem,
Here, charge enclosed by the gaussian surface,
From equations (i) and (ii), we get
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, C
2 = 2C
1
When the capacitors are connected in parallel,
Total capacitance will be, C
P = C
1 + C
2 = 3 C
1Energy stored in the capacitor,
When the capacitors are connected in series,
Energy stored in the capacitor,
Given that, energy stored in both the cases is same.
That is,
Hence, the result.
