2. (a) Verify Gauss\'s law for an infinite sheet of charge occupying the xy plan

Question

2. (a) Verify Gauss's law for an infinite sheet of charge occupying the xy plane (with charge density sigma per unit area). Do so by explicitly computing the electric field at coordinate z. Compare with the answer you would get using Gauss's law. (b) Suppose the sheet of charge is finite along the y direction: so it occupies the region -L < y < L (but all x, so it is an infinite "strip" of charge). Explicitly compute the field at points (0,0,z) for all z. At what value of z (in terms of L) does the Gauss's law result cease to work? What can you say about the field at distances |z| >> L?

Explanation / Answer

2a) According to Gauss law eletric flux in any closed surface is defined by the total charge divided bypermittivity.

In this case we consider X-Y plane and that means for symmetry we take Z=0. The infiinite plane has charge density=sigma/area (given). Now we consider the infinite plane has electric field which is normal to the plane and function of Z axis only. Therefore its direction depends on sigma(+ or -). If we consider a cylindrical gauss surface then total charge in the enclosed infinite plane is Qe= sigma*A where A is the area of the cylinder. For net flux the ends and sides are considered but sides are parallel so for two ends net flux= 2EA

As the electric field is uniform it is not related to Z. Then using Gauss law net flux= Qe/permittivity=2EA

or sigma*A/epsilon =2EA [epsilon=permittivity

or electric field E=sigma/2epsilon

which proves the gauss law that electric field=total charge/permittivity.

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