physics problem, field of an infinite plane

Question

Without doing any math, explain why the field of an infinite plane, uniformly charged, must be constant. Likewise, explain why the field produced by an infinite, uniformly charged wire must be proportional to 1/r.

Answer 1

You usually use Gauss's Law and Gaussian surfaces to derive these fields: For a constant field E = ε₀Q/A for any surface that is perpendicular to the field lines.

For an infinite plate, the E field goes directly away from the plate with area of σ C/m² , If you draw a cylinder of radius, r and height h through the plate, the field lines go through the end caps. So, E = ε₀σπr²/(2πr²)

or E = ε₀σ/2 or E is constant.

For an infinite wire with charge λ C/m, you draw a cylinder along the wire. Now the E field is radially outwards through the sides of the cylinder. E = ε₀hλ/(2πrh) or E = ε₀λ/(2r) or E is proportional to 1/r