Chapter 26, Problem 26.53P

The general form of Gauss’s law describes how a charge creates an electric field in a material, as well as in vacuum:

${\displaystyle \int \stackrel{\to }{E}\cdot d\stackrel{\to }{A}=\frac{q_{\text{in}}}{ϵ}}$

where ϵ = κϵ₀ is the permittivity of the material. (a) A sheet with charge Q uniformly distributed over its area A is surrounded by a dielectric. Show that the sheet creates a uniform electric field at nearby points with magnitude E = Q/2Aϵ. (b) Two large sheets of area A, carrying opposite charges of equal magnitude Q, are a small distance d apart. Show that they create uniform electric field in the space between them with magnitude E = Q/Aϵ. (c) Assume the negative plate is at zero potential. Show that the positive plate is at potential Qd/Aϵ. (d) Show that the capacitance of the pair of plates is given by C = Aϵ/d = κAϵ₀/d.