Chapter 20, Problem 86P

Two square plates of sides ℓ are placed parallel to each other with separation d as suggested in Figure P20.86. You may assume d is much less than ℓ. The plates carry uniformly distributed static charges +Q₀ and −Q₀. A block of metal has width ℓ, length ℓ, and thickness slightly less than d. It is inserted a distance x into the space between the plates. The charges on the plates remain uniformly distributed as the block slides in. In a static situation, a metal prevents an electric field from penetrating inside it. The metal can be thought of as a perfect dielectric, with $\kappa \to \infty$. (a) Calculate the stored energy in the system as a function of x. (b) Find the direction and magnitude of the force that acts on the metallic block. (c) The area of the advancing front face of the block is essentially equal to ℓd. Considering the force on the block as acting on this face, find the stress (force per area) on it. (d) Express the energy density in the electric field between the charged plates in terms of Q₀, ℓ, d, and ϵ₀. (e) Explain how the answers to parts (c) and (d) compare with each other.

Figure P20.86