Week 5: Resistance 157
Advanced Problem 8.
You are given a square parallel plate capacitor of sideLand plate separationdand a slab of
dielectric material with relative dielectric permittivityǫrthat exactly fills the volume between the
plates if fully inserted. At the moment, however, the slab is insertedonly a distancex. The capacitor
has aconstant voltageV 0 connected across it that cando workadding charge to or taking charge
away from the capacitor as the slab is inserted or removed (!).
a) Find the potential energy of the partially filled capacitor, as a function ofV 0 , L, d, ǫ 0 , ǫrandx.b) Draw a simple picture involving the probable bound charge distribution on the partially in-
serted slab and the plates. Do you, based on this distribution, expect the slab to be pulled
into or expelled from between the plates?c) Find the amount the potential energy of the capacitor changeswhen one inserts the slab an
additional amount ∆x. Does the energy increase or decrease? Is this result surprising given
your gut level physical expectation from the picture? (Don’t worry, your gut is correct...)d) To find the “missing energy”, determine the amount of work doneby the voltage source as
one inserts the capacitor an additional amount ∆x. Note that this is related to the additional
charge that flows onto the capacitor at constant voltage and represents thedecreasein the
potential energy of thevoltage source.e) By using
Fx=−dU
dx
whereUis thetotalpotential energy of the voltage source and capacitor, find the force on the
partially inserted place. Does the force pull the dielectric slab in (to fill the plate volume) or
does it push it out from between the platesafter all?