Someone argues: "Plates of width and gap are held at a constant by a battery while a…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter · Electric fields in matter

The Boundary-Value Problem, Ch Energy and force in dielectrics

Problem

Someone argues: "Plates of width

b=0.080\,\mathrm{m}

and gap

d=0.80\,\mathrm{mm}

are held at a constant

V=1500\,\mathrm{V}

by a battery while a dielectric slab (

\chi_e=0.80

) filling the gap slides in a distance

x

. The stored energy is

W(x)=\tfrac12C(x)V^2

with

C(x)=\varepsilon_0b(L_p+\chi_ex)/d

, so

dW/dx=\varepsilon_0\chi_ebV^2/(2d) > 0

: the energy rises as

x

increases. A system always moves so as to lower its stored energy, so the field must be pushing the slab back out, i.e.

F=-dW/dx < 0

." ((a)) Find the error in this reasoning. ((b)) Give the correct magnitude and direction of the force on the slab, and state whether it depends on

x