Week 4: Capacitance 145
Dielectric Response of an Insulator in an Electric Field
Now that we understand whateachatom in an insulating material does when the material is placed
in an external field, let’s try to understand what the materialas a wholedoes – in particular, what
happens to the electric field inside, which is now thesumof the external field and the field produced
by all of those dipoles!
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−−StopSbottom+σ−σρ = 0E 0 S tArea AFigure 46: A lattice of atoms polarized by an external electric field.In figure 46, we see an imaginary lattice of atoms, all polarized by an external field in the direction
indicated. Note well that we’ve erased thedetailsof even our simple model – we represent each atom
as a neutral object with a small dipole moment where “some” chargeis split by “some” distance
by the generalprocessderived and discussed in the previous section. We’ve drawn severalpossible
Gaussian Surfaces inside the material.
Now let use Gauss’s Law. On theinside, if we draw any Gaussian SurfaceSlarge enough to
contain “many atoms”, since the atoms are neutral the average charge inside will bezero^45.
Note that even where it contains an extra charge or two of either sign by splitting an atom,
those charges are almost always paired with charges above or belowon the neighboring atoms and
the bulk remains neutral, with an average charge densityρ≈0. The interior atoms, then, do not
directly modify the average field.
This isnottrue on thesurface. If we draw a Gaussian surfaceStopso that it just contains the
upper half of the polarized atoms we see that it contains a nonzero positive charge; inside a similar
surfaceSbottomon the lower surface there is an equal and opposite negative charge. These charges
make up asurface charge layerwith a surface charge density±σbthat is directly proportional toE,
the net field in the medium.
Note Well:I put a subscript “b” onσto indicate that this kind of “surface charge” produced
by the polarization ofneutral insulator atoms or moleculeswhere the plus and minus charge
is “bound” together and not “free” to move as it is in a conductor is generally referred to asbound
(^45) If it contained an integer number of whole atoms, it would be exactly zero. If the surface cuts through atoms to
include or exclude some of their charge, the surplus charge is limited to be some fraction of the charge on the atoms
on the surface. But the number of atoms on the surface scales with the characteristic length scale of the volumeDlike
D^2 where the volume inside the surface scales likeD^3 , so theaveragecharge scales smoothly to zero as the volume
gets larger.