146 Week 4: Capacitance
charge. We will only consider boundsurfacechargeσbin this course as the most common important
case, but in principle one can generate bound bulk charge distributionsρb.
In contrast, the charge we have discussed up to this point is primarily is “bare”, isolated, normal,
unbalanced charge, the charge that is directly producing electric fields or potentials that we have
evaluated various ways. In contexts where both are present, wewill usually differentiate them by
means of “f” and “b” subscripts: A net free charge might be referred to asQf and a net bound
charge might be referred to asQb. Now, back to the thread of our discussion.
Let us understand this in this particularly simple case, where the upper and lower surfaces are
conveniently perpendicular to the field and the cross-section of the material is rectangular. The
total dipole moment of the system is given by the total charge on the upper or lower surface, times
that thickness (recall that all the charges in between sum to zero). That is:
psystem=Qsurfacet= (σbA)t=P V=P(At) (245)(all in the direction of the field) or clearly:
σb=P (246)This argument is actually more general than one might suspect – if you think about it in terms of
calculus you can see why it would be true for less conveniently shapedobjects in a uniform field and
how it might be changed to accomodate an angle between the polarization density direction at a
surface and the normal to the surface there. In any event, themodifications of the field we deduce
from this below are reasonably general and hold for arbitrary objects in nearly arbitrary fields^46.
+ + + + + + + + + + + + + + + +− − − − − − − − − − − − − − − −E 0+ + + + + + + + + + + + + + + +− − − − − − − − − − − − − − − −E 0Er+σ−σE = E 0 /εrFigure 47: The polarized material generates areaction fieldErthatopposesthe applied field and
partially cancels it, making the total field in the material smaller. A dielectric material thusreduces
the applied electric field inside the material.
(^46) Truly advanced students might look ahead at a book on electrodynamics and learn about how this statement is
not precisely true and how polarization density itself bothsatisfies certain partial differential equations and how our
entire picture at this level relies on alinearresponse that is at best an (often quite good) approximation.