we've seen that if there is a material that
is polarized or carries polarization p we
can think of this as if for electrical problems
there is a sigma which is p dot n over the
surface so i can write some plus charges here
minus charges here where n is the normal going
out of the surface or out of the material
and charge density which is minus divergence
of p let me explicitly write a function of
r and this we call bound charge because these
are bound charges they are not free to move
and we what we saw is that we can calculate
the potential at some point or the electric
field from these charges for example the potential
was given as one or four pi epsilon zero and
integral minus del prime dot p r prime i am
writing r prime because r prime i am taking
as source point r minus r prime d v prime
plus one over four pi epsilon zero integration
p r prime dot n prime over r r minus r prime
d s prime and the corresponding electric field
e will be given as minus grad of v r so this
is a general strategy
however the materials that we are going to
deal with are going to be polarizable that
is when they are put in external feel if i
apply a field to it or put it in presence
of other charges they are going to create
a polarization and what we would like to know
is develop a general method to calculate electric
fields or the effects of these media these
media polarizable media or presence of charges
ah and the polarizable medium together what
kind of fields or potential does it give rise
to so in this context we're going to do introduce
something called the electric displacement
that facilitates solving of such problems
to understand displacement let us then write
that if i have electric field e due to the
presence of some charge distribution which
i'll call rho and a dielectric medium which
i am showing by red this is the dielectric
medium this charge distribution could be inside
it this is different from the bound charges
then del dot e divergence of e is going to
be minus this is rho r upon epsilon zero additionally
there is field due to the bound charges so
that's going to be given as minus divergence
of p that is a bound charge over epsilon zero
i can rewrite this equation as divergence
of epsilon zero e plus p is equal to this
is plus rho of r and this to distinguish it
from the bound charges arising out of polarization
i am going to call it rho free and this quantity
in brackets epsilon zero e plus p i am going
to call d or displacement 
and this satisfies the differential equation
that divergence of displacement is equal to
rho free 
if you like this is the gauss's law for d
now what is the advantage of taking displacement
you see charge free is something that i know
what i have supplied or something that we
bring in so it is good to have a quantity
that is related to that charge rather a quantity
which is induced for example if i bring a
medium like this one here and put it in front
of an electric field there's going to be induce
charges on this and i do not know what those
induce charges are and therefore i would like
to develop theory in terms of charges which
i do know and which are rho free
so we have got one quantity displacement which
is related to the free charge like this so
del dot d is equal to rho free on the other
hand the electric field is still given rise
to by charges whether bound charges or free
charges and therefore as we saw earlier can
be written as minus v r i could have also
explicitly written this as integration rho
free at r prime r minus r prime divided by
r minus r prime cubed d v prime plus integration
minus divergence of p r prime that's the bound
charge divided by r minus r prime cubed r
minus r prime d v prime plus due to surface
charges the point is that it is coming out
of certain charges so none the less what we
are seeing is that electric field at r is
arising out of these static charges and therefore
curl of e is zero what that also means that
we can calculate e from a potential v r which
we have already seen right
so i have two quantities whenever i have a
medium with polarization whenever i have a
medium both polarization so this is polarization
and some free charges rho r prime i have two
equations one is with the displacement divergence
of displacement is equal to rho free r and
curl of e is zero now recall helmholtz theorem
that says that to calculate any quantity i
need its curl as well as divergence here i
have two quantities d and e i know divergence
of d and curl of e lets ask a question is
divergence of or curl of d is zero
if curl of d was zero and it was known it
becomes like a problem of calculating electric
field then i know the divergence and curl
of d i can write d as a gradient of a potential
and do all the calculations the way we did
earlier but this is not necessarily true because
curl of d will be equal to curl of epsilon
zero e plus p which is epsilon zero curl of
e plus curl of p this is zero so i am left
with curl of p look at the picture on the
left of your screen suppose i have a material
let's take this to be y direction this to
be x direction and i have a polarization which
is in the y direction and changes 
so suppose i have p of x is equal to some
p naught e raise to minus x by a y unit vector
you will immediately see that curl of p is
not zero
the point i am trying to make is curl of p
need not be zero and therefore in general
i can write divergence of e rho free bound
charges del dot p over epsilon zero but if
there is a polarizable medium where p itself
depends on e i do not know this so let me
write this we do not know this a priory and
therefore now we we have to develop techniques
to solve problems in presence of a dielectric
medium before we do that let me now do some
calculations for electric displacements so
that you have an idea about what it is like
remember earlier we had solved a problem where
i had given you a dielectric cylinder with
height h radius r r much much much better
than h and p going up this way in this case
we showed that this was equivalent to the
cylinder with positive charges on top which
sigma equals p over epsilon zero negative
charges on the bottom with charge being minus
p upon zero and the electric field therefore
is in this direction this is e if we neglect
fringe effects on the side which i am showing
here if i neglect fringe effects e outside
anyway is zero p is zero so displacement d
will also be zero
as a second example for displacement let's
take a polarized sphere that we had talked
about earlier with polarization being uniform
and let's say this z direction and what we
showed was that this is equivalent to a sphere
with positive charge here and negative charge
on the lower side with this being given as
p cosine of theta and this gave an electric
field inside which is going down so if p is
p naught z then sigma theta was equal to p
naught cosine theta and the electric field
is minus p naught by three epsilon zero z
and this implies that displacement inside
is going to be epsilon zero e plus p which
is two thirds p zero z or two thirds p outside
e is that of a dipole where p is four pi by
three r cubed r is the radius p z and polarization
outside is zero and therefore d is simply
one over four pi p dot r sorry there is a
three here r minus p over r cubed
so in this lecture we have introduced a new
quantity called electric displacement which
is epsilon zero e plus p its divergence is
equal to three charge in the system and its
curl is not zero and it is related to to and
and we have calculated some ah displacement
for certain examples
