we are looking at in the previous lecture
a medium or material that has some polarization
density and therefore a net dipole moment
and we looked at physically and said that
gives rise to surface charge as well as bulk
charge exactly what this charges are what
do they produce is what we want to see this
lecture in this lecture finally that we want
to see due to this charges is the field produce
or the potential they produce our part of
learning about what kind of charges give rise
to is going to be through this quantities
only because this are quantity we are interested
in so let us take some material which is polarized
right and it carries polarization p then the
potential we solve in the previous lecture
at are due to this polarization is going to
be given as one over four pi epsilon zero
p at r prime dot r minus r prime over r minus
r prime cube d v prime
d v prime means i am integrating over the
volume over which this charges is given now
to understand this we will let me use this
trick if i take gradient with respect to the
prime variable of r minus r prime this you
can easily show i will leave this an excise
for you come out to be r minus r prime divided
by r minus r prime cubed and therefore i can
replace this quantity here in circle by red
by this quantity here and write v r as one
over four pi epsilon zero integration p r
prime dot gradient with respect to this prime
variable one over r minus r prime d v prime
now let's do some manipulations p r prime.dot
gradient of one over r minus r prime i am
going to write as divergence of p r prime
divided by r minus r prime minus this is divergences
to prime variable prime divergence of p r
prime divided by divided by r minus r prime
all right this this is a better identity in
one d it is very easy to see in three d also
is very easy to see
and you can this work it out and therefore
i get the potential v r to be one over four
pi epsilon zero integration to divergence
with respect to time variable p r prime divided
by r minus r prime d v prime minus one over
four pi epsilon zero integration divergence
with respect to prime variable of r prime
divided by r minus r prime d v prime remember
we are looking at this polarize material i
can use divergence theorem for this first
term and write this as one over four pi epsilon
zero integration one over r minus r prime
p r prime dot n prime n prime is this unit
vector out of the volume which is the direction
of the surface element d s plus one over four
pi epsilon zero integration i will take the
minus sign inside minus dell prime dot p r
prime divergence of population density divide
by r minus r prime d v prime
let me write this in slightly different form
and you see the what this quantities mean
one over four pi epsilon zero if there was
a surface charge sigma on a similar volume
right if there was a surface charge sigma
and bulk charge row and sigma i will write
the charge density as one over r minus r prime
sigma r prime integrated over the surface
plus one over four hours non zero integration
row r prime over r minus r prime d v prime
d s prime d s prime what does they tell you
this tells you that as for as electrostatic
properties electrostatic potential or electric
field are concerned
this charge density sigma is equivalent to
into this quantity and the bulk density row
is equivalent to this quantity here divergence
of p so i am going to interpret in terms of
as percent as electricity or electrostatic
properties are concerned that given a bulk
material with some charge polarization density
p r n dot p at the surface is like a surface
charge density and sense this charges due
to polarization or not free to move i'm going
to call this bound charges and similarly the
divergence of p r i don't have to have time
here is equivalent to ah bulk charges row
part r again this are not be charges i am
going to call this bound you see this is consistent
with the physical picture we develop earlier
that the the polarization gave rise to a surface
charge as well as bond charge and you can
see divergence of actually how p varies in
space if it varies then gives rise to boundary
charged
p was a constant there will be no bound bulk
charge but there is still be surface charge
so as i electric charge potential is concern
p is equivalent to bound charge surface as
well as bulk and therefore the electric field
will also be which which is very inter potential
will also be equivalent to as if being rise
to by this bound charges let's look at some
example suppose i have a cylinder of height
h radius such that r or is much much much
greater than h so it's like a very thin disc
and it carries polarize a constant polarization
p then as far as electrostatic potential or
field is concerned this is going to be equivalent
to a surface on a bound charge and p is a
constant there's no bound charge inside
however on the upper side is and coming out
so p dot n it will give me positive charge
over with upper surface with sigma being sincere
p and r the same direction sigma b equal to
p dot n which is p and the lower surface will
give negative charge because here n in this
direction and p is in this direction so it
will give me negative charge minus sigma equals
minus p so this thing shown in red out here
is equivalent to two large plates with plus
and minus surface charges therefore the electric
field inside is going to be in the direction
opposite of p the magnitude be p over epsilon
zero let's example one
let's take example two suppose instead i have
a thin rod again with p be given inside with
this length being much much much larger than
the radius here a here then i can consider
this equivalent to against p is a constant
there is going to be no charge inside no bulk
charge inside but the upper and lower ends
again give me p as surface charge density
and since this is very a small compare area
on the sides very small compared to the length
i can consider this to be equivalent two be
charges p times pi a square separated by a
distance p a square l and the field lines
are going to be look like this in this approximation
as at third example let's take a sphere which
carries a constant polarization for a sphere
the n is same as unit vector r in spherical
coordinate n is same as unit vector r and
polarization p is p z then the surface charge
sigma is going to be p.dot n which is p z
dot r and if you call from the lecture on
this spherical coordinates this is p cosine
of theta on this side n is like this your
side so what you see is sigma is positive
as for as cosine theta is positive it goes
smaller and smaller as you go towards cosine
theta equals zero becomes précised the zero
on the equator and then becomes negative on
the lower side with the dependence p cosine
theta
how about the bulk is bound bound row is divergence
p which is zero because p is a constant now
this is equivalent if you recall equivalent
to a problem we solve earlier which i had
sigma a cosigner zero cosine theta and what
did that give me an e field in the opposite
direction which magnitude towards sigma zero
word three epsilon zero and outside it becomes
the field of two slightly separated charged
sphere positive and negative here and become
like a dipole filed which i didn't do and
solve that example but now you can see that
ah where we derive it it was two opposite
charge sphere like this way is shifted which
for outside sphere this becomes a dipole
so this will also give me the previously the
same field it will give me a constant e inside
which is equal to p over three zero and outside
will give me a dipole field of a dipole p
dipole movement p which is four pi by p r
cube b so what we have seen in this lecture
is how i can think of the polarization as
a charge distribution is divergence give me
a the bulk charge distribution bulk charge
density and stock product within where n is
the normal going out of the dielectric gives
ne the surface charge density and form that
i can calculate potential and field
