Field Inside a Dielectric Video Lecture - Civil Engineering (CE)

Section four point two three the field
inside a dielectric from Griffiths
introduction to electrodynamics second
edition this is Jonathan Gardner if you
appreciate what I'm doing be sure to
like and share the video comments can go
in a video response or in the comments
below and if I go too quickly you can
just rewind and play it again so this
section deals with the difference
between pure dipoles and physical
dipoles if you remember and in the long
run when you're far away from a physical
dipole it behaves exactly like a pure
dipole but when you get close to it the
field is a lot different and if you if
you think about like if you shrunk
yourself to atomic size and we're
looking inside any material
there's polarization going on then the
electric field varies wildly depending
on whether you're close to a negative
charge or a positive charge but just
like the arguments for density when we
know matter is composed of point
particles we can still treat materials
as if they're made out of you know
uniformly dense material that it has
like in the infinitesimal limit has a
uniform density so here I'm going to
explain to you why that's a perfectly
good assumption for fields inside of a
dielectric that even though the actual
field inside the dielectric of any given
point could vary wildly if we're dealing
with you know averages of large enough
spaces it's very consistent it can be
very consistent so the argument
basically goes like this so let's take
any material which have some
polarization going on and we're going to
look at point P which is right there and
if we blow up point P we're not going to
look at a actual point here we actually
look at a small sphere okay radius R
centered at Point P okay
and the the electric field at the point
P is actually going to be the average of
the electric field of everything going
on the outside of point P so all this
stuff out here plus the electric field
of the stuff going on in the inside so
let me write that out there so the
electric field at point p is going to be
the electric field due to the outside
stuff at point p plus the electric field
due to the inside stuff at point p okay
so let's calculate the electric field
due to the outside well the potential
due to all the stuff corn in the outside
can be written out as 1 over 4 PI
epsilon nought times the integral of all
the stuff on the outside this including
this point or the sphere on the inside
of the polarization of that material dot
R hat over R squared D tau
okay so that's the outside and this
integral includes everything but this
sphere okay so that only gives us a part
of the electric field now we need to
think about what's going on do the stuff
on the inside of the sphere well if you
did problem three point four one if you
didn't go back and do it it's it's a
good one to have to fight with for a
while you would have found out that the
average of the fields so if you take
this to this volume and you average the
field over that volume okay and you take
any point configuration for a charge
configuration inside the sphere okay
then the average of the inside average
is going to be equal to negative one
over four PI epsilon naught times the
dipole moment of each given particle
which is just the distance from the
center there to the particle let me draw
that what that would that would look
like so if we had a single particle here
then the dipole moment would be this
vector times the charge of that particle
multiple particles we just add them up
and we get the total dipole moment
because we're just
those dipole moments together over the R
cubed okay know that the the sphere
we're looking at okay so this is the the
inside is basically just the total
dipole moment times this stuff okay
which we can rewrite the total dipole
moment as that's just going to be the
polarization of that material big P
times the volume right so the total
dipole moment is the polarization times
the volume because polarization is the
dipole moment per unit volume okay so
plugging that in the electric field the
average electric field on the inside is
equal to negative one over three epsilon
naught times the polarization of the
stuff within the sphere now the
polarization is it's an average okay
because this is the total dipole moment
we divide that by the total volume and
we get the polarization it's constant
the polarization does not change within
that sphere and so we found that the
potential due on the inside of a sphere
of constant polarization is equal to so
the potential on the inside is 1 over 4
PI epsilon naught integral of the
polarization dot R hat over R squared D
tau I'm sorry I feel foolish that's just
over the sphere so insight this is the
inside integral ok now when you take
this Plus that this is everything
outside this is everything in sight you
see that these equations that they've
just merge together perfectly okay and
so you're left with the result that the
potential for any given point if you're
averaging over these spheres will give
you will be determined by this formula
of the polarization dot R hat over R
squared D tau okay so the the thinking
here is kind of settled with the two key
parts the the first key part is finding
the average of the electric field okay
and it happens to be that the dipole
moment the sum of the dipole moments of
all the particles in there it doesn't
matter where they are whatever their
dipole moment is you add it up and you
get the average of the electric field at
the center of the point which is what
we're interested in and the second key
point is that the potential on the
inside and the potential on the outside
so the facility outside is determined by
what's going on on the outside and then
the only thing we're missing is it
potentially the inside well the petition
on the inside looks exactly like the
potential on the outside except for the
dipole moment at that point right the
average dipole moment at that point I
guess is the or the the density of the
dipole moments at that point and so the
the potential at any given point even
though that point may be close to some
particles that have you know if you if
you get in there with your SuperDuper
microscope you'll see the electric field
to vary wildly but the average the the
average of these things are very very
consistent and you can do the same
argument for cubes or loops ellipsoids
I'm not going to do it if you do do it
I'd like to see how you do it but the
end result is that even though we're
dealing with real matter we can this
this polarization is actually very very
useful I'm very very descriptive what's
really going on I hope that helps let me
know in the comments if you're still
confused the the reasoning is kind of
subtle so anyway thanks and bye