by now we have learnt that electromagnetic
phenomena 
electromagnetic phenomena is described completely
by maxwell's equations so let's write them
they are the gauss's law that tells you divergence
of e is equal to rho over epsilon zero curl
of e which is paradise law is equal to minus
d b d t curl of b depends both on the regular
current j plus displacement current which
is arising out of the change in electric field
with respect to times epsilon zero d e d t
and the forth equation is divergence of b
is zero
let us now see explore this equations a better
in this lecture and see what they predict
for example if i were to take a situation
in free space ah let's see in ah ah i mean
some space where all these fields are coming
and there's no current here no charge here
only fields are coming from outside so then
i would have if i would describe e n b divergence
of e is zero because i am taking no current
inside so this inside this volume curl of
e would be equal to minus d b d t curl of
b will be equal to mu zero epsilon zero d
e d t and divergence of b would be zero so
let us see when these fields come in what
happens so let there be an electric field
e inside which is changing with time
so this e which is changing with time so let's
write that d e d t is not equal to zero will
give rise to a magnetic field 
it gives rise to magnetic field which will
also change with times coming out of something
which is changing arbitrarily in time so this
magnetic field b t is also changing with time
and this changing magnetic field so that means
d b d t is not equal to zero this change in
magnetic field implies e t again it gives
rise to another field that e t gives b again
and so we keep on going on all right where
do we stop
for example when i consider the problem of
displacement current in a capacitor in the
previous lecture we said this change in electric
field between the plates is giving rise to
a magnetic field and we stop did not even
consider that that magnetic field which is
changing gives rise to another field again
similarly in a problem where we were calculating
the induced electric field applying for by
applying for a faraday's law for example in
a solenoid where the current is changing we
stop by by calculating the electric field
and we did not bother that electric field
is changing and that will give rise to another
magnetic
where do we stop how do we decided that so
we were using in all this something called
the quasistatic approximation where we stopped
after calculating the magnetic field or electric
field and did not go beyond to the next step
so i want to spend some time in understanding
this quasistatic approximation so let's see
what we did for example we did something in
which is solenoid whose carrying a current
which was time dependent i t and therefore
the magnetic field inside this b was changing
and that we said by faraday's law gave rise
to an electric field going around um direction
i am just making arbitrarily could be the
other way similarly in the other example what
i did i took a parallel plate capacitor and
i said there is a current which is coming
in which is i t this current is time dependent
and therefore electric field in between electric
field in between e changes the time which
gives rise to a magnetic field circular magnetic
field which is coming up but we stopped at
that magnetic field we did not go beyond that
that that may also give rise to further an
an an additional electrical field why did
we stop there what gives us the right to do
so we will analyze that in this lecture under
quasistatic approximation these two calculations
have been done under that approximation and
we want to understand when we can apply this
so do this let us again give rise look at
the equations time dependent equations that
tell us that this fields give rise to each
other so we have curl of e which is minus
d b d t curl of b which is equal to mu zero
epsilon zero d e d t and let me write mu zero
times epsilon zero as one over c square d
e d t where c is the speed of light because
light is electromagnetic wave and seen naturally
into all this
so we are going to write it as one over c
square now let me assumed that the length
scale in the problem that we are solving in
this is l what it means is if i am looking
at the capacitor that the problem i saw in
the previous lecture the capacitor may be
of the size of you know a few centimetres
ten centimetres or something i may be observing
fields at a distance of few meters so length
scale would be a few meters two let's assume
that the time scale is of the order of tau
again what we mean by that is let's suppose
i am applying a signal or electric field which
is changing in time the time period is omega
then for that that signal let me write this
e which is e zero sin omega t then the time
scale tau would be of the order of two pi
over omega that is the scale we are talking
about
and third related quantity would be that maybe
the system is moving with some velocity and
there velocity is of the order of a l over
tau so these three things are there and let
us see under what circumstances how should
l and tau or all this thing should be related
so that we can apply this quazi static approximation
i will write these equation again curl of
b is plus mu zero epsilon zero d e d t which
is one over c square d e d t and i have curl
of e which is equal to minus d b d t so with
the time length scale l time scale tau and
velocity or speed v i can write this equation
roughly as curl of b is like b divided by
l magnitude of b divided by l that's the magnitude
curl of b and this is roughly equal to one
over c square d e d t so it will be magnitude
of e divided by tau
similarly on the other equation i have magnitude
of e divided by l of the order of magnitude
of b divided by tau and let's see how we play
out of this let's take that example of a solenoid
in the solenoid there is current i t changing
in with time scale tau and therefore this
change in magnetic field gives rise to the
electric field so we use this equation and
i am observing it at some distance l so we
going to say that e is going to be of the
order of l over tau times p whatever the magnetic
field is this will be the induced electric
field so let me write this induced
this induced electric field is also changing
in time and therefore this gives rise to this
new magnetic field so let's write it b induced
that is the further induced due to this change
in electric field b induced divided by l is
going to be of the order of one over c square
that e induced divided by tau so b induced
is of the order of l over tau c square e induced
let me remind you again since this is a new
slide i am in this region where there is a
solenoid i am looking at some distances l
the induced electric field and then consequent
induced magnetic field due to this b inside
which is changing in time
so we had all ready found that e induced is
of the order of l over tau times is b t that
is changing originally in the solenoid and
therefore b induced is going to be of the
order of l over tau c square times l over
tau times b t which is l over tau c square
b t so let's see what is happening i have
an original b t which i will call b zero t
this gives rise to an induced field let me
call it e one t which gives rise to of field
b one t and then further like this this is
related to b as l over tau b and this is related
to the original b as l over tau c square b
so b one t is l over tau c square p t right
so if b one t is of the order of l over tau
c square b zero t if l over tau c is much
much much less than one then b one t can b
ignored and that is why you can we ignore
and that is why we kept ignoring it because
you're talking about distances much much smaller
right then tau c c is the very large c is
a order of ten raise to eight meters per second
is actually three times ten raise to eight
meters per second
so unless you are really talking large distances
or very short time period so that c tau is
very small the the the b one is going to be
almost negligible i can apply similar argument
the other way the other way was we took this
capacitor in which this current i t was coming
in and there was this magnetic field being
produced so originally i had e t let's call
it e zero t which is giving rise to ah b one
t which in turn again will give rise to another
e one t and so on again i can now argue that
b one t because i have dell cross b is equal
to mu zero epsilon zero d e zero d t i can
argue from this that b one t i am going to
have b one t over l is equal to one over c
square tau e zero t which gives me b one t
of the order of l over c square tau e zero
t
and again using curl of e equals minus d v
d t i get e one which is equal to l over tau
c square e zero t so again we see that if
l is much much much smaller than tau c i can
stop at this order and see what the effect
is finally if l over c tau is very small that
means now let me write l over tau as some
speed divided by c then the v or c is much
much much less than one then also i can use
this kind of approximation for example it
could be the case of one charge particle moving
with speed v in that case all replaced l by
tau which is equal to this speed now another
manifestation of this displacement current
is going to be precisely this example if there
is a charged particle which is let's say moving
with respect to moving the speed v then if
i had this equation curl of b equals mu zero
j only the free current
and if i want to take stokes theorem around
a region where the charge particle has pass
forms of charged particles up here then i
would have b dot d l by stokes theorem is
equal to mu zero i but i is zero through this
area because charge particle is not there
it is moving somewhere else and therefore
this would imply that b is zero on this curl
whereas in reality it is not that arises because
the displacement current because of this charge
there is this electric field since charges
moving electric field also changes with time
and that produces that magnetic field i will
give that as an assignment problem
