Hello.
So we discussed the electromagnetism before
the advent of Maxwell.
And we have seen the Faraday's law howah time
varying magnetic field leads to some curl
in the electric field; and we have seen how
that has modified the equations relevant to
electromagnetism.
And we have written down all 4 equations relevant
to electromagnetism before Maxwell.
Now, what made Maxwell modify these equations
in the context of electromagnetism?
Let us consider that situation.
So there was an inconsistency, in order to
fix that inconsistency, reconcile everything
properly.
Maxwell has introduced a new term in the Ampere's
law, let us see what kind of situation led
to that inconsistency .
So, if we consider the curl of the electric
field , that is given by minus del B del t.
Now if we take the divergence of this quantity
then , its this which is nothing, but the
divergence of minus del B del t and that is
since the divergence and the time derivative
they commute, we can take the time derivative
out .
So it becomes the negative time derivative
of the divergence of the magnetic field and
the divergence of the magnetic field is 0
so this quantity has to be 0 . But now divergence
of a curl is always 0.
So this quantity on the left-hand side must
be 0, on the right-hand side we have found
this to be 0 so everything is consistent so
far . 
How about considering the other way around?
If we take the curl of a magnetic field that
is mu naught times the current density.
Now if we take the divergence of this quantity
, that is mu naught times the divergence of
the current density.
And so, divergence of a curl is always 0,
the left-hand side must be 0.
But what is the reason?
Sorry please delete this partdeletewhere I
have said the left-hand side must be 0 . So
we have a divergence of curl here andthat
equals to the divergence of current.
Now we can see that the divergence of curl
is 0 so we have the left-hand side to be 0
, but there is no reason that the divergence
of the current density would go to 0 , this
may not be 0.
Do we have an example where the divergence
of the current density is not 0 ? Let us consider
1 example , let us consider the case when
a capacitor is being charged .
So we draw a circuit here, its a parallel
plate capacitor and its connected with a battery
like this.
So the capacitor is getting charged , its
a closed loop.
During the charging of this capacitor, a current
I that is the function of t flows into this
circuit and when the capacitor is fully charged,
there is no current flowing in the circuit.
The capacitor does not allow any current anymore.
Now the current is flowing through the wire,
but not through the capacitor because in between
the capacitor with plates, we have some dielectric
material that does not allow any current to
flow through it . And now let us consider
an Amperian loop like this . 
For this Amperian loop, we can write that
the closed integral of B dot dl.
This can be given as mu naught times the current
enclosed using Ampere's law in integral form
and the current enclosed in this case is the
surface integral of J dot da where da is the
area cross-section of thisah current carrying
wire.
Nowthis is valid for any surface . We can
consider this loop so 
which area do we consider that isenclosed
by this Amperian loop?
We can consider the flat area here, but we
can also consider the area of a balloon like
this.
That is also allowed and so, if we consider
the flat surface shaded here then the current
enclosed is for the flat surface 
is the amount of current flowing through the
wire.
But if we considered the curved surface 
like the balloon , then it does not enclose
any current whatsoever becausethis curved
surface is going through the capacitor and
inside the capacitor there is no current even
when the capacitor is getting charged.
So the current is 0 and that brings in an
inconsistency.
What is the closed integral of B dot dl?
Now Maxwell fixed this inconsistency.
How did he fix this?
The problem is thatif we consider this equation
here, this one the divergence of a curl is
always 0.
So the right-hand side should also be always
0, but if the right hand side is not 0 of
this equation we have to make it 0.
Maxwell tried to make that right-hand side
0.
How did he do that?
He did that by applying the continuity equation.
So on the right-hand side, we have the divergence
of the current density which is according
to the continuity equation, minus del rho
del t . And rho using Gauss law can be given
as epsilon naught divergence of the electric
field minus del del t of that comes here.
That means, this quantity is equal to minus
the negative divergence of epsilon naught
del E del t because the time derivative and
space derivative they commute with each other.
Now if we have this, then now if we write
the curl of the magnetic field B as mu naught
J plus mu naught epsilon naught del E del
t and now if we take the divergence of this
equation, that will give us divergence of
the curl of B which is mu naught times the
divergence of J plus mu naught epsilon naught
times the diverge.
So the divergence of let me take epsilon naught
inside not outside .
So we can write mu naught times the divergence
of epsilon naught del E del t . Now from the
continuity equation, we can clearly see that
the right-hand side becomes 0 and the left-hand
side is also 0 because it involves the divergence
of a curl . Now thisequation, this equation
is consistent with every physical principle
. And this was due to Maxwell . This is the
way Maxwell fixed this inconsistency .
So this term here that represents a current
density.
We can write that as J d, its called the displacement
current equals epsilon naught del E del t
. 
After adding this, Maxwell's term to Ampere's
law, the Maxwell's equations get modified,
let us write down.
So the equations relevant to electromagnetism
get modified and all 4 equations are now called
Maxwell's equations.
Let us write down those Maxwell's equations
.
The first equation becomes in its differential
form, the divergence of electric field equals
the volume charge density over epsilon naught.
The second equation is the divergence of magnetic
field is as always 0, the third equation is
about the curl of electric field that is Faraday's
law which gives us minus the rate of change
of magnetic field with time .
And the fourth equation as modified by Maxwell
gives us curl of B equals mu naught times
the volume current density plus mu naught
times the displacement current that is epsilon
naught del E del t .
Now let us consider an example . After finding
after writing down the Maxwell's equations
. 
Imagine 2 concentric metal spherical shells,
one is big and the other one is small with
a common center here.
The smaller one is with radius a and the bigger
one has radius b . 
And the inner one with radius a that carries
a charge Q that is a function of time, its
not constant over time.
And the outer one that also carries a charge
exactly of same amount, but opposite in magnitude
that is minus Q.
The space between them is filled with some
ohmic material with conductivity sigma . 
So we will have a radial current flowing from
the inner cylinder to the outer cylinder and
this current can be given as sigma times the
electric field and if we work out the electric
field using Gauss law for this simple system,
we can find out that that current density
would be 1 over 4 pi epsilon naught sigma
times this times Q over r squared along r
cap direction.
So, we can write I the total current equals
minus d Q dt which is integration over J dot
da and that becomes sigma Q over epsilon naught.
Now, this is a spherically symmetric configuration
that we have at hand.
So the only directionthe magnetic field can
point is radial.
Then we can write the divergence of B that
equals 0.
And that means, closed surface integral over
B dot da that also goes to 0.
Now if B was uniform over the surface, then
it would have the magnitude of B times 4 pi
r squared.
Now what would be 0?
B must be 0 in this case.
Otherwise,this integral cannot be 0.
So, how could B be 0?
In this situation, there is a current so current
should develop some magnetic field, but there
is no magnetic field in this situation.
How could that happen ? That could happen
because if we consider the displacement current
then, that is of the amount epsilon naught
del E del t which is nothing but 1 over 4
pi and we have the expression forthe electric
field, from that we can write the time derivative
of Q as Q dot over r squared r cap.
This is the expression for the displacement
current and this is exactly equal and opposite
to the current that we haveminus sigma times
q over 4 pi epsilon naught r squared.
Soand in r cap direction.
So the displacement current density is exactly
equal and opposite to the real current density
that we have, the current density volume current
density that we havecalculated earlier.
Compare this equation and this equation and
that tells us why there is no magnetic field.
The displacement current exactly cancels the
conduction current and therefore, we expect
no magnetic field in this example .
