Let us move on to electromagnetic induction.
Let us consider 3 situations. In situation
1 we have this kind of a rectangle, this kind
of a region where there is a magnetic field.
And here is a current carrying loop with resistance
at one end like this. And the velocitythis
loop is pulled out of this magnetic field
region with velocity V this is the situation.
Now, if we have the magnetic field pointing
into the screen, then 
if we pull out this loop as we have discussed
earlier, we will have a current along this
direction and Faraday, Michael Faraday performed
an experiment and saw exactly the same the
current was along this direction.
Then he performed another experiment where,
there was similar kind of a region with magnetic
field. The magnetic field points into the
screen again this time and there is a loop
like this with some resistance at the end
here. And now he made this loop move into
the opposite direction towards the left and
when he did that, he found there was also
a current flowing. But the current direction
reversed .
So, the current would this time flowearlier,
let us see in which direction the current
should have flown? So, we have considered
earlier this case and the this was the direction
of the current the way we have drawn it. Then
the current in this case would be in the opposite
direction. And in the third experiment what
he did was. He had a magnetic field here in
this region, and the magnetic field was changing
with time. It was not constant over time rather
it was changing with time .
And the loop was fixed the loop was not being
pulled out or pulled in to the magnetic field.
Even then because the magnetic field was changing,
the area was fixed still the magnetic flux
was changing, and that gave rise to a current.
In this loop, that is pretty interesting.
So, the flux is what we need? Not the motion
the change in flux is what we need to induce
a current in this kind of where this kind
of a loop? Its not the motion of the loop
itself, that is clearly proved from this experiment.
And the electro motive force as we have written
it earlier as minus d phi dt, that would give
the amount of the current provided we know
the resistance in the circuit. Now if we have
a change in magnetic field. So, what we learned
from this experiment? was changing a change
in magnetic field induces an electric field.
Otherwise there would be no current .
And how do we express this the electro motive
force? as we have done it earlier a closed
integral over E dot dl, that gives us minus
d phi dt. And closed integral over E dot dl
it could be written as. So, minus d phi dt
if we have the magnetic field changing, then
it will be minus integral of del B del t dotted
with da.
So, this surface integral will give us E dot
dl. And now we canconvert this closed line
integral using stokes law, into curl of the
electric field and the and the surface integral
over that. Which is equal to minus integral
of del B del t dot da and this expression
to be true in general, for any electric and
magnetic field we must have the curl of the
electric field equal to minus del B del t
.
This is very important, earlier we knew that
the curl of electric field was always 0. Now
we have so that was in the context of electrostatics.
Now in the context of time varying magnetic
field, if if there is time varying magnetic
field present, we see that the curl of electric
field is no longer 0. There is a finite curl
of the electric field that is the the negative
of the rate of change of magnetic field with
time. And this is known as the Faraday's Law
.
So, we can conclude that whenever for whatever
reason the magnetic flux through a loop changes,
we develop an emf and that emf will be developed
in the loop. Now we can write another law
namely, Lenz Lenz's Law that tells us nature
always tries to avoid the avoid a change of
flux .
So, that will tell us along what direction
the induced current should flow? So, that
the change in flux is minimized. So, ifthe
flux of magnetic field is changing due to
a loop is being pulled out. Then the current
will be induced in such a direction. So, that
the magnetic field is enhanced although the
area is getting shortened. And the other way
around. Ifthe magnetic field is changing in
its magnitude then a current will be induced
in such a way that it tries to restore the
original magnetic field. It will not besuccessful
completely, but that is how nature will react
to the situation.
Let us develop the idea of inductance after
this. And in the context of inductance we
will start with mutual inductance. What do
we mean by mutual inductance? Let us consider
that there is one current carrying loop here,
that carries a current I 1, we call it loop
1. And there is another loop at the vicinity
of loop 1, this is loop 2 .
So, the magnetic field on loop 2 due to loop
1. So, the magnetic field due to loop 1. Let
us call it B 1 can be evaluated to B mu naught
over 4 pi I 1 is a constant closed integral
dl 1 cross r cap over r squared closed because
we have a closed loophere in this picture.
And if we now try to find the flux due to
this magnetic field on the second loop. So,
we call it phi 2 flux at the second Loop the
magnetic flux due to this current, that will
be B 1 dot da .
So, if we nowfor simplicity write down that
phi 2 equals M 2 1 times the current which
we can do? Because everything else is a constant
of the geometry. M 2 1 is a constant of the
geometry, only I the current is something
that we can change in loop 1 and that leads
to a flux magnetic flux in loop 2 everything
else,that we have in this integral here,or
the integral over the area element all these
things are constants of geometry.
So, M 2 1 can be given as a constant of proportionality.
And the flux in the loop 2 depends on its
proportional to the current in loop 1. Which
means we can write phi 2 as integration over
B 1 dot d a 2 that is equals to integration
over curl of A 1 dot d a 2. Is nothing but
closed line integral of the vector potential
dot dl 2.
So, this M 2 1 is called the mutual inductance
and we have thisphi 2 the flux expressed in
terms of the vector potential a closed integral
over the vector potential. How do we express
the vector potential ordinarily? A 1 can be
expressed as mu naught I 1 over 4 pi, integration
dl 1 over r. And this is the expression for
the vector potential.
And if this is the expression for the vector
potential then phi 2 can be expressed as following
what we have developed so far? mu naught times
I 1 over 4 pi closed integral ok. Here we
will have closed integral because we have
a closed loopcarrying I 1 current.
So, closed integral of a closed integral over
d l 1 over r dot dl 2. So, the mutual inductance
M 2 1 can be expressed as mu naught over 4
pi, double closed integral d l 1 dot dl 2
over r. So, we can see that M 2 1 depends
completely on geometry. And nothing else and
because there is a dot product involved and
dot product is commutative M 2 1 would be
equal to M 1 2 .
It does not matter, whether we have dl 1 first
or dl 2 first? as long as we are evaluating
the same integral. After establishing thislet
us write down the emf on the second loop.
That would be the rate of change of flux through
the second loop in time, that is minus d phi
2 dt which is minus M d I 1 dt ifwe induce
this change of flux due to change of current
in the first loop. Then this way we can write
it. And this is the idea of mutual inductance,
there is also an idea of self-inductance.
That means, due to the flow of a current in
one loop and the change of the magnitude of
the current there is a change in flux in the
same loop itself. And that will also createan
electromotive force. So, that kind of a phenomenon
is called self-inductance.
So, if we have phi flux, then this can be
written as L times I in the context of self
induction, where L is the constant due to
the geometry which is called the self-inductance,
and the emf epsilon can be expressed as minus
L d I dt .
