So far in our study if you have been following
all the modules in a linear fashion you will
see that we have been studying things which
are not dependent on time We started with
charge configuration that was essentially
static right and then we talked about the
electro static field that would arise out
of this Electro static would mean that the
electric field is independent of time There
is no time variation anywhere right and when
we let the charges move to form the currents
Currents can only be formed when you have
a charge movement and charge movement would
mean that there is some time variation involved
in the charge configuration However as long
as the current was steady right we did not
have to worry too much right because the magneto
static fields that were generated they were
all again independent of time So this is the
reason why we would call both electro static
fields and magneto static fields as static
fields okay
These are static in the sense that there is
nothing interesting is happening in terms
of time The moment you start varying the current
going through a circuit then very interesting
things will begin to happen and to understand
that we need to introduce time dependency
to our expressions and we do that by discussing
first what is perhaps the most important experimental
discovery in the nineteenth century called
Faradays law upon which the entire diffuse
of electromagnetic is built and once we discuss
that we are in a position to understand what
happens when time varying fields are present
okay
So we will begin by discussing Faradays law
which gives us the link between electric and
magnetic fields and then we will discuss some
applications of Faradays law and tell you
what exactly this Faradays law is useful for
and finally we will talk about a different
form of link between magnetic field and electric
field that cannot actually be related from
just Faradays law but it can be understood
in a broader sense of Faradays law okay
So we will begin by considering what happens
when you have a current that is carried by
a particular loop okay So we have a current
let us assume that we actually have a loop
okay
.
So we draw a loop here 
and this loop is carrying a current i okay
or let us assume that initially the current
loop is not carrying any current Ii that would
make Faradays law very interesting okay Now
if the loop is not carrying any current I
and we will assume that this gap that we have
is very small so we can consider this entire
loop as a proper loop closed loop okay If
this loop is carrying no current then what
happens There is no magnetic field that is
generated
But is it possible that if I have this loop
and then I have a magnet okay nearby So I
have a magnet which is giving out magnetic
fields everywhere right so this are the fields
of the magnet and if this magnet is there
and there is a loop that is sitting here will
there be any current in the loop See we know
Ampere’s law states that when there is a
current carried in a conductor there will
be a magnetic field that is generated
We have seen the applications of this Ampere’s
law all along in the last few modules Now
we turn around and this question if I have
a loop and if a loop is subjected to a magnetic
field external magnetic field will there be
a current in the loop It turns out that 2
people worked on this problem actually many
people worked but 2 of them were very famous
and both almost simultaneously got the correct
answer
The correct answer being yes there will be
a current in the loop provided the magnetic
field is changing with time okay If we take
this magnet and introduce the loop okay and
both of them are constant with respect to
each other that is they are not varying with
their positions then there will be no current
induced okay When will current be induced
You keep the magnet where it is and then take
the loop in and out or you keep the loop where
it is and then take the magnet and push it
back and forth
When you are doing this then the magnetic
field that is associated with or the magnetic
field that is linking with the loop changes
and this change in the loop or change in the
flux linking with the loop will induce a current
okay So this is what essentially 2 people
Faraday and Henry found out Henry was U S
and Faraday was English but it so happened
that Faraday was the first one to publish
widely his findings and therefore all these
laws and the subsequent development is given
credit to Faraday okay
Henry is honored by giving the units for inductance
while Faraday has been a very influential
scientist on Maths well who finally unified
all of the electro magnetics okay So to reiterate
what we were discussing yes it is possible
for the loop to get induced by a current okay
whenever there is an external magnetic field
linking to the loop changes this is very important
the magnetic field linking or the magnetic
flux linking must change okay
So when that happens there will be current
induced in the loop and what should be the
direction of the current induced in the loop
Would it be along clockwise direction or would
it be along the anticlockwise direction Well
the answer to that is conveniently comes from
what is called as Lenz's law and this Lenz
law states that the induced current would
generate a magnetic field which would oppose
the original magnetic field
Say if it opposes in the sense that if it
subtracts from the original magnetic field
then the amount of this one that would be
induced would actually be lessen Now when
current is flowing through a conducting wire
right it would actually give rise to a certain
electro motive force or in our words EMF or
potential okay In fact Faradays law does not
really talk about current getting induced
but rather what he says or what the law states
is that there is a EMF produced in the coil
whenever there is a time rate of change of
magnetic flux okay
If the magnetic flux is changing with time
then there is an EMF induced EMF is called
electro motive force and this was sort of
the force that people considered which was
responsible for the movement of current okay
So it is not really the current that we are
actually inducing although which is what happens
but it is more importantly the EMF that we
are focused on So there is an EMF induced
whenever the magnetic flux linking to that
circuit is changing
Now why is there a minus sign here The minus
sign is precisely Lenz's law Minus sign is
put so as to make us remember that the currents
must be induced in such a way that the EMF
actually reduces right The net flux change
will be reduced So if you choose a current
direction in 1 sense the EMF could actually
or the current would generate its own magnetic
field which would add to the original magnetic
field which means that the EMF would increase
right
However you can actually do this in a continuous
loop kind of a thing right So you have a magnetic
field induced current induced current gives
you the induced EMF and the induced current
will generate a magnetic field which will
add to the original magnetic field increase
the EMF add more magnetic field and this process
can go on which obviously will not happen
in nature
What actually happens is the current induced
in the coil would actually be in a direction
that would reduce the magnetic flux linking
to the circuit okay So in mathematical terms
this is all that is there for Faradays law
okay There is an EMF induced in a coil or
a circuit which is related to minus time rate
of magnetic flux linking the coil That is
what Faradays law states To put it into slightly
better form for us to understand we express
magnetic flux in terms of the magnetic field
itself or magnetic flux density B right
What is the magnetic flux linking to a particular
circuit This would be equal to integral of
B dot d s this is something that we saw when
we discussed inductance as well So there is
a magnetic flux linking to a circuit and that
would be given by B dot d s right So you can
substitute for that So there is d by d t of
magnetic field and that would be getting converted
into del by del t of B dot d s okay
I have chosen to take this d by d t inside
into this B assuming that whatever the surface
I am integrating would remain constant okay
So this is the case where the surface or the
coil is actually stationary while the magnetic
flux that is linking to it actually changes
How could it be possible Well in the coil
and the magnet experiment you place the coil
do not move the coil but you take the magnet
and move it up and down right
So when you move the magnetic field when the
magnetic field recedes there will be decrease
in the magnetic field causing a current and
when the magnet approaches the coil there
will be increase in the magnetic field again
causing a current right So that depends on
how what velocity the magnet is actually moving
As I said there are 2 aspects to this one
right
I keep the magnetic field as it is right I
won't change the magnetic field but I will
let the coil move around right So let the
coil move into the magnetic field and out
of the magnetic field There will be again
an EMF induced okay Will this EMF be the same
as the EMF induced in the other case Is this
also called Faradays law Interesting questions
to think about okay We will talk about that
second situations slightly later okay
So for us we can substitute for now what is
the magnetic flux okay and then say what is
the EMF Of course EMF also can be expressed
in terms of the electric field correct EMF
is integral over the closed loop or whatever
the loop that you are considering E dot d
l that is the EMF over the closed loop and
this must be equal to minus del B by del t
dot d s integral over the surface S which
is bounded by this closed curve C okay So
this is your Faradays law in vector notation
You have a line integral okay of the electric
field and that line integral of the electric
field which forms the EMF must be equal to
the time rate of magnetic flux that is changing
with the circuit okay So before we discuss
other aspects of this let us lookay at some
examples of Faradays law okay Let us understand
what directions to choose and what we need
to do with this one okay So as a simple example
first let us consider the case of a loop okay
.
Sorry my loop is not very nicely written so
let me just write down the loop like this
I have this loop with the gap being very very
small okay and there is a magnetic field which
is uniformly perpendicular to the loop okay
as the value of B which is given by some B
0 e to the power minus alpha t Thus the magnetic
field associated with this loop is actually
in the z direction perpendicular to the loop
So if this is my loop then the magnetic field
is in this direction and it is actually decreasing
with time okay So this is the magnetic field
Now what is the EMF that is induced If I take
this as my specified direction for the EMF
okay what would be the EMF that is induced
okay Now before we actually lookay at this
one let us lookay at a voltage source the
conventional form of writing a voltage source
is that a current leaves voltage source the
positive terminal of the voltage source correct
So the current actually leaves the positive
terminal of the voltage source and if we have
chosen the induced voltage in this way then
there must be a current that is leaving here
Remember this current is not the current that
is present in the coil That is in the sense
that is not initially present it is the one
that is because of the magnetic field that
is changing So I will write this one as some
I ind to indicate that this is the induced
current okay and we want to find whether the
induced current will be clockwise in the same
direction or it would be anticlockwise
If it is anticlockwise then the terminals
for induced voltage would also have to change
correct So this is what we want to find out
now Before you can do that one here is a point
here is a typical idea to remember Let us
assume a certain polarity for the surface
right so this an open surface the surface
normally can actually point upwards or downwards
If they allow the surface element normal to
point in the direction okay such that if I
curl around my fingers using the right hand
rule the thumb should point along the B region
I mean B fields right
So if this the convention that we have been
following right so if I have the current in
the loop then the thumb points in the B direction
and the loop surface area normal also now
points in the direction of B field right So
my field actually points in the direction
of z Therefore the surface normal also points
in the direction of z and this would have
been the magnetic field that is generated
provided there was a current I in the loop
correct
There is current I in the loop that current
would have generated a magnetic field along
z direction okay If my induced current is
in the same direction as that of the current
I marked in blue here it would generate more
magnetic field right it would keep on generating
more magnetic field and the EMF would actually
keep on increasing So before you solve Faradays
law problem I would suggest you make these
simple pointers okay
You establish the surface normal in such a
way that the normal should point in the direction
of B field and then establish the direction
of the current I that would actually give
you the magnetic field in the applied direction
okay In any case you have to only note 2 things
Either your magnetic field is in this way
or the magnetic field is in the other way
right in the opposite direction okay
Now apply Faradays law EMF induced must be
minus d psi by d t correct And if you assume
that the magnetic field B is uniform through
the surface and the surface actually has a
cross sectional area of S then this can be
written as minus S d B by d t okay And EMF
of course is a scalar quantity so there is
no vector out there So you can just take the
magnitude of the B and then integrate this
or whether the part of the B and then integrate
this one removing the vector part to this
So if I do d B by d t what do I get I get
minus alpha into B 0 e power minus alpha t
There is already a minus alpha so that will
cancel with each other I get alpha S B 0 e
to the power minus alpha t This is the EMF
that I am actually getting Now with this EMF
what should be the direction of the current
If I take the direction of the current to
be along I terminals for the induced voltage
that would have been reversed correct
If I take the induced current along the same
direction and this induced current would leave
the positive terminal then the induced voltage
would have been in this polarity The bottom
one would actually have the plus sign and
the upper part of the conductor would have
the minus sign And the induced current would
have been in this direction
However I know that induced current cannot
be in this direction why Because if the induced
current is in the same direction as I then
the total magnetic field will actually add
And that is what we do not want okay So the
induced current is not in the direction of
I rather the induced current is actually in
the direction opposite to I okay So that the
magnetic field produced would actually be
going negative and therefore the total magnetic
field would be reduced
So the induced voltage will be positive with
the upper terminal as plus and the lower terminal
as minus okay So this is what you would obtain
from EMF Although we should realize that this
are not really critical because in most cases
the magnetic field changes happen in such
a way that the EMF would be positive and negative
It is essentially AC fields that would be
applied and therefore the polarity of the
induced voltage keeps changing okay from plus
to minus minus to plus plus to minus and minus
to plus
So there is really no point in worrying too
much about which direction the induced voltage
is there unless there is a very specific reason
for this When we discuss dia magnets we talked
about in the module of dia magnets we talked
about the induced magnetic moment that would
be opposing the B field There we have to careful
in writing down what is the direction of the
magnetic moment okay So here we don’t really
have to be that careful
However whatever we discussed for dia magnets
the justification for magnetic moment actually
comes from Faradays law and Lenz's law okay
With this let us consider second example okay
this example is the mirror example of the
first example The reason I am putting this
down is to just indicate which direction the
current would be induced okay
So again I have a loop okay this loops are
increasingly not becoming circular but imagine
that this is actually a circular loop and
then the magnetic field is still again in
uniform along the z direction and it is given
by z hat B 0 but this time I am going to change
the file and say e to the power k t so it
is exponentially increasing magnetic field
okay So what would be the EMF induced Again
the magnetic field would be generated provided
the current I will be in the anticlockwise
direction correct
The surface area normal will also be along
the z direction for this assumption right
and find out what the induced voltage is So
if this current would have been present in
the loop the current would have actually generated
the magnetic field along z direction The normal
to the surface is also along the z direction
But EMF when u calculate turns out to be minus
k B 0 s e to the power k t right So the EMF
is turning out to be negative
What does it mean It means that now if you
lookay at the current direction and if you
assume the current direction to be clockwise
then it would be generating the magnetic field
in the negative direction but my EMF would
also be negative right So because of that
the induced current must be in the same direction
as this one right So if the induced current
is in the same direction then it would generate
a positive magnetic field
So that the overall EMF would actually be
reduced So what would happen to the science
of the induced voltage That would be minus
plus v induced voltage okay So in the previous
case d B by d t was negative and in this case
d B by d t is positive So the sign conventions
or the current directions for these two would
actually change okay So that is what we have
As the third example consider an alternating
field applied okay
.
In this we are going to apply a magnetic field
of z hat B 0 sin omega t okay So during the
first half at least the magnetic field is
actually increasing which means that d B by
d t will be in this particular case d B by
d t will actually be cos omega t and cos omega
t is decreasing but it is still positive right
So during this half cos omega t is decreasing
but this is actually positive and what would
be the EMF induced EMF induced will be minus
omega so d B by d t is actually omega cos
omega t
So it would be minus omega cross sectional
area is B 0 cos omega t okay So again the
induced current directions must be in such
a way that initially it would be along I in
counter clockwise direction so that the EMF
is reduced Now in the second half of the region
where d B by d t actually goes down the current
direction should actually be in the counter
clockwise direction okay As you can see the
polarity of the EMF would actually be alternating
from plus minus to minus plus okay
.
As another example okay consider a inhomogeneous
magnetic field okay let us assume that B field
is given by some z directed field okay Except
that this would be changing with distance
okay So cos pi r by 2 b cos omega t okay However
as far as the right hand side of Faradays
law is concerned I am only interested in minus
d B by d t right And this minus d B by d t
assuming uniform cross section for the loop
will be given by minus omega sin omega t right
And there is also the other part of B 0 cos
of pi r by 2 b in here correct Now what is
the EMF produced EMF produced is integral
of this minus d B by d t over the surface
right What is the surface The surface integral
will be the d r d phi surface integral right
D s will be equal to r d r d phi integration
over d phi gives you 2 pi integration over
r should give you a different value So it
would be actually be cos of pi r by 2 b r
d r integral from 0 to a if a is the radius
of the loop or b is the radius of the loop
let us take
So this would actually be equal to from 2
pi sin omega t and there is some 4 b square
by pi square pi by 2 minus 1 or if you take
a is equal to 1 in this particular case So
if you do all this calculation you will see
that the EMF is actually given by some omega
8 b square divided by pi pi by 2 minus 1 and
in terms of B this would be B sin omega t
because the differential of cos is sin okay
So this is the EMF that is induced
What is interesting in this is that except
that this was an exercise for finding the
flux of a non homogeneous magnetic field What
is interesting is that if we turn off the
time variation of the magnetic field and we
replace this cos omega t by a constant value
there will be still no magnetic flux link
that is changing with time and hence there
will not be any current induced correct
There will not be any current induced the
current will only be induced only when the
magnetic field is changing and that magnetic
field must be changing with time it can be
for homogeneous field or inhomogeneous field
does not matter okay As long as time variation
is there only then there will be current induced
okay Again the current direction will be different
for the first half the current direction will
be in 1 count a clockwise or count a clock
wise and for the other one it would be clockwise
direction okay
So this change would be happening depending
on what your sign for d B by d t is okay So
if you are given the values of B and omega
you can actually find out what will be the
total EMF that is induced but the point here
is that homogeneous field or inhomogeneous
field both will give you induced currents
as long as they are changing with time
