Good morning. The last 25 or so lectures we
have been talking fields that are not varying
in time. First we introduced the electric
field coulomb’s law, the divergence theorem
poisson’s equation and then we introduced
the static magnetic field talked about ampere’s
law the biot savart law. And we have established
a large number of equations which seem to
all describe a single consistent picture.
So, for context I am going to write down those
equations. First of all we have the electric
field is one over four pi epsilon naught volume
integral row. This is the charge density at
another point r prime and this electric field
is at a position r.
So, r minus r prime divided by r minus r prime
cubed d v prime. This is our basic coulomb’s
law. If I write the same equation for magnetic
field, we have magnetic field is equal to
mu naught over four pi volume integral j of
r prime. This b is b of r prime of r sorry,
j of r prime cross r minus r prime divided
by r minus r prime cubed. It is the same dependence,
but in one case the electric field is due
to a scalar multiplying this operator. The
magnetic field is due to a vector j cross
this same operator. Now we know that r minus
r prime over mod r minus r prime cubed is
equal to minus the gradient of one over and
this quantity I tend to call minus gradient
one over r 1 2. It is just notation.
In fact it is not even standard notation.
The correct way of saying it is one over the
magnitude of the vector r minus r prime. So,
from this definition we went to potentials.
We defined the scalar potential phi of r which
was 1 over 4 pi epsilon naught volume integral
row of r prime divided by r 1 2 d v prime.
It is a scalar field because row is scalar
and r 1 2 is again a scalar field for the
magnetic field. We defined a vector potential
a of r which was equal to mu naught over pi
volume integral j of r prime divided by r
1 2 d v prime. Symmetry is there again the
scalar source creates a scalar potential.
The vector source creates the vector potential.
Both of them are connected through the r 1
2 field. The way potential and electric field
connect is electric field is equal to minus
the gradient of potential. The way the vector
potential in b connect the magnetic field
is equal to curl of the vector potential.
This is the difference. Magnetic field is
intrinsically at right angles to everything
whereas the electric field is intrinsically
in the direction of everything. This is essentially
saying electric field is a central force.
That is the force is along the line joining
the location of the source and the location
of the observer along that line.
Whereas, the magnetic field is not a central
force because you have the line joining r
and r prime and you have the source current
the magnetic field is created in a direction,
that is 90 degrees to both. Because, the electric
field is a gradient we could show that curl
of electric field which is minus curl of gradient
of phi was 0. Because, the magnetic is a curl
divergence of b is divergence of curl of a,
is 0. The symmetry continues since b is a
curl it is the divergence that goes to 0.
Since e is a gradient, it is the curl that
goes to 0. Now, there is a very remarkable
thing about the electric field which is that
if you take the divergence of the electric
field, the divergence of the electric field
is dependent only on the charge at the point
where you are measuring the electric field.
Even though the electric field may be due
to charge in all sorts of places when you
write the divergence of the electric field
it is equal to charge at the same point divided
by epsilon naught. This same remarkable feature
is also present in the magnetic field. If
you take the curl of the magnetic field the
magnetic field may be created due to currents
in many different places but the curl of the
magnetic field only responds to the current
at the place where you are measuring the magnetic
field namely mu naught j of r. So, the divergence
of the electric field is a function only of
charge density at r itself not at r prime.
Curl of b is dependent only on j at r not
at r prime.
All of this is essentially a complete description.
It is a complete description for vacuum and
it is a complete description for known charges
and known currents but we also have materials
and when we have materials, we have that row
is equal to row free plus row bound. Meaning,
the charges we actually place and the charges
that are induced in materials. Similarly,
currents j is equal to j free plus j bound.
The currents we actually place and the currents
that are induced in materials. Once you have
this the row bound and the j bound are not
known. They are the materials’ response
to the applied electric field. They can even
be non-linear. Similarly j bound is the materials’
response to magnetic field. It is almost always
non-linear. But they are material properties
and supposing we knew that we could define
we could take this row bound put it into the
definition of epsilon and we could define
a new field which we call d is equal to epsilon
e and then we would find that our equation
for d divergence d was equal to only row free.
It is not a function of row bound at all.
Similarly if we take this bound current and
put it into our definition of mu, well h is
equal to b over mu then we find that curl
of h is equal to j and only the free current
corresponds to h. So, the symmetry is remarkable.
Actually for every equation in magnetostatics
there is a corresponding equation in electrostatics.
For every equation in electrostatics there
is a corresponding equation in magnetostatics.
Finally we can also write down what are the
boundary conditions.
The boundary conditions we had were d normal
is continuous, e tangential is continuous
in magnetic field case b normal is continuous,
h tangential is continuous. They are switched
in the sense that e and b are the real fields.
They are the fields that respond to the total
charge of current free and bound, e tangential
is continuous but only b normal is continuous.
The reason is the curl, the e and b are 90
degrees in related by a 90 degree shift in
direction.
So, if e tangential is continuous you should
expect b normal to be continuous. That 90
degree effect comes in and it is normal and
h tangential that are continuous. We have
a force equation. For a charge it is q e plus
q v cross b and you can see the force only
cares about electric and magnetic fields.
It does not care about d and h, d and h are
conveniences. They are imagined because we
need to get rid of dealing with bound charges
and bound currents. But the real fields are
e and b. So this, I think is a summary of
everything we have done in the first half
of the course. And you can see that really
the equations are quite pretty and it looks
very much as if electrostatics and magnetostatics
are going hand in hand, they are really talking
about very similar things. So, what is there
to add?
Now let us take a thought experiment. I have
a magnetic field. Let us say it is a uniform
magnetic field b along the z direction and
I have a charge q 
and I have another charge q prime. Now these
charges are at rest. Because, they are at
rest if you look at the force equation the
force on charge q is zero because of the magnetic
field. Because the charge q is not moving,
but it is non-zero because of the electric
field, there is an electric force e. Now I
have not drawn this properly. Then we put
the charge this way. So, the electric field
is not along b.
Electric field is in that direction and because
of that electric field there is a force. So,
the force is equal to q, q prime over four
pi epsilon naught and if this distance is
d or r 1 2 it Is equal to r 1 two divided
by r one two cubed. So, we know this is coulomb’s
law. Everything makes sense. Now, supposing
I, as the scientist was doing this experiment,
gets into a car and this car comes moving
with a velocity v. The charges are not moving
but the car is moving and when I come here
out of the window I stick out my measuring
instrument and measure force on the charge.
Now common sense tells us that the fact that
I have a car does not change the amount of
force felt by the charge. I could be in a
rocket. I could be in the next galaxy. Whatever
force the charge is feeling is the same. But
if you look at my point of view sitting in
the car as far as I am concerned, I am stationary.
I am inside the car. This charge is moving
with a velocity v in the opposite direction.
Since the charge is moving with a velocity
in the opposite direction, there is a force
q v cross b. So, the force according to me
sitting in the car will call this force car
is well there is the electric field, q q prime
over four pi epsilon naught r 1 2 over r 1
2 cubed right. But there is also this magnetic
field force which is plus q v cross, b naught
which is along z. So, I can put z hat. Now,
this force is the same. It is a new force
and that is very strange because what this
is saying is if I measured the force while
I was walking around if I measured the force
sitting in a car, if I measured the force
sitting down I am going to measure different
forces. Now right at the beginning of the
course I am made a very important point. All
the theory of electricity and magnetism is
a theory.
The only real thing is the force that we can
measure because it is the force that deflects
a needle. It is the force that actually causes
your electronics to respond. So, it is the
forces that are real. The electric field is
not real. The magnetic field is not real,
h is not real, d is not real, phi is not real,
a is not real. The only thing that, real is
f because this is the only one we can measure.
And what I am saying is i am going to measure
different things depending on what I am doing.
Even though the charge is not in my car charge
is staying where it is, but if I am moving
relative to the charge the charge experiences
different force, Now about 300, 400 years
ago Galileo, the same Galileo who got into
trouble with the pope and looked at the moons
of the Jupiter that same Galileo proposed
a fundamental principle of physics. He said
whatever the scientist is doing cannot change,
the laws of physics, if the charge is moving,
yes. That makes a difference.
But if I am moving it, should not make any
difference and he thought about that and he
said that this amounts to saying that if I
take any law of physics 
and when I say physics, I mean engineering
also. Any law of physics is invariant under
translation, what that meant was if I have,
if I have measured some law of physics in
x y coordinates and if I then say x is really
equal to some w plus v t then my w coordinate
y w coordinate is something else. This is
w, this y w coordinate is going to move.
It is going to move with the velocity minus
v. Now, whatever law of physics I propose
for x y should also hold in w y because this
coordinate system is my imagination. The electron
does not know about coordinate systems. Electron
does not know about what is the centre of
the universe. It does not know about origin.
It does not know about any of these things.
It does not know about x y z also. It is just
doing. What it is doing because it is experiencing
a force and that force cannot depend on where
we drew our axis and it cannot depend on whether
the axis is moving with a steady velocity.
So, it said any law of physics is invariant
under translations, but this law of physics
is not invariant on a translation. The force
measured if I am standing still and the force
measured, if I am sitting in a car are different,
either Galileo is wrong or this equation is
wrong. Well Galileo turned out to be wrong
but he turned out to be wrong in a very sophisticated
way.
Einstein improved on his invariance statement
and made it even better. But the philosophy
behind this invariance we believe to be completely
correct. That is nothing in this universe
is concerned about what the scientist is doing.
It is only concerned about what the object
we are studying are doing. So, somehow these
two things must be saying the same thing.
So, how do we do that? I mean how can we make
the force be the same whether we have movement
or we do not have movement.
Let me redraw the picture. I have my magnetic
field b. I have z this is x this is y and
let us say I have two charges q q prime. Now
what I am going to do is I am going to say
that supposing I allow electric field to be
different because electric field is just something
I have defined. Only thing that is real is
force. Supposing I will have my electric field
and magnetic field to be different to be defined
differently depending on whether I am sitting
still or I am moving. So, in that case I will
define f is equal to q e at rest and I will
define f is equal to q e plus q v cross b.
But it is not the same e and the same b. So,
I will give it a dash when moving.
Now, we already know that the magnetic field
came out of relativity. If you go back and
listen to my earlier lectures, you know that
you can derive the magnetic field using coulomb’s
law and Einstein’s theory of relativity.
So, the magnetic field is basically okay,
it is already got current built into it. So,
when you move if currents happen to change
magnetic field changes. It is the electric
field. Therefore that has to change. These
are saying the same force. Then, it must be
that q e is equal to q e prime plus q v cross
b. So, when I move there is a change in my
electric field such that the changed electric
force plus q v cross b is equal to the original
electric force or you can look at this formula
and say the modified electric field is equal
to the original electric field minus q v cross
b sorry minus v cross b. Now this is very
non-intuitive. It does not agree with coulomb’s
law. As far as coulomb’s law is concerned
e is e, e has been defined and we defined
it here. There is no scope for putting in
a velocity in here. Yet somehow when you look
at the required electric field the required
electric field should change so that there
is an adjustment for velocity. In the presence
magnetic field the electric field somehow
modifies itself. Now this is just coming from
taking electrostatics magnetostatics and the
most fundamental requirement we have about
science. The most fundamental requirement
is that the universe did not create itself
for us. It does not change its laws depending
on whether I am walking or I am standing.
It does not know about me. If it created a
law called coulomb’s law, it created it
without any regard for human beings. We as
human beings are looking at these laws, but
the laws do not change because we look at
them and the laws do not change depending
on how we look at them. If we just put in
that requirement we are getting this change
in the definition of electric field and it
is a change that is not compatible with coulomb’s
law. Something new has appeared and it is
very strange.
Now let us just assume that this is so and
let us see what it would do and there is a
very standard experiment that you would have
read about in your electricity and magnetism
course earlier which is you have a rail stationary
rail and you have a moving rod. This rod moves
with the velocity v and let us say there is
a magnetic field b uniform and to make things
interesting we can put a light bulb. So, normally
if this rod is stationary we already know
that nothing happens. There is no battery
and therefore the light bulb is off. Now what
happens if this rod is moved? Now according
to what that equation tells us if this rod
moves then inside this rod there is an electric
field. How much is that electric field? It
is equal to the original electric field was
zero. The new electric field is this v cross
this b and with a minus sign. So, the new
electric field is in this direction e prime.
Now what will this e prime do? What it will
do is, it will push electrons. The presence
of this e prime means a lot of electrons will
come there leaving behind a lot of positive
charge here and this negative and positive
charge build up. What it will do is, it will
cause a current to flow and this light will
glow. Needless to say, this experiment has
been carried out. You probably have done it
in your labs and the light does glow. If you
take a circuit and you have a sliding bar
conducting bar of course and if there is a
uniform magnetic field and this bar moves,
there is an induced electric field and it
is completely outside whatever coulomb’s
law had talked about. There is no charge that
was there to create this electric field. After
this electric field was created, a charge
build up happened which caused the current.
But the charge build up is notional because
the moment the charge builds up it flows away
as current. This electric field is only there
because galileo demanded it and it is called
motional electric field. Now this electric
field was detected first by faraday and that
is why in fact the law that comes out of it
is called faraday’s law and he did it in
different ways. Basically what we were doing
was something slightly different. He took
a circuit and he applied a magnetic field
and he found that if you changed the magnetic
field a current would flow.
That is, again you put in a bulb. If you changed
the magnetic field made it stronger, made
it weaker a current would flow. Both of these
could be captured by one statement and the
statement was an induced e m f electromotive
force is created when the magnetic flux 
through the circuit changes. That is to say,
if you take this circuit or this circuit and
you calculate surface integral b dot d s.
This is what we define as magnetic flux phi
m. If you take if you calculate the magnetic
flux and if phi m is not a constant in time
if it changes with time an induced e m f is
created. Furthermore it is also observed that
this e m f has a direction that opposes the
change in phi.
What do I mean by opposes? Well a current
is induced the moment does an induced e m
f. For example you got an electric field,
so a current flows, now imagine in your mind,
this current is flowing. As a result of this
current flowing there is going to be a magnetic
field. What is the direction of that magnetic
field? Well the direction of that magnetic
field you apply your right hand rule. The
direction of that magnetic field is like this
it is downwards. Now if you look at the flux,
the flux is increasing because the velocity
is in this direction b dot d s integrated
is increasing but if you take into account
this magnetic field as well b is reducing,
areas is increasing, b is reducing. So the
direction of the current is that, direction
which is tending to reduce the change in magnetic
flux had the electric field been the other
way. Then you would have created a magnetic
field upwards which would mean not only does
the area increase, but so does the field strength
which means phi magnetic would increase even
more.
But what faraday’s law is saying is the
change tries to oppose the induced e m f tries
to oppose the change so much. So, it is trying
to keep magnetic flux constant. Now this is
an observation. Faraday actually did experiments
and observed. This on the other hand is just
a theory based on how we believe the universe
works and it is surprising, but true that
both of these are say exactly the same thing.
In fact if you take just this statement and
you ask, what is that statement saying? Well
you can look at this. In one second this rod
would have moved to a new position. It would
have gone a distance of v metres all right.
Now let us also assume that this magnetic
field is not straight upwards. Let us say
it stay up like this. Now, the velocity has
moved v metres. Therefore the area has increased
by this length l v metres square. Now the
area you know has a direction. It is a normal.
So, the area direction is l hat cross v hat.
It is either this or minus sign of this depending
on whether you make it point upwards or downwards.
Now, the amount of magnetic flux has to do
with b dot the area. So, what will be the
amount of the magnetic flux that has increased?
It is a constant magnetic field. So, the amount
of flux that came through this part is constant.
It is only this part that has given me new
magnetic flux. So, how much is that? Well,
it will be l cross v. That is the area with
if this is l and this is v it is downwards
dot b. That is the amount of flux that is
piercing this new area.
Now you can rewrite this. It is triple product.
So, any triple product can always be written
as you can rotate. So, it is l dot v cross
b. So, you can see that same v cross b has
come. Now I would really like to take l as
one metre because when I want to talk about
electric field, it is a local quantity. It
does not know about distances. So, I would
say l is one. So, this v cross b represents
the amount of electric field picked up if
the rails are one metre apart. So, the same
v cross b is also saying that this v cross
be is also saying that, this is rate of change
of magnetic flux phi m if rails are one metre
apart. So, you can see that the same statement
is coming back at us.
We got here not from faraday. We got here
simply by looking at coulomb’s law and saying
coulomb’s law looks funny because if we
sat in a car and moved we got different answers.
So, we said it must be that the electric field
is changing. It cannot be the magnetic field
that is changing because the magnetic field
was derived based on relativity itself. So,
magnetic field is safe. Our definitions took
into account moving things but the electric
field did not. Electric field had coulomb’s
law. So, the electric field must change and
now we find that if you take a picture like
this, this v cross b is nothing but the rate
of change of magnetic flux if the rails are
one metre apart.
We will come back to that one metre apart
business later. So, in a sense now both of
these statements are saying the same thing
and what are they saying? They are saying
that an induced e m f is present if the magnetic
flux changes.
And if you take that length l into account,
the induced e m f, induced e m f remember
is a voltage, okay? The induced e m f must
be equal to integral electric field dot d
l and e m f is a voltage. Electric field is
the force per unit charge. If you want to
get to voltage you have to relate it by e
is equal to minus grad phi. Now, this is a
suspect equation but that it where it comes
from. So, I am taking volts. I am taking a
derivative in space. So, volts divided by
length is field. So, if I have an e m f then
the e m f is related to electric field by
integrating in distance. So, I have my final
answer now. This is the magnetic flux. This
is the e m f and the induced e m f is trying
to resist the change in magnetic flux. So,
it is saying e dot d l is equal to minus,
it is trying to resist d d t of b dot d s.
There is only one thing that is important
to note. This integral is not really from
here to here, e m f is not so much a point
to point thing. It has to do with a circuit
because after all you are talking about a
surface and the surface connects to a complete
closed circuit. So, you cannot talk about
an e dot d l till you define the full circuit.
Therefore, this is a closed integral e dot
d l and this is nothing but faraday’s law.
I want to emphasize faraday’s law is really
derivable from coulomb’s law. It is that
simple.
It is a measured phenomenon but we started
from coulomb’s law just required that what
galileo said namely what charges do in the
presence of electric and magnetic fields in
the presence of other charges and other currents
cannot depend on how fast we are moving. The
moment you require that you find that the
electric field has a correction term to it
which is v cross b, you take that v cross
b and you find that v cross b is nothing but
change in magnetic flux and you can write
down this equation.
And that rate of change of magnetic flux if
rails are one metre apart is basically saying
that if you took one metre then this is nothing
but e because, it is integrated over unit
distance. So, the per meter amount of electric
field, that is induced is equal to minus v
cross b but if we integrate over the whole
circuit then that v cross b is generalized
to minus d d t of b dot d s. This is a completely
general equation. It is relativistically correct.
It is remarkably we have not been able to
find any case where this equation is not true
and yet we got to it just starting from coulomb’s
law. Now little bit of mathematics and we
are done.
Loop integral e dot d l is equal to minus
d d t of surface integral b dot d s. Now I
know everything about loop integral of something
dot d l because we have already encountered
it in ampere’s law. This piece is nothing
but surface integral. Let me call this c.
This is the surface connecting c of curl of
e dot d s. Now that should give us a warning.
We have already had a proof that curl of e
is 0. Why because, we had e was equal to minus
gradient of phi which implied curl of e is
identically zero and yet a loop integral of
e dot d l is nothing but surface integral
of curl of e d dot d s. Now on the right hand
side there is a d d t of a small integral.
For the moment let us keep that surface constant.
So, we have got a circuit a battery resistors
etcetera and it is just a stationary circuit.
So, the surface is a constant surface. In
that case this time derivative is not talking
about how the surface is changing. Its talking
about how the field is changing.
It is quite an important point because you
people are EEE students. So, you will be doing
a lot of cases where it is the surface that
is changing rather than magnetic field. For
example, if you have inside a machine when
the rotor moves surfaces move, so the motional
e m f is very important to you. But let us
look at the case where the surface is stationary
in which case the surface is not a function
of time. Then this time derivative cannot
act on s. It can only act on b. So, it becomes
minus integral over the surface time derivative
of b. Now, this quantity is a function of
time. So, it makes sense to write d d t of
that quantity. But, b is a function of x y
z and t. So, I cannot write d d t here because
if I write d d t here what does d d t mean?
When I write something like d b d t what I
really mean is move along some x of t y of
t z of t and compute d b d t. But that is
not what I am going to do because x of t y
of t z of t do not exist. This is a, they
are dummy variables. They are what II used
to construct my d s. So, these are all zeroes.
I mean they are constants.
So, what it means is that in this dependence
x, y and z are kept stationary, when I do
this time derivative I only take the derivative
with respect to t. That means I have to do
a partial derivative with respect to b. 
I have a surface integral on both sides. So,
I want to combine them. So I get surface integral
curl of e plus del b del t dot d s is equal
to 0. Then this whole derivation I have not
made any assumption about what c is. C could
be anything, c could be large, c could be
a square a circle anything and as I told you
before, if i have any kind of integral of
this type arbitrary surface v dot d s is equal
to 0, it has to imply that v is zero.
Because if it’s not true, then I can go
to wherever v is not 0 and do a small surface
in only that part where v is not zero and
I will get a non-zero answer. It must be true
that v is zero everywhere which means this
relationship which is actually an integral
over a surface is true at every point of the
integrant. So, I can write down my final equation
which is faraday’s law. Curl of e is equal
to minus del b del t.
So, this is the integral version of this equation
and this is the differential version of this
equation. So, let us see, let us sort of summarize
and see where we have reached because this
is quite a packed lecture. What we have done
is we started with a set of equations divergence
of e is equal to row over epsilon naught,
curl of h curl of b equals mu naught j. Curl
of e is equal to zero. Divergence of b is
equal to zero. This is where we started and
what we have found is that this equation is
not correct because if you look at this equation
the correct form of this equation is curl
of e is equal to minus del b del t. It is
not surprising we did not find it before because
up till now we were not taking into account
time rate of change of things.
So, it is only when we started looking at
how things change in time that we started
noticing an error in these equations. I can
see that it makes a big difference because
it was only this equation that allowed us
to say therefore electric field is equal to
minus grad phi. Why is that? Because if you
take loop integral of b dot d l, it is equal
to surface integral over the same surface
over a surface connecting this loop curl of
e dot d s. And if curl of e was zero it gives
me zero and from this idea we had that if
you have any point one nay other point two
if you take any two different ways of going
from one to two. You could do an integral
one to two plus an integral two to one of
e dot d l and it will give you zero.
It gives you zero because loop integral of
e dot d l is zero which meant integral from
point 1 to point 2 of e dot d l did not depend
on how you got them. It was completely independent
and because of that you were able to define
a function that depended only on the beginning
point and on the ending point and then we
would take this beginning point and put it
at infinity and say we defined a function
that is a function of position. So, everything
depended on this equation being correct. Our
entire electrostatics hinged on this and now
electrostatics is wrong.
Now if you do this problem you can do integral
1 to 2 and 2 to 1. It is equal to loop integral
e dot d l which is not anymore equal to 0.
It is equal to minus d d t of surface integral
b dot d s. The electric field is no longer
derivable from a potential and that is the
main difference. That is what is changed everything
here yet and I would like to repeat this again
and again. Faraday’s law comes out of coulomb’s
law. Faraday’s law is not a new law.
It is not an overstatement to say that if
you know coulomb’s law and if you know relativity
all the other equations are derived, magnetic
field can be got from relativity. Faraday’s
law comes from shift invariance. Similarly,
when we generalize ampere’s law it also
comes from the same kind of symmetries. Coulomb’s
law requires that we generalize it and some
how we have to add something new to get an
improved definition of electric field. We
will do that in the next class and complete
our understanding of faraday’s law.
