WEBVTT
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Hello, so we have now come to the end of the
course. This is going to be my last lecture;
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trying to wind up whatever we had started
about the electric field and magnetic field.
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In our last lecture, we had taken one example
and discussed the electric field and magnetic
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field transformation. We had said that, what
appears to be a purely a magnetic field or
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an electric field in the frame may appear
to be combination of electric field and magnetic
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field in a different frame.
So, whether you term a field as electric or
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magnetic or partly electric or partly magnetic,
it could depend on the frame of reference
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from which you are observing. Today what we
will try to do is try to analyze the situation
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little bit more and try to see how we can
think about the presence of these fields or
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change these fields, by taking one simple
example of a current carrying wire or current
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carrying conductor. Then eventually will show,
I will not really be able to show because
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you know this is actually beyond the scope
of this particular lecture series, that Maxwell’s
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equations which are supposed to be the basic
equations in electrodynamics, they remain
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unaltered even after special theory of relativity.
So, there those equations do not change once
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that special theory of relativities also introduced.
So, they are invariant under special theory
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of relativity. So as I said this particular
aspect will not be able to be prove, as it
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is beyond scope of this particular course,
but we will just take one equation and try
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to sort of convince I would say, try to convince
that for one equation will sort of say that
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you know, you can sort of see that Maxwell’s
equation is expected to be obeyed. So, this
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is what we are going to do in our last lecture
today.
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So, this is what we have said we recapitulate;
we discussed an example using electric and
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magnetic field transformation in our last
lecture.
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To get a physical feeling how the fields could
change in different frames, we discuss a new
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concept today which you call as a current
density four-vector. As I said, we want to
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discuss little bit more physically by taking
current carrying conductor but before we come
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to that situation, we will define what we
call as a current density four-vector.
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Before we do that, let us first take traditionally
what we mean by a current density. In physics
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generally, current density is considered a
little more fundamental than the current and
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current density is always a vector quantity.
This concept of current density emerges more
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from the fundamental aspects of how the motion
of charge carriers takes place in a particular
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wire or particular conductor. So, let us assume
that we have n charge carriers per unit volume
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in a given conductor or given wire which is
carrying current. Now of course, we know that
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is actually the electrons which carry the
current which are negatively charged particle.
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But somehow traditionally the direction of
the current has been always defined to be
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opposite direction of the flow of the electron.
So, if the electrons sort of flow in the negative
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x-direction, we define the standard traditional
current in the positive x-direction. As we
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said let us assume that there are n charge
carriers per unit volume and each one of them
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is carrying a charge of plus e. So, I am just
assuming that the charge carriers are positive,
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though as I have said that actually the charge
carriers in a real conductor is posed to be
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electrons which are negative charge carriers.
But just for defining the concept of current
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density, I take this as positive charge carriers.
Now we evolve the concept of what we call
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is the drift velocity. Actually once we apply
electric field, these charge carriers start
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accelerating under the influence of the force
which is created by this particular electric
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field, but because of the scattering that
these electrons or the charge carriers, whatever
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they are, freeze within a metal or within
a conductor.
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They are not able to move too much ahead,
they go little bit ahead and then they get
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scattered; then they again accelerate and
then they get scattered and if you remember,
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what we can define that under the influence
of electric field, these electrons will eventually
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develop what we call as the drift velocity.
It is something like the motion of air, breeze,
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the molecules in air may be moving with very
large speeds, but we do not feel it.
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Only when the overall air drifts, which may
be drifting with very small speed in comparison
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to the actual speed of the molecules; that
we feel that there is a breeze. So, that is
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what is the drift of this whole air. So, similarly
is the concept of electrons; the electrons
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could be moving in very large speed inside
a conductor, but if there is one electron
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is way, another electron is moving this way.
So, eventually you do not see any drift; only
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when there is net flow of electron in a particular
direction, then only we see the current and
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that happens when we apply an electric field
and then we define what we call as a drift
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velocity.
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Now current density vector J is defined in
terms of this drift velocity as follows. J
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is equal to ne times drift velocity, where
as I have said n is the total number of charge
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carriers per unit volume, E is the charge
of each of the carrier which I am assuming
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at the moment to be positive; actually it
makes no difference if we take for an electron.
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We have to take care of science properly;
otherwise, this equation which defines the
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current density inside the metal or inside
the conductor.
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Now if the situation is viewed from a different
frame, then obviously we expect that the velocity
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of the electron would not be same, because
we have discussed number of times that the
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velocity is actually frame dependent quantity.
So if I go to a different frame, the drift
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velocity that you are seeing in a given frame
may turn out to be different; it is very well
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expected. Also I would like to emphasize that
even the current density or charge density,
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not sorry current density, the charge density
would be different and that is mainly because
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of the length contraction.
Because once we have come to the case of relativity
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theory, we will find that there is a contraction
of length; and that itself would change the
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charge density, and if charge density change,
the drift velocity change. So when I look
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into a different frame of reference, the current
density is also expected to change. Therefore,
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I must look on in relativity, how this current
density will change if I change my frame of
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reference; that is what is the question that
we are going to answer now.
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That is what I said; let us examine this concept
from the point of view of relativity. For
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that particular thing, let us assume comparatively
simpler picture. Let us not bother about the
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drift velocities and other things. We will
just take comparative, very very simple picture
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and try to see how this current densities
will transform if I go from one particular
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frame of reference to another frame of reference.
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So, let us assume that we have a volume which
contains all the positive charges; each one
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of which has a value of q which moves with
a velocity u. So, let us assume that all of
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them move with the same velocity because there
is no question of drift here; I do not want
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to confuse, will about drift little later.
So, let us assume that they all of them moving
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with the same velocity u as seen in the given
frame S; obviously, when these charges move
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in a particular direction, this will constitute
current or eventually current density.
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Let n be the number of charge carriers per
unit volume as seen in this frame. We have
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just now said that this n is also likely to
be frame dependent; I would discuss this point
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little more in detail little later. So, we
have a frame; as we are always telling, we
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have to be consistent in our frame. So, there
is an observer sitting in s frame, which absorbs
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the velocity of the charge carrier to be u
and first he calculates the total charged
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number density or charge density, if you call
it as number density; then that is the number
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of charge carriers per unit volume. He absorbs
that number to be equal to n.
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Now let us define a four scalar; we have used
the concept of four scalar earlier. This is
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the quantity which does not change, when we
change our frame of reference. See like we
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have defined, when from velocity to momentum;
at the time, we multiplied by four scalar
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which we called as the rest mass of the particle.
Similarly here, we define a four scalar n
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naught which does not change on frame of reference.
This of course is going to be different from
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n and this is the number density; that is
the number of charge carriers per unit volume
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in a frame of reference in which these charges
are at rest.
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See like rest mass, we define that this is
the mass what we call in a frame of reference.
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When the particular mass is at rest, take
analogy from the particular concept. Similarly
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we define a number density, total number of
charge carriers per unit volume as evaluated
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in a frame of reference in which these charges
are at rest; this n that a person is going
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to find out in S is going to be different
from n naught as we will see just now.
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Now the charge density in the frame S is higher
from the proper one; this number n, I can
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call as a proper number density, n naught
sorry, this n naught we can call as a proper
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number density because this is evaluated in
a frame of reference in which these charges
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are at rest. Now what I insist that, the n
that I am evaluating in S; S is the frame
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in which the charges are actually moving.
In this frame, the number density would be
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different from its proper density n naught
mainly because of this length contraction;
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and this length contraction always occurs
along the relative velocity direction. So,
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let us look in to this particular picture.
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Let suppose we have a wire or a conductor
like this and let us assume that this is in
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the x-direction. Now they are charge carriers
here; that is the negative charge carriers
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whatever are. Here we have assumed positive,
so let us assume positive. Let us assume that
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they are moving in a particular direction
x. Now if I view this particular thing from
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a different frame of reference, then in that
particular case and let us assume that this
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particular frame of reference; let us say
as prime or whatever is that particular frame
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of reference. A person sitting in that frame
views these charge carriers, then that particular
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person would find that this length is contracted.
I am assuming that this particular frame is
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actually moving in the same direction as x.
So, we know that standard length contraction
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formula. So according to this person, the
length will turn out to be smaller and the
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dimensions would not change. And this length
will become smaller by a factor of gamma u;
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new length or the length as seen in different
frame of reference will be gamma u times the
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original or the proper length. Therefore,
the volume of this particular material will
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also go down by gamma u, because no other
dimensions change. So, if you have let us
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say now the dimensions are a, b, c, then only
one of them becomes gamma a and b and c remains
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same.
So when I take the volume, volume goes down
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by a factor of gamma u and therefore charge
density, the total number of charge carriers
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per unit volume go up by a factor of gamma
u. So, that is what I say that if I have come
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to this particular frame of reference S in
which I find that these charges are moving,
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then this number density that will be evaluated
by the frame will be different from its proper
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number density because of the length contraction
and therefore, this particular factor will
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get modified or get multiplied by a factor
of gamma u.
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Hence the current density if I have to write
in S frame, I have to write the total number
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of charge carriers per unit volume which I
know; I have earlier written as n, will actually
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be gamma u times and not where n naught is
actually four-scalar. This factor is n and
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this n is gamma u times n naught. Using the
definition of charge density that we had used
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earlier, this equation we can write that current
density is total number of charge carriers
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per unit volume, which is gamma u times n
naught multiplied by the q multiplied by the
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velocity; I am assuming all of them are moving
in the same velocity.
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So, this will be the current density as seen
in this particular frame of reference. So,
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the only reason why I am writing this equation
in this particular form is because I am using
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a four scalar here. If I would have written
just n, this n would have changed if I would
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have want different frame of reference, while
this n naught is going to be the same in all
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the frames. Now instead of number density
sometimes we define charge density. This charge
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density is very simple; just multiply whatever
is the n naught multiply by the charge which
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is q.
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So, this is a very simple definition; most
of the time we talk in terms of charge density
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rather than number density. So we define the
charge density rho and like we have defined
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a proper number density, we can define a proper
charge density rho naught as follows. So,
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rho naught whatever was the number density
multiplied by charge; it is very simple it
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is the charge density. The total number multiplied
by charge. Similarly, the charge density in
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a particular frame is the total number density
as seen in that frame multiplied by q. So,
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this rho is equal to n q and rho naught which
is the proper charge density is n naught q
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and because as we have discussed n is equal
to gamma u n naught.
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So, this particular factor rho naught and
this equation rho can be written as gamma
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u rho naught, because this is n naught q.
This n I can write this as gamma u n naught,
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n naught if I observe here this becomes rho
naught. So, this becomes gamma u rho naught.
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So, rho charge density in a particular frame
of reference can be written as gamma u rho
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naught and only thing that I insist, that
this rho naught is also a four scalar because
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n naught is a four scalar. And q the charge
in relativity we do not expect this to change,
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first we change the frame of reference. This
point we have not specifically mentioned earlier,
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but let us emphasize it now. The charges do
not change once we change the frame of reference.
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Therefore, this quantity rho naught is also
a four scalar.
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So, we can write the current density now in
terms of this equation J is gamma u rho naught
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u is what we have written here earlier; except
this gamma u n naught. This n naught, I am
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sorry, have been written in terms of rho naught;
that is all have done in this equation. So,
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this is J is equal to gamma u rho naught u.
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Now I am in a position to define current density
four-vector. If you remember in one of our
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earlier lectures, we have talked about the
velocity four-vector. We have said that U
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x, U y, U z, they are not the components of
the four-vector or the velocity four-vector.
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See remember P x, P y, P z are the first three
components of the momentum four-vector, but
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U x, U y, U z are not the component or not
the first three components of velocity four-vector.
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In fact, what are actually the first three
components? They are gamma u U x, gamma u
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U y gamma u U z. This aspect we have discussed
much more in detail when we were discussing
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the velocity four-vector.
So, this velocity four-vector, the components
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are given gamma u times U x; this is the first
component. Second component is U y, third
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component is U z. This is iC; all have to
be multiplied by gamma u. So, strictly that
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mean that the first component of velocity
four-vector will be gamma u U x; second will
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be gamma u U y; third will be gamma u U z;
and the fourth component will be I gamma u
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C. This is what we have defined earlier. Now
I can use this particular thing to define
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charge density four-vector; all I have to
do is to multiply by a four scalar which I
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can do by multiplying by rho naught, because
we have earlier said that 0rho naught is actually
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a four scalar.
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So, we define the components of current density
four-vector by multiplying this u four-vector
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by a four scalar which is rho naught. This
has been multiplied by rho naught. All other
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components are exactly same; only thing this
has been multiplied by rho naught. Now we
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realize, if you look back at the equation
of current density this I can write component
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wise; I if write component wise, it will become
J x or the x component become J x gamma u
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rho naught U x. The y component of this equation
becomes J y is equal to gamma u rho naught
19:45.489 --> 19:51.929
U y and things like that.
So, once I write here rho naught gamma u U
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x, this will become J x. Once I say rho naught
gamma u U y, this will comes J y; rho naught
19:58.690 --> 20:04.100
gamma u U z, this will comes J z and I am
retaining this particular thing writing this
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rho naught gamma u as rho which is the charge
density. This becomes i times rho multiplied
20:11.100 --> 20:18.100
by C. So what we see, that the first three
components of current density four-vector
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are actually the real current density as seen
in that particular frame; it is like momentum,
20:26.389 --> 20:30.119
momentum four-vector. The first three components
are the momentum as absorbed in a particular
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frame of reference.
Similarly current density components are the
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first three components of the current density
four-vector and the fourth component, like
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in the case of momentum four-vector depended
on the energy of the system. Here the fourth
20:45.210 --> 20:50.600
component of the current density four-vector
depends on the charge density. So, the fourth
20:50.600 --> 20:56.340
component is rho. See like in that particular
case of momentum, it was I E upon C; the fourth
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component dependent on the energy. Here the
fourth component depends on the charge density.
21:01.200 --> 21:08.200
Now once we have found this particular four-vector,
we know how to transform it. If I go for particular
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frame of S to any other frame S prime and
if I know the relative velocity between S
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and S prime to be V, I can always transform
this particular four-vector components into
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any other frame by using this standard matrix
equation which we have been doing earlier
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number of times. So, the current density will
obey the following transformation rule if
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we go to another frame S prime.
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This is the current density transformation.
So we have in S frame, these are the components.
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The current density is J x, J y, J z; these
are the x, y and z components of the current
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density. The fourth component is I rho upon
C where rho is the charge density as seen
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in the particular frame of reference. I want
to find out the current density and the charge
21:59.850 --> 22:05.539
density in a different frame as prime frame
of reference. And let those current densities
22:05.539 --> 22:12.429
be give or the components of the current densities
in as prime be given by J x, prime, J y prime,
22:12.429 --> 22:17.919
J z prime, and let the charge density be given
by rho prime.
22:17.919 --> 22:24.919
Then these J x prime, J y prime, J z prime,
rho prime would depend on J x, J y, J z and
22:26.710 --> 22:33.710
rho by this particular matrix equation transformation
equation. Here of course, this gamma, beta,
22:35.080 --> 22:39.940
all these depends on the v the relative velocity
between the frame between S and S prime; the
22:39.940 --> 22:45.609
standard way we have been dealing in relativity
so far. So, now, I can expand this particular
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thing; we have done it a number of times.
So, we have to spend too much of time in expanding
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this particular thing. I will just write the
explanation of this particular equation which
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I shall deal in the next class.
22:59.460 --> 23:06.460
This is what it becomes. J x prime becomes
equal to gamma times J x minus v rho, J y
23:07.789 --> 23:14.789
prime remains equal to J y, J z prime remains
equal to J z, and the charge density now will
23:15.539 --> 23:20.590
be different in a different frame S prime
as given by this particular equation rho prime
23:20.590 --> 23:27.590
will be gamma times rho minus V J x upon C
square. So, this is somewhat like your x,
23:28.940 --> 23:33.669
y, z and if you want to remember; somewhat
like x, y, z, t-transformation where x prime
23:33.669 --> 23:39.070
turned out to be gamma x minus V t. So, it
is a similar type of equations if you go back
23:39.070 --> 23:44.940
in that particular case J x prime is gamma
times J x minus V rho and rho prime is rho
23:44.940 --> 23:48.429
gamma minus V upon C square J x.
23:48.429 --> 23:55.429
I can always write the inverse transformation
equation. For that particular thing as we
23:56.509 --> 24:03.509
have said, change prime to unprimed quantities
and vice versa; make v as minus V. So, this
24:03.619 --> 24:08.059
is what happens. As the inverse transformation,
it means that if I know the current densities
24:08.059 --> 24:13.369
and the charge densities in S prime frame
of reference, I can find out the current density
24:13.369 --> 24:19.960
and the charge density in S frame of reference
by using these inverse transformation equations.
24:19.960 --> 24:26.960
So, this is what has happened to my transformation
of current density. Now let us come back to
24:32.479 --> 24:36.979
the situation of current carrying conductor.
24:36.979 --> 24:43.979
My idea of using this particular example of
giving current carrying conductor is only
24:44.200 --> 24:51.200
to look little more deeply into the origin
of these electric fields and magnetic fields
24:54.879 --> 25:00.159
as we change from one frame to another frame
of reference. And this particular example,
25:00.159 --> 25:04.710
generally reasonably illustrative because
many times we have a question; that suppose
25:04.710 --> 25:09.059
there is a particular frame of reference in
which we see only magnetic field and you say
25:09.059 --> 25:13.769
that you go to a different frame of reference.
Now in addition to this magnetic field, you
25:13.769 --> 25:18.070
also have an electric field. So, from where
this electric field has come? Because once
25:18.070 --> 25:22.769
we say that the equation, the Maxwell’s
equation which are the basic equations do
25:22.769 --> 25:26.429
not change the relativity.
So, what is the origin of this electricity
25:26.429 --> 25:31.820
field? So, let us what we are trying to do
is by giving this particular example to go
25:31.820 --> 25:36.669
little deeper into this particular aspect,
try to somewhat understand these things. So
25:36.669 --> 25:43.669
now, let us come to a realistic conductor
and let us assume that it is carrying a current
25:43.739 --> 25:50.739
in plus x-direction as seen in frame S. So,
there is a frame S in which you see a current
25:50.899 --> 25:57.899
flowing in a particular wire or a conductor
along the x-direction. Let us assume that
26:00.440 --> 26:05.289
charge carriers are actually the electrons.
It means actually the electrons must be drifting
26:05.289 --> 26:12.289
in minus x-direction and that is the reason
that the current is actually in the plus x-direction.
26:12.779 --> 26:16.840
As we have said, the direction of the motion
direction of the drift of the electron is
26:16.840 --> 26:20.749
opposite to the direction of the conventional
current.
26:20.749 --> 26:27.749
Now when we talk about the current we do not
talk about this so clearly but let us be very
26:29.649 --> 26:36.279
very specific. I am looking this particular
thing from a frame S and in this frame S,
26:36.279 --> 26:40.029
I assume that the positive charge carriers;
remember the conductor overall is electrically
26:40.029 --> 26:45.669
neutral. It is a slightly different situation
from what we have discussed now. I am talking
26:45.669 --> 26:50.749
now a realistic conductor. See, if there was
no electric field, this particular conductor
26:50.749 --> 26:55.379
was electrically neutral and the negative
charges are there, the positive charges both
26:55.379 --> 27:02.379
are there. Now these electrons tend to be
somewhat free or approximately free and when
27:03.159 --> 27:06.779
we apply these electric fields, these electrons
are the one which actually give rise to the
27:06.779 --> 27:10.340
current.
But those positive charges which are sort
27:10.340 --> 27:17.340
of immobile which are ions which have been
left behind; they remain at rest, that is
27:17.399 --> 27:22.960
what is the normal picture of current which
is always told when I was in high school.
27:22.960 --> 27:29.679
So, I am looking at the same frame in which
this picture is correct. It means I am assuming
27:29.679 --> 27:33.559
that the positive charges are at rest and
it is the negative charges which are drifting
27:33.559 --> 27:38.940
in negative x-direction to give rise to a
current in the plus x-direction. That is what
27:38.940 --> 27:45.940
I have said here; in this frame, the positive
charges are at rest and electrons drift in
27:46.629 --> 27:53.629
negative x-direction giving a current in positive
x-direction, alright. Now let me assume we
27:57.969 --> 28:03.320
all of us know that, if there is a current
carrying conductor; let us assume that is
28:03.320 --> 28:08.479
a long conductor. This conductor will generate
electric magnetic field around it.
28:08.479 --> 28:13.779
So because of this particular current, there
is a magnetic field which is created around
28:13.779 --> 28:20.779
it. We know how to find out the direction
of the magnetic field if the current going
28:22.749 --> 28:29.749
like this and you can find out. The magnetic
field lines of forces may be circular with
28:33.899 --> 28:38.219
the wire passing through the center of the
circles. These are well known; actually it
28:38.219 --> 28:43.539
is not discussed in detail. What I am also
insisting is that all is generating only the
28:43.539 --> 28:50.090
magnetic field without any electric field.
So, there is no electric field and the reason
28:50.090 --> 28:51.409
there is no electric field because they are
positive charge carriers as well as negative
28:51.409 --> 28:55.639
charge carriers and in a given volume whatever
small elements we have taken, we assume that
28:55.639 --> 28:58.929
there are same number positive charge carriers
and same number of negative charge carriers.
28:58.929 --> 29:05.929
So, the net, let me just write it here. If
we take a small element in a wire, in this
29:13.309 --> 29:17.359
element negative charge carriers or positive
charge carriers, both are same. Therefore,
29:17.359 --> 29:21.820
if you take a small section of the wire, you
do not find net charge density here and therefore,
29:21.820 --> 29:25.429
there is no electric field which is generated
outside. There is only a flow of electron
29:25.429 --> 29:30.690
and at all the times, this charge neutrality
is maintained in a small section of this particular
29:30.690 --> 29:36.139
wire. Therefore, there is no electric field,
but there is a magnetic field. So, this is
29:36.139 --> 29:40.239
picture with which we are quite familiar in
our high school, when we have discussing various
29:40.239 --> 29:47.239
laws related to magnetism; at that time we
have been talking about this type of behavior.
29:48.090 --> 29:55.090
So, this is what I said. Let the negative
and positive charge densities in frame S be
29:55.469 --> 30:02.469
rho e and rho p; this e symbol I have said
for negative charge carriers, and this p symbol
30:03.940 --> 30:10.940
I have said for positive charge carriers.
So, let us assume that the charge density
30:11.599 --> 30:18.599
for negative charge carriers is rho e and
for positive charge carriers is rho p; of
30:19.009 --> 30:24.049
course, because this is charge density. So,
this is negative because the charge is negative,
30:24.049 --> 30:30.029
the number density is not negative. Number
density has to be always positive. While positive
30:30.029 --> 30:36.309
charge density is positive because this charge
is positive, and because we are saying that
30:36.309 --> 30:40.440
there is no electric field in this particular
frame of reference, I expect that if I take
30:40.440 --> 30:46.159
any small unit volume, necessarily a small;
in fact it is a very small volume. You always
30:46.159 --> 30:49.769
find that this rho p plus rho e is equal to
zero.
30:49.769 --> 30:54.119
So, there is no charge density in a smallest
volume; a small volume which you take along
30:54.119 --> 30:58.570
the wire. I am not writing in the differential
format sector because that is not the idea.
30:58.570 --> 31:04.789
We want to make thing simple; so just want
to say that I expect that this rho p plus
31:04.789 --> 31:09.999
rho e must be 0 in this particular frame because
I do not see any electric field, I see only
31:09.999 --> 31:15.799
the magnetic field. There is a current, there
is a current density, but there is no electric
31:15.799 --> 31:22.799
field. This current and current density causes
the magnetic field. Now we realize that because
31:24.389 --> 31:29.859
in this particular frame, it is only the negative
charges which are moving. So the current density,
31:29.859 --> 31:34.429
the net current density is being caused only
by the motion of negative charge carriers;
31:34.429 --> 31:39.479
positive charge carrier would not contribute
to any current density which is obvious, because
31:39.479 --> 31:42.950
they are not moving in this particular frame
of reference.
31:42.950 --> 31:49.950
So, this is what I said. In this frame S,
the current density is solely due to the electrons.
31:50.649 --> 31:57.139
Therefore, net current density I can write
as equal to J xe; I am writing only x component
31:57.139 --> 32:01.139
because current density current I have been
told that is only in the x-direction, current
32:01.139 --> 32:08.139
density also in the x-direction. J xe I am
using to mention the x component of the current
32:10.489 --> 32:17.229
density for negative charge carriers and as
we have said and J x is the overall net; the
32:17.229 --> 32:22.479
total current density. As we have said because
positive charges are immobile, so this J x
32:22.479 --> 32:29.479
is solely because of J xe and that will be
given by rho e multiplied by the drift velocity
32:30.989 --> 32:36.119
of the electrons or negative charge carriers;
negative charge carriers we mean electron.
32:36.119 --> 32:41.820
So, this is what it will be given as rho e
times U de where rho is the charge density,
32:41.820 --> 32:48.359
U de is the drift velocity we have already
defined; this particular thing is exactly
32:48.359 --> 32:49.979
the same thing.
32:49.979 --> 32:56.979
Now let us go to a different frame of reference
S prime in which the drift velocity of electron
32:56.999 --> 33:01.659
is found to be zero. So, I have already found
out what is the drift velocity. If I know
33:01.659 --> 33:06.389
the current density, I know what is the drift
velocity; if I know the drift velocity, I
33:06.389 --> 33:12.049
can always go to a frame of reference, an
inertial frame of reference which has their
33:12.049 --> 33:16.889
same relative velocity with respect to S as
the drift velocity of the electrons.
33:16.889 --> 33:23.889
What it means is that, if we have wire in
which these electrons are moving in this particular
33:31.330 --> 33:38.330
direction; this is plus x-direction. Let us
consider one electron which has exactly the
33:38.919 --> 33:45.409
same speed; let us not bother about that.
Just consider a drift velocity and assume
33:45.409 --> 33:50.320
that overall electrons are drifting in this
particular way. I go to a particular frame
33:50.320 --> 33:54.200
of reference which has exactly the same velocity;
of course this frame of reference also has
33:54.200 --> 34:00.299
move in this particular direction so that
they find that overall, there is no drift
34:00.299 --> 34:05.969
of electrons in this particular frame of reference.
So in this particular S prime frame of reference,
34:05.969 --> 34:10.540
I am defining this particular frame of reference
S prime as one frame of reference which of
34:10.540 --> 34:14.750
course is inertial frame of reference, because
drift velocity is supposed to be constant
34:14.750 --> 34:20.589
for a given electric field. So long, current
density is constant; the drift velocity is
34:20.589 --> 34:26.409
also constant. So, I am looking at this particular
aspect from a frame of reference S prime in
34:26.409 --> 34:33.409
which electrons are not found to be drifting.
So that is what I said, let us observe this
34:38.320 --> 34:43.649
wire from a frame S prime in which the drift
velocity of the electron is found to be zero.
34:43.649 --> 34:49.899
Let us see what an observer in S prime frame
would notice; would it notice that the current
34:49.899 --> 34:56.899
has become zero and therefore, there should
not be any magnetic field. No, that is not
34:57.690 --> 34:59.920
a correct picture.
34:59.920 --> 35:06.070
Actually in this frame, the positive charges
will be observed to be moving in plus x-direction.
35:06.070 --> 35:11.260
See remember, it was in S frame that your
electrons were drifting at the positive charges
35:11.260 --> 35:17.590
were stationary. But once I have changed my
frame of reference to make the electrons stationary
35:17.590 --> 35:21.820
in that particular frame of reference; in
that particular frame of reference, the positive
35:21.820 --> 35:28.820
charges will move and because as we have said
here, this velocity is in the negative direction.
35:29.380 --> 35:34.340
So, the positive charges which were mobile
earlier which would appear to a person sitting
35:34.340 --> 35:41.340
here to be moving in positive x-direction.
Therefore, an observer in S prime would feel
35:43.550 --> 35:48.950
that all these positive charge carriers are
moving in plus x-direction. So, there is a
35:48.950 --> 35:53.810
current in that particular frame of reference
but that current is being caused by the positive
35:53.810 --> 36:00.810
charge carriers and this positive charge carrier
will actually produce a magnetic field. Therefore,
36:04.610 --> 36:11.610
a magnetic field is expected to be present
also in S prime frame of reference. What about
36:12.000 --> 36:14.620
electric field?
36:14.620 --> 36:21.060
If you look at these equations, these are
the transformation equation which I have written
36:21.060 --> 36:24.700
from S to S prime frame of reference. Of course
in this case, we have x to be negative; lets
36:24.700 --> 36:29.840
forgot about it and I am just writing in the
number form. If you look at this particular
36:29.840 --> 36:34.220
equation, of course E x prime will turn out
to be equal to Ex, but E y prime will depend
36:34.220 --> 36:40.750
on V B z. E z prime will depend on V B y.
Depending upon which point you are looking
36:40.750 --> 36:47.750
or you are looking at the field which point;
of course the current in the x-direction.
36:48.100 --> 36:55.100
You will always find; you may always find
at least a component of B y or B z or both.
36:55.940 --> 37:01.900
Even though E y and E z are zero because we
have said in S frame there is no electric
37:01.900 --> 37:06.320
field. But there is a magnetic field and this
magnetic field has to be in some direction;
37:06.320 --> 37:11.820
of course it cannot be in x-direction because
the current flows in the x-direction. And
37:11.820 --> 37:17.290
the magnetic field is being caused only by
that current flow and I am assuming this to
37:17.290 --> 37:24.290
be infinite wires or sought of infinite wire.
So therefore, at least one of these will be
37:24.390 --> 37:29.310
non-zero. It means in this particular frame
in addition to the magnetic field, the observer
37:29.310 --> 37:36.140
would also find electric field. This is going
to happen if my transformation equations are
37:36.140 --> 37:40.710
correct.
So, these electric fields are present in the
37:40.710 --> 37:47.710
particular frame of reference. From where
are they arising? Earlier we have said that
37:47.780 --> 37:54.560
their charge densities were just neutralizing
at each small volume of the wire and therefore,
37:54.560 --> 38:01.070
there was no net charge in any small section
of the wire and therefore, there was no electric
38:01.070 --> 38:08.070
field earlier. From where the electric field
is coming; so now we will realize, that is
38:09.290 --> 38:11.690
what we will do just now, to show that it
was too earlier in S frame. In S prime frame
38:11.690 --> 38:18.690
of reference, if you take a small section
of the wire; now you will find it to be charged
38:18.820 --> 38:23.560
and if you find it to be charged, it is this
charge which will generate this electric field.
38:23.560 --> 38:28.590
So, electric field will be present in this
frame of reference because that particular
38:28.590 --> 38:34.580
section; I mean sections of wire being charged.
How do I do that? For particular I must know
38:34.580 --> 38:41.580
a charge density transformation; that is what
I am going to do next.
38:42.020 --> 38:49.020
For that let us evaluate the charge densities
in S prime frame of reference. I have just
38:49.380 --> 38:56.380
now evaluated; I just now found out the transformation
equation relating to the charge densities.
38:56.920 --> 39:03.360
Let us do the same thing here, find out what
will be rho prime if I know rho and S, what
39:03.360 --> 39:06.180
will be rho prime.
39:06.180 --> 39:11.630
Of course, relative velocity of between the
frames I have taken specifically equal to
39:11.630 --> 39:16.340
U de; of course I have reserved symbol U instead
of V. Though when we originally defined charge
39:16.340 --> 39:21.490
densities we had used symbol V, but I am using
U because V we have as I will say always reserved
39:21.490 --> 39:26.180
for the relative velocity between the frames.
Of course in this specific example, I am taking
39:26.180 --> 39:31.820
V is equal to U de because only in that particular
case, the electrons or the negative charge
39:31.820 --> 39:38.260
carriers will be at rest. So, this is my transformation
equation for the negative charge carriers
39:38.260 --> 39:45.260
rho e prime is equal to gamma and rho e minus
U de because V is equal to U de J xe whatever
39:46.360 --> 39:53.060
was the current density divided by C square.
So, this is gamma rho J xe I can write as
39:53.060 --> 39:58.040
rho e into U de. So, this rho e I have written
here, this becomes U de square divided by
39:58.040 --> 40:04.820
C square. This rho e I can take it out, this
will become gamma rho e multiplied by one
40:04.820 --> 40:11.820
minus U de divided by C square; put this is
also squared; U de is the drift velocity of
40:11.960 --> 40:17.140
electrons as was seen in S frame. So this
will be the charge density of electron which
40:17.140 --> 40:24.140
will be seen in S prime frame of reference.
What will happen to the charge density of
40:24.890 --> 40:31.890
positive charge carriers? I will write exactly
the same equation, gamma rho p minus V J x
40:34.060 --> 40:41.000
p divided by C square where J x p is the current
density of positive charge carriers in S frame.
40:41.000 --> 40:46.940
But I know that this J x p is zero because
the charges are not moving in S frame of reference;
40:46.940 --> 40:51.860
therefore, current density of that particular
frame has to be zero. Therefore, this quantity
40:51.860 --> 40:58.860
will be zero and this I can write as gamma
rho p. As we can seen that the charge densities
41:00.950 --> 41:06.180
as seen in S prime frame of reference are
not same because of this transformation; therefore,
41:06.180 --> 41:11.760
this is not equal to this. They were earlier
same; the magnitudes were earlier same, of
41:11.760 --> 41:17.420
course signs were different. Their magnitudes
were same in S frame, but in S prime frame
41:17.420 --> 41:23.870
of reference even their magnitudes have become
different. So, let us evaluate the net charge
41:23.870 --> 41:28.810
density in S prime frame of reference which
is just the sum of these two which earlier
41:28.810 --> 41:29.760
was zero.
41:29.760 --> 41:36.540
So, I have just used this particular equation
which is from this particular transparency
41:36.540 --> 41:43.540
this plus this; this is what I have written
here. This gamma rho E would cancel out here.
41:44.870 --> 41:51.870
Remember this rho E was equal to minus rho
p in S prime frame of reference. So this rho
41:51.950 --> 41:57.840
e has been changed as minus rho p here. So,
once I do that this gamma rho p and this will
41:57.840 --> 42:04.840
cancel out and this will become gamma rho
p U de square upon C square. So, we find that
42:05.950 --> 42:11.440
there is a net charge density in this particular
wire. So in S prime frame of reference, it
42:11.440 --> 42:17.620
will find that the wire is charged’ if you
take one particular section of the wire, it
42:17.620 --> 42:24.620
will be charged and if it charged of course,
we generate electric field.
42:25.280 --> 42:30.600
Now what happens to charge conservation? Some
of you may ask that, does it mean to say that
42:30.600 --> 42:36.940
this wire overall has become electrically
non-neutral. Originally we have said that
42:36.940 --> 42:41.870
the conductor was electrically neutral. Now
we are saying that there is a net charge density.
42:41.870 --> 42:47.540
Does it mean to say that, it is no longer
electric neutral? What happens; from where
42:47.540 --> 42:53.030
we got the additional charges? Is the charge
conservation obeyed or not? The question is
42:53.030 --> 42:54.880
that see normally you will not have, strictly
speaking, infinite wire. You will always have
42:54.880 --> 43:01.550
a loop. So once you are supplying the current,
now there will be current flowing and then
43:01.550 --> 43:03.950
eventually the current has to flow back into
a particular direction.
43:03.950 --> 43:10.060
So, whatever is the situation, you will always
have situation somewhere where you have wire
43:10.060 --> 43:17.060
and this wire eventually has to close. Here
there has to be some current source. Now as
43:20.810 --> 43:24.130
you can see, the current direction in this
section of the wire is going to become different
43:24.130 --> 43:29.390
from whatever is here. If I am looking at
positive x-direction, the charges are going;
43:29.390 --> 43:35.080
let us say the current is going on this particular
direction on this way, while here going this
43:35.080 --> 43:39.910
way. So inverse section of the wire if I am
finding it positively charged; the other section
43:39.910 --> 43:44.970
of the wire I will find negatively charged.
So, overall the wire will still remain to
43:44.970 --> 43:49.780
be electrically neutral. So nothing happens
to that particular thing once we realize that
43:49.780 --> 43:54.940
thing. But if you take a specific section
the way we have happen defining this particular
43:54.940 --> 43:59.500
thing, we feel only because of this particular
wire that comes, because of this positive
43:59.500 --> 44:05.230
charges which are being in that particular
frame; we use the particular charge carrier,
44:05.230 --> 44:10.610
this particular wire as a charged charge now.
44:10.610 --> 44:17.610
Now with this particular thing as I said,
this is probably the last section that we
44:18.510 --> 44:24.290
have wanted to cover formally; I want to just
introduce how the Maxwell’s equation we
44:24.290 --> 44:30.620
expect them to remain invariant. In fact if
you remember the electric field and magnetic
44:30.620 --> 44:36.680
field transformation, we had obtained from
the force equation; the Lorentz force equation
44:36.680 --> 44:41.950
and saying that this force must obey a force
transformation law. In fact, they can also
44:41.950 --> 44:48.950
be derived by maintaining that the Maxwell’s
equations are unaltered or unaffected when
44:49.840 --> 44:53.910
I change my frame of reference. We do not
change any magnetic equations unlike we have
44:53.910 --> 45:00.710
changed many of the classical mechanics equation;
the equations pertaining to electromagnetic,
45:00.710 --> 45:05.230
electromagnetic theory.
They do not change, must be changed by frame
45:05.230 --> 45:08.380
of reference. The equations which are the
basic Maxwell’s equation, four Maxwell’s
45:08.380 --> 45:13.260
equation, they do not change. So, I will not
proving general in general which is beyond
45:13.260 --> 45:17.410
the outcome of this course which is required;
if you want a general proof much more, many
45:17.410 --> 45:23.690
more details. We will just take one simple
example and convince you and I am always trying
45:23.690 --> 45:28.780
to say that I am trying to convince that Maxwell’s
equations are expected; nothing like that,
45:28.780 --> 45:35.060
nothing more than that. So, this is what I
have said in relative to the Maxwell’s equations
45:35.060 --> 45:40.740
are unaffected; we shall not give the general
proof, we are just trying to convince you.
45:40.740 --> 45:47.340
So, these are the four Maxwell’s equations
which are very very standard equations. The
45:47.340 --> 45:53.420
first equation in essentially what we call
as a Gauss’s law which talks of divergence
45:53.420 --> 46:00.420
of electric field in terms of the charge density.
This is a similar Gauss’s law for the magnetostatics
46:02.530 --> 46:07.080
which is divergence of B equal to 0 and this
quantity 0 because it always says that there
46:07.080 --> 46:14.080
is no monopole. This is what is the curl of
electric field which is called Faraday’s
46:15.170 --> 46:20.890
laws of electromagnetic induction and this
is Ampere’s law with the connection of Maxwell
46:20.890 --> 46:26.970
pertaining to the displacement current which
gives you current B, so the two equations
46:26.970 --> 46:32.700
are having divergence.
There are two equations which are having curl,
46:32.700 --> 46:38.440
and these are the four equations which we
call as Maxwell’s equation. So, I will take
46:38.440 --> 46:43.810
one of these equations; this particular equation,
the third equation and out of that I will
46:43.810 --> 46:49.800
take one component and try to convince you
that this does not alter once I go for a frame
46:49.800 --> 46:55.750
S to S prime frame of reference. So, this
is what I am trying to do in this particular
46:55.750 --> 46:59.710
way as a last part of this particular course
46:59.710 --> 47:06.710
So we said, we shall take third equation and
try to write this in S prime frame. But before
47:07.230 --> 47:13.740
we do that, let us expand this curl into individual
components; remember it has a curl here. There
47:13.740 --> 47:18.040
is a curl of electric field and here you have
derivative of time, derivative of magnetic
47:18.040 --> 47:25.040
field. So, let us first write this into the
component form. Then this components will
47:25.550 --> 47:31.160
of course be in the form of x, y, and z and
on the right-hand side, you will have time
47:31.160 --> 47:37.000
derivative. Now I will change these x, y,
z to x prime, y prime, and z prime; this t
47:37.000 --> 47:40.830
to t prime.
Assuming that x prime, y prime, z prime, t
47:40.830 --> 47:45.030
prime, are related to x, y, z, t by Lorentz
transformation because that is what is the
47:45.030 --> 47:51.930
relativistic transformation. And try to see
that I can write exactly similar equation
47:51.930 --> 47:56.790
in S prime frame of reference where E will
be replaced by E prime and B will be replaced
47:56.790 --> 48:01.960
by B prime and this curl will be all with
respect to x prime, y prime, z prime, and
48:01.960 --> 48:06.890
this t will be with respect to t prime. First
I may have to write that, I have show that
48:06.890 --> 48:11.240
is Maxwell’s equation remain invariant.
Of course, we can do similar type of things
48:11.240 --> 48:16.610
for all other equations. We know how the charge
density transforms; we have just now discussed.
48:16.610 --> 48:22.240
We have also known how current density transforms,
that also we have discussed. So, we know the
48:22.240 --> 48:25.330
transformation of all these equations. So
in principle, you could have taken any equation
48:25.330 --> 48:30.640
and try to proof it. But as I said I am not
doing that in general, just taking this particular
48:30.640 --> 48:35.860
equation trying to convince you. So, let us
first expand the curl del cross E.
48:35.860 --> 48:42.420
So, left hand side of that equation had just
curl of E; that is del cross E. Of course,
48:42.420 --> 48:46.020
you can use any method of expanding it; some
people will try to write in form of determinant;
48:46.020 --> 48:52.240
then try to expand it. I have written in a
particular vector form which I find generally
48:52.240 --> 48:57.430
much more easy to remember but in any way
one prefers for this you can write; this particular
48:57.430 --> 49:03.670
way also we can expand this curl. So, this
del operator as we call can be written as
49:03.670 --> 49:09.960
i del del x plus j del del y plus k del del
z, where I, j, k are the unit vectors. Then
49:09.960 --> 49:16.200
cross E i can write in the vector form as
i E x plus j E y plus k E z; again i, j, k
49:16.200 --> 49:20.650
are unit vectors in the x-, y-, and z-direction,
this standard way.
49:20.650 --> 49:25.870
So, I have to expand this curl; it means I
have to take, this cross this, this cross
49:25.870 --> 49:29.990
this, this cross this, then plus this cross
this, plus this cross this, plus this cross
49:29.990 --> 49:36.720
this, then this cross this, this cross this,
this cross this. If I take first term i del
49:36.720 --> 49:43.720
del x, when you take i cross i, I will get
zero; i cross j will give me k; and i cross
49:44.460 --> 49:51.020
k will give me minus j. Similarly j cross
i will give minus k; j cross j will give me
49:51.020 --> 49:55.870
zero; and j cross k will give me i. Here also
k cross k will give me zero.
49:55.870 --> 50:01.270
So all I wanted to say that normally if you
would have expanded this, you will have got
50:01.270 --> 50:05.770
nine terms; because one has to operate on
three, second also has to operate on three,
50:05.770 --> 50:10.830
third also has to operate on three; we will
get nine terms. But out of those, three terms
50:10.830 --> 50:15.480
will be terms zero. One because of i cross
i, another because of j cross j, and third
50:15.480 --> 50:22.220
will be because of k cross k. So, eventually
you will be landing only with these six terms.
50:22.220 --> 50:26.520
I am not giving the details of these things,
I can think it and mark out simply and try
50:26.520 --> 50:31.780
to convince yourself that whatever I have
written is probably correct.
50:31.780 --> 50:37.500
So, these are my cross products and all I
have done is expanded into this particular
50:37.500 --> 50:44.500
form. As I have said that this explanation
you could have your doubt in any other way
50:44.570 --> 50:51.570
which is comfortable to you. So all I have
written is six terms and content kept those
50:51.660 --> 50:58.660
terms along the x-direction here, y-direction
here, and the z-direction here. So, this del
50:59.530 --> 51:04.690
cross E i can writen like this; that this
has to be equated to the right-hand side which
51:04.690 --> 51:11.690
had del B del t.
51:12.130 --> 51:18.940
The x component and y component and z component
of this equation, in fact can be written as
51:18.940 --> 51:25.940
i del B x del t plus j del B y del t plus
k del B z del t. Then this x component can
51:36.950 --> 51:41.390
be equated to x component of the curl that
we have obtained; y component can be equated
51:41.390 --> 51:48.390
to the y component of the curl that I have
found out; z component can be equated to the
51:48.770 --> 51:53.330
z component. So, I will get three equations
which I am writing in the next transparency.
51:53.330 --> 51:57.930
So, these are collecting them; we have collected
all the components and equated to the right-hand
51:57.930 --> 52:04.930
side. So, I get these three equations. Now
I will use only one of this equation; this
52:05.890 --> 52:11.660
particular equation, the second equation.
Try to transform into S prime frame of reference
52:11.660 --> 52:18.650
by using this standard equation corresponding
to the partial derivatives. I will not go
52:18.650 --> 52:23.010
into the details because this requires lot
of time but as I say my idea is only to convince
52:23.010 --> 52:27.610
you and not to give you too much of details
of the mathematics, which I say that is beyond
52:27.610 --> 52:30.610
the scope of this particular course.
52:30.610 --> 52:36.400
So, we will use the standard partial differentiation
formula and use it along with the Lorentz
52:36.400 --> 52:38.790
transformation.
52:38.790 --> 52:44.730
What is this particular formula? This del
del x can be written as del x prime del x
52:44.730 --> 52:51.350
del del x prime plus del y prime del x del
del y prime plus del z prime del x del del
52:51.350 --> 52:56.890
z prime plus del t prime del x del del t prime.
This is the standard partial derivative formula.
52:56.890 --> 53:01.910
Then you have therefore variables on which
it is dependent. I know from Lorentz transformation
53:01.910 --> 53:08.520
x prime is equal to gamma x minus V t, y prime
is equal to y, z prime is equal to z, t prime
53:08.520 --> 53:14.720
is equal to gamma t minus Vx upon C square.
So, let us examine this particular term. If
53:14.720 --> 53:19.930
I take del x prime del x, this is x prime,
this is partial derivative; it means all other
53:19.930 --> 53:25.090
variables I have take them as constant. So,
t will turn on constant. Once I take the partial
53:25.090 --> 53:29.580
derivative of x with respect to x prime with
respect to x, so I will just get gamma. This
53:29.580 --> 53:36.580
term will give be zero. So, this becomes gamma
into del del x prime. Similarly del y prime
53:37.920 --> 53:43.110
del x if I take partial derivative with respect
to x, it means y has to be treated as constant.
53:43.110 --> 53:49.180
So, this will give me zero, del z prime del
x will give me zero. Here del t prime del
53:49.180 --> 53:55.840
x, t prime does depend on x. Once I take partial
derivative, t has to be taken as constant.
53:55.840 --> 54:02.840
I take derivative, so I will get minus gamma
t V upon C square. So this is gamma not t,
54:02.950 --> 54:09.950
minus gamma V upon C square. So, minus gamma
V upon C square into del del t prime. So,
54:10.470 --> 54:15.470
this operator del del x can be replaced by
this operator once I change my frame from
54:15.470 --> 54:19.380
S to S prime frame of reference.
54:19.380 --> 54:24.560
I can write similar equations for y which
will give me very clearly del y is equal to
54:24.560 --> 54:29.060
del y prime. This you can write to work it
out yourself and exactly, similarly I will
54:29.060 --> 54:35.180
get del del z is equal to del del z prime.
54:35.180 --> 54:40.160
Again del del t will be different from del
t prime. I use exactly the same thing, again
54:40.160 --> 54:45.700
you can work it out yourself; del del t will
turn out to be equal to gamma del del t prime
54:45.700 --> 54:52.700
minus V del del x prime. So now, I know how
to replace these partial derivatives from
54:53.060 --> 54:56.490
S frame to S prime frame of reference.
54:56.490 --> 55:03.310
So, these things I will substitute in this
particular equation. So this del del z prime
55:03.310 --> 55:07.480
del del z, I will replace by whatever we have
valued it earlier. Of course as delta z is
55:07.480 --> 55:11.140
concerned, it is just that z becomes z prime.
Here there is a partial derivative with respect
55:11.140 --> 55:17.460
to x. So this once I change to del x prime,
I will get another term. Similarly when this
55:17.460 --> 55:21.600
partial derivative with respect to time, I
will get another term which I am doing in
55:21.600 --> 55:23.460
the next class we will see.
55:23.460 --> 55:28.360
I am rather going fast, but idea as I have
to say is to convince you. So, this equation
55:28.360 --> 55:34.730
as I said remain identical; del del x I am
replacing by this equation. So, once I go
55:34.730 --> 55:39.150
to prime frame of reference, this will become
del E z del x prime minus V upon C square
55:39.150 --> 55:46.150
del E z del t prime. This del t also gets
changed and this becomes del B y del t prime
55:46.510 --> 55:53.510
minus V del B y del x prime. So, this equation
now becomes in this particular form. What
55:54.790 --> 55:59.480
I will do, I will collect terms correspond
to partial derivative with respect to x and
55:59.480 --> 56:04.830
with respect to partial derivative with respect
to t.
56:04.830 --> 56:11.830
So if I do that, this equation becomes in
this form. This is interesting because normally
56:12.550 --> 56:16.540
if a similar equation has to be obeyed in
S prime frame of reference.
56:16.540 --> 56:23.540
I expect del E x prime is equal to del z prime
minus del E t prime by del x prime is equal
56:32.420 --> 56:39.420
to del B y by del t prime. This is an equation
which I expect to be true because this is
56:45.960 --> 56:49.590
exactly the same equation. I am sorry there
should be a prime here. Because this what
56:49.590 --> 56:53.910
I expect if the same equation I am able to
write in S prime frame of reference, I will
56:53.910 --> 56:59.020
be able to say that things are consistent
in S prime frame of reference. This is the
56:59.020 --> 57:03.520
equation which I have written here. So if
I write this equal to this, this happens to
57:03.520 --> 57:09.150
be equal to E z prime and this happens to
be equal to B y prime, I know that these equations
57:09.150 --> 57:14.810
will be identical in S prime frame of reference
and this is precisely what I know is true
57:14.810 --> 57:17.810
from electric field and magnetic field transformation.
57:17.810 --> 57:24.810
And of course, E x must be equal to S prime.
So, from this I can write these equations.
57:26.140 --> 57:32.260
So, these equation will be consist ant in
S prime provided we have these equations valid.
57:32.260 --> 57:36.340
I know from electric field and magnetic field
of transformation that these equations are
57:36.340 --> 57:41.410
correct. So, I expect this equation to be
valid also in S prime frame of reference.
57:41.410 --> 57:45.500
So, what I have only done is shown only for
one equation. You can try to show for other
57:45.500 --> 57:50.480
equation; you have to know somewhat more mathematics
and try to convince yourself that Maxwell’s
57:50.480 --> 57:56.860
equations are actually consistent in all both
the frames. They remain unaltered; we do not
57:56.860 --> 58:01.950
change Maxwell’s equation once I go from
S frame to S prime frame of reference.
58:01.950 --> 58:06.490
So, this is my summary. We discussed current
density four-vector. We tried to analyze the
58:06.490 --> 58:10.690
current carrying conductor in two frames.
Especially discussed how the electric field
58:10.690 --> 58:16.630
is generated in that particular frame of reference
and given a hint how Maxwell’s equation
58:16.630 --> 58:21.480
remain unaltered. So, this happens to be the
end of the course. Best of luck all the best.