Greetings, we will in this class conclude
the discussion on classical electrodynamics
which is a brief overview of Maxwell's equations
and how it connects to the special theory
of relativity. What we will be discussing
here is, examine the trajectories of charged
particles in electromagnetic fields, but as
seen by 2 different observers. One who is
in this frame which is a black frame and the
other who is in this blue frame which is moving
at a constant velocity with respect to the
black frame.
I want to emphasize that this velocity is
constant. So, both are inertial frames of
references; there are no pseudo forces which
are involved. But, what is electric field
and magnetic field for one observer is not
the same electromagnetic field for the other
observer and that is the new thing that we
are going to learn.
The equation of motion of course, is given
over here that the rate of change of momentum
is force; so it is a completely classical
Newtonian idea between cause and effect. The
force that we are talking about is the electromagnetic
force namely, the Lorentz force and this is
something that you need over and above Maxwell's
equations, other than the boundary conditions
that are required to solve problems of charged
particle dynamics in electromagnetic fields.
The governing idea is that the speed of light
does not change from one inertial frame to
another. What changes is time and in fact
also, space intervals; so these are the governing
ideas. Einstein was a very young person when
he formulated this in 1905; he must have been
about 26 years old I would think so, something
like that and this connects the special theory
of relativity with the with laws of electrodynamics.
So, like I mentioned in the previous class,
as a result of Lorentz transformations x y
z and t which is time for the observer in
the unprimed frame of reference, must transform
to x prime, y prime, z prime, t prime. These
transformations are essentially the Lorentz
transformations that we are talking about,
not Galilean and therefore, t prime is different
from t. The trajectory of the particle is
given by how the function the position vector
changes as the function of time. Whereas,
for the observer in the primed frame the trajectory
will be given by how the position vector in
the primed frame changes as the function of
time in the primed frame which is t prime
which being different from t.
The connection comes from the fact that what
is E and B for the first observer is a different
mix of E and B to the second observer , which
is E prime and B prime which is not the same
E prime; is not equal to E and B prime; is
not equal to B; that is the essential idea
and therefore, the observer in the second
frame of reference can get his trajectory
by setting up an equation of motion in the
primed frame which is d p prime by d t prime.
Notice that you have the primed symbols appearing
over here because the times are different
and then, automatically every other parameter
also has got a different meaning.
Not only that because, E and B transform to
E prime and B prime, the force is given by
F prime rather than F because, E is no longer
E, it is E prime; v is no longer v, it is
v prime; B is no longer B, it is B prime;
but then, in the primed frame you are still
talking about the same electromagnetic interaction
namely the Lorentz force; there is no pseudo
forces.
The solution to this will be given by the
position vector in the primed frame as a function
of t prime and now If you solve this equation
of motion in the unprimed frame and carry
out transformations to the primed frame by
Lorentz transformations and then plot r prime
as a function of t prime, you will get the
trajectory in the primed frame. The other
way of getting the trajectory in the primed
frame is to set up the equation of motion
in the primed frame and express the solution
r prime as a function of t prime in the primed
frame. Now, these are again completely 2 different
methods.
The first method is a Lorentz transformations
of the coordinates. The second method involves
the relativistic transformations of the electromagnetic
field from E B in 1 frame to E prime B prime
in the other and then setting up a different
equation of motion. If we have done this correctly,
then the solution obtained using this method
must agree with the solution obtained using
the other method. The two should generate
the same curves in their respective spaces.
So, this is the test that r prime is a function
of t prime obtained using Lorentz transformations
of the coordinates x y z t or the relativistic
transformations of E B to E prime B prime
and solving a new equation of motion generate,
you know, Congruent space curves giving the
trajectories. So, here we have the Lorentz
transformations which we have discussed already
in unit 6 and these are the transformations
of the Electromagnetic field.
So, what is E x is E x prime to the other;
but, what is E y is not E y prime to the other.
E y prime is made up of a mix of E y and B
z and then there is a scaling factor which
comes from this gamma . Likewise, E z prime
is not the same as E z, but it is a mix of
E z and B y. The mixing coefficient is this
velocity and then on top of it, there is a
scaling factor which is this gamma. So, E
prime is not the same as E; rather it involves
superposition of components of E with components
of B. Let us look at how the magnetic field
transforms.
Now the magnetic field: what is B x B y B
z for the observer in the unprimed frame of
reference, is a different field given by B
x prime B y prime B z prime. These 3 components
give you the magnetic field in the primed
frame and B x prime is equal to B x all right,
but B y prime is not equal to B y; it is a
mix of B y and E z and B z prime is a mix
of B z and E y.
These transformation equations which tell
you how to go to E prime B prime from E and
B again are statements of the unity of the
electromagnetic phenomenon you are actually
constructing the superposition adding pieces.
One piece is E y the other piece is B z and
you mix E y with B z and get E y prime.
Now when do you mix quantities? Only when
they are the same; otherwise, you cannot.
Of course, the coefficients velocity and everything
are such that they take care of the units
and dimensions of what you are adding, so
that everything will have consistent dimensions.
You always add a term to another only when
it has got the corresponding dimensions not
otherwise.
So all these things are properly balanced
and that some of these are, some details that
I think are good exercises for you to work
out for yourself, just to make sure that everything
is agreeing appropriately. Over here mind
you, that there is a square of the speed of
light in the denominator and that will also
take care of the units, dimensions, everything
. you So, you have to work this out in full
details to convince yourself that the relationships
are dimensionally correct.
I like to see the unity of electromagnetic
phenomenon expressed by these equations very
much. In fact, I feel that this is a stronger
statement of the unity of electricity or magnetism
than Maxwell's equations because, Maxwell's
equation express the curl of E, a function
of E in terms of a function of B and vice-versa.
Over here, you actually mix them and you do
know right from your high school days that
you mix things only when they belong to the
same kind. So, this is in a certain sense
a stronger statement of the unity of the electric
and the magnetic phenomena which was achieved
by Maxwell and then this becomes a precursor
to the search for unification of the physical
interactions.
So, as the nuclear weak interaction is already
unified with the electromagnetic interactions,
that is what is called as a electro weak interaction
and it belongs to the general formalism in
physics in which frontier research is being
carried out to understand the unification
of all the fundamental interactions.
We will consider some examples now. These
are constructs of linear super positions.
So, that is the very big feature over here
and many years ago I was discussing this in
the physics 101 course here in IIT Madras
when 2 students, Satish and Venkatesh came
up and we had some discussions which led to
development of an educational software which
was subsequently published in this issue of
Resonance, Volume 9, Number 7. It was published
much after Satish and Venkatesh actually left
IIT Madras because, for some reason the software
was developed but we never got the time to
write it up and then the technology became
obsolete. The graphic software which they
had developed was no longer being used so
Chaithanya Das and Srinivas Murty rewrote
it later and then we published it in 2004.
So, what I will do is, I will demonstrate
this particular software. What this does is,
it solves the problem of charged particle
dynamics in electromagnetic fields using 2
different methods: one in which this solution
is obtained in 1 frame of reference then you
carry out the Lorentz transformations of x
y z t to x prime, y prime, z prime, t prime;
you get a new position vector r prime and
you plot it as a function of t prime.
The second method which is completely independent
of the first method is, it employs the transformation
E B to E prime B prime and I already wrote
those transformation relations in the previous
slide. Then you set up a new equation of motion
with the new E prime and B prime. It is this
one; this is the new equation of motion which
is set up with a new E prime and B prime and
then it is this equation which is solved to
get the trajectory r prime as a function of
t prime. So, these are completely 2 different
methods. Then there is a graphics package
which projects r prime as a function of t
prime and in fact, these projections can be
made with the screen identified as the x y
plane or with the y z plane or the z x plane.
So, you can rotate the coordinate system,
so that you,... because, on a screen you can
see the trajectory only on a 2 dimensional
flat surface. But, you can chose that flat
surface to be either the x y plane or the
y z or the z x so all of these features are
incorporated in this very nice educational
software which was first prepared by Satish
and Venkatesh and subsequently by Chaithanya
das and Srinivas Murthy. This software can
in fact be downloaded from my website at this
link , so that you can run this these programs
yourself. But, we will show you a few test
cases and what we will do is consider an electron;
we have the its charge and mass and we will
place it in different electromagnetic fields
and I am going to invite Gagan and Jobin to
demonstrate this for you, because they know
how to run this software much better than
what I do.
Gagan and Jobin will you please come over
and demonstrate this software?
So this is the first case we are going to
demonstrate, in which all the components of
electric field is actually 0 and we have a
B x which is 0 point 1 weber per meter square;
0 point 1 that is measured in terms of webers
per meter square
Per meter square
And the initial velocities are given v x is
4 point 6 into ten raise to ten raise to 7
And v y is 2 point 65 into 10 raised to 8.
This is the initial velocity of the electron
Of the electron
Of the electron
Yes and this v relative that is 2 into 10
raise to 8
This is
So this is the velocity of an observer of
a second observer
Yes
Who is moving with respect to the first observer
Yes
At 2 into 10 to the 8
Meter per second
Is that meter per second
Meter per second
So then that is fairly high speed because
3 into 10 the 8 is about the speed of light
so this is at 2 third speed of light, right?
2 thirds yes
Yes
So, this is the fairly high relativistic speed
and which is two thirds the speed of light;
so let us put this electron in this field.
So this is the software which was developed
by Satish Venkatesh Chaithanya das and Srinivas
Murthy. What is this CAPE IIT M? That is computer
aided physics education I believe; computer
aided physics education at IIT Madras; so
that is the CAPE IIT M comprise alright. What
you doing now Jobin?
This we are just putting the electric field
along x axis
Y axis
And these are the parameters that you chose?
Yes
So mention your specifying the value of the
electric field and
Shift tab
Magnetic field 0 point 1 1 0
All right
And then
B x is 4 point 6
B y and B z are both 0
4 point 6 into 10 raise to 7 that is the x
component of velocity
This is the initial velocity of the electron?
Electron yes
And v y is 2 point 65 into 10 raise to 8 2
pint 65 into 10 raise to 8
65
So all right,... so the initial velocity of
the electron has got an x component and a
y component and 2 point 65 into 10 to the
8 and what is v z?
It is 0
V z is 0
Yes
And that will be displayed it is 2 into 10
raise to 8 meter per second.
And enter into you particle parameter so this
Gagan's electron.
Yes this is the charge of the particle.
Yes
Minus 1 point 6 into 10 to the power minus
19
So coulombs
Yes
And next is the charge mass of the electron
Mass of the electron
That is 9 point 1 2 10 the power 31 kilo gram
Yes now go alright
Continue
So what is the program doing? It has already
finished the calculation?
Yes
Yes
And now you get
In the particle is initially the particle
is here at the origin.
X equal to 0 y equal to 0 and z equal to 0.
Now let us run this
This is the trajectory that you see .
As observed from the s frame or the rest frame
you can say.
This is the projection on the x y plane.
Yes
And this is on the y z plane
Yes yes
This is the projection of that trajectory
on now this on the z x plane.
Z x plane yes
And what is this?
This is the trajectory of the particle what
the observer and the prime to reference frame
c.
Now the same trajectory as would be seen by
another observer who is moving with respect
to the first observer at the velocity that
you specified earlier .
Yes
And that trajectory will be different and
it is shown to be what it is but there are
2 panels over here what is that?
One is using this field transformation method.
And another using point transformation method.
So on the left panel you get the solution
not by solving any equation of motion you
have already solved that equation for the
first observer.
Yes
And you just carry out the Lorentz transformations
of x y z t to x prime, y prime, z prime, t
prime and plot r prime as a function of t
prime that is given in the panel on the left.
Left
Whereas what you have on the panel on the
right is you setup a new equation of motion
not with E and B but with E prime and B prime.
Integrate F prime equal to d p prime by d
t prime and then plot r prime as a function
of t prime and at least on the y z plane the
trajectories are Congruent. So, thank you
Gagan and Jobin and then let me carry on this
discussion.
So, essentially what we did was to compare
the observations of 2 observers; 1 in the
black frame and the other in the blue frame;
the blue frame moving at a constant velocity
with respect to the black frame and we obtained
the solutions using completely different techniques.
One using Lorentz transformations of the x
y z t to x prime, y prime, z prime, t prime
the other in which we set up a new equation
of motion with the Lorentz force being F prime
which is charge times E prime plus v prime
cross B prime and then integrating that.
We found that the 2 solutions are quite Congruent
to each other these are the some of the cases
that we considered to illustrate this and
thanks to Satish, Venkatesh, Chaithanya das
and Srinivas Murthy and also to Gagan and
Jobin for showing this nice software to us.
I will conclude by giving a very brief summary
of writing the equations of electro dynamics
and tensor notation. It is not something on
which I will spend any bit of time but, it
is something that you should be familiar with
because, once you have seen this you will
then be able to read literature on this subject
from any source.
So, you need a little bit of familiarity with
the notation; there is no new physics in it
there is only a new notation. Let me just
introduce you very briefly to the notation.
We do know that the electromagnetic field
is expressible in terms of potentials; you
can derive it from potentials. In tensor notation
what we do is, write Contravariant vectors
in this form. So, x mu takes these 4 values
x 0, x 1, x 2, and x 3 so mu takes 4 values
0 1 2 3 and the covariant notation is appears
as a subscript over here the Contravariant
notation appears as a super script over here.
So this is the relationship between the Contravariant
and the covariant notation. The signature
that I make use of is this 1 minus 1 minus
1 minus 1. This is what distinguishes the
space time continuum from an ordinary extension
of the Euclidian space to 4 dimension. So,
it is not an ordinary extension of the Euclidian
3 dimension space to 4 dimension. This is
the space time continuum coming out of the
special theory of relativity and this is the
signature of this metric g.
You can go from the covariant notation to
the Contravariant notation in a simple fashion.
By using this matrix notation, you can just
use this as matrix - like matrix algebra and
these are the relations which you use to get
the electromagnetic field from the potentials.
As you can see here you have the derivatives
of A mu and these derivatives the A mu itself
is a 4 vector .
The 3 components of what we call as a magnetic
vector potential and this is the electro electric
scalar potential together they give you 4
components which are A mu which are A 0 A
1 A 2 and A 3 and these are derivable you
know when you take the derivatives of this
electromagnetic potential, you get the corresponding
electromagnetic field.
So the electromagnetic field is now written
over here the right hand side gives you the
derivatives of the potentials and the left
hand side gives you the components of the
electromagnetic field . The notation is sometimes
made a little more [gibberish] you can write
this del by del x mu as del mu and this one
as del mu notice that there are covariant
and Contravariant indices to keep track of.
So now these are the Lorentz transformations
that we had I have used a barred frame rather
than a primed frame but, it is the same thing.
I just needed the different notation to express
the fact that you have 1 frame of reference
which is moving with respect to another. This
is the transformation which gives the Lorentz
transformation and you can go from the parameters
in one frame to the other by carrying out
these transformations and you can write them
in a very compact manner by this transformation
matrix.
Lambda - this is the upper case lambda and
this is the transformation matrix and you
can go from 1 frame the normal frame without
the bar to the corresponding parameters in
the other frame which has got a bar or the
primed frame if you like.
So, on the left hand side I could have used
a primed a mu prime instead of a bar mu; it
is the same thing. The electromagnetic field
in fact, is very conveniently expressed as
an anti-symmetric tensor which has got a very
simple form. Notice that there is some sort
of a symmetry about the diagonal but, this
is more like anti-symmetry because the terms
over here have got opposite signs. So if you
go equidistant away from the diagonal then
the corresponding terms have got opposite
sign. So, this is t 2 3 and this will be minus
t 2 3. So, t 3 2 is equal to minus of t 2
3; so when you inter change the indices over
here you pick up a minus sign that is what
makes this tensor to be anti-symmetric.
So in this notation you can write the electromagnetic
field the electromagnetic field in our usual
vector notation is made up of E x E y and
E z so they are appearing over here; this
is E x this is E y and this is E z and in
terms of this anti symmetric tensor the electromagnetic
field is expressed by this 4 by 4 matrix which
is an anti-symmetric tensor. Again, you have
0s along the diagonal just like this typical
anti symmetric tensor.
Again you have an anti-symmetry over here
so if f 0 1 is E x over c, then when you interchange
0 and 1 so instead of the element in the second
column and first row, you go to the element
in the second row and the first column, you
get minus of F 0 1 so this will be minus E
x over c. Likewise, you will have other components
to be given by the corresponding anti symmetry.
So this is just a matter of notation. Essentially,
what you have written is the electromagnetic
field; there is nothing new in it here. This
is the electromagnetic field over here; so
you have the x component over here, you have
got the y component over here, you have got
the z component over here, then you have got
the magnetic field over here; so the B z is
here the B y is here and the B x here.
So this is just the electromagnetic field
there is nothing new in it and the compact
transformation relation can be written for
a first rank tensor which is a 4 vector now
instead of a 3 vector. Now, this is the transformation
rule for the first rank tensor the electromagnetic
field is a second rank...
Yes
Yes
The corresponding transformation rule for
the second rank tensor is given by this because
there are 2 indices to play with in this case.
So, this is the transformation rule for the
second rank tensor; this lambda we have already
defined.
And Maxwell's equation, the same Maxwell's
equations that you have seen earlier, are
written in a very compact manner and in this
particular form; so this is a set of equations
these are the Maxwell's equations; J mu is
the density, this is the current density 4
vector so it has got the charge density as
well as the current density which we have
met already in the vector notation in which
Maxwell's equations we developed earlier.
Let us just satisfy ourselves that this notation
gives us consistent relationships which we
have seen earlier in the ordinary vector notation.
So, you have the Maxwell's equations and you
have got the current density 4 vector and
if you just take the particular case, put
mu equal to 0 then, all I have done over here
is to put mu equal to 0. So, this is del by
del x nu of F 0 nu and this index nu which
appears twice is the one over which you carry
out the summation. This is sometimes called
as the Einstein summation convention. You
must sum over this index nu going from 0 to
3. So, let us do that; so the term corresponding
to nu equal to 0 is here; this is nu equal
to 1 term, this is nu equal to 2 term, and
this is nu equal to 3 term and Maxwell's equations
take this form.
Now, let us see if it is the same as we have
seen earlier because we do know that these
are the electromagnetic field terms. So, F
0 0 we have already seen is 0 this is coming
from the anti-symmetric tensor for the electromagnetic
field F 0 1 is E x over c, F 0 2 you have
to take it is derivative with respect to x
2 and then F 0 3 is e z over c and you have
to take it is derivative with respect to x
3 and you already know what these terms are.
So, just take those derivatives term by term
and what you get means the first term will
gives you 0 because F 0 0 is 0. So, from the
remaining 3 terms you get del E x by del x,
del E y by del y, del E z by del z you have
got a one over c here you get mu 0 c times
rho on the right hand side but this c you
could move to the right side so you will get
mu 0 c square and that is 1 over epsilon 0.
So, what you get essentially is that the divergence
of E is rho by epsilon 0. In other words,
this equation gives you the complete family
of Maxwell's equations and you can put in
the other values of mu and do this as an exercise
for yourself and satisfy yourself that all
of Maxwell's equations can be written in a
very compact form in this tensor notation
which you will find being used in a lot of
literature on classical electrodynamics; so
it is good to be familiar with that.
So with that I will like to give you a set
of references which you could use to read
these topics further. Berkeley physics course
is one of my favorite sources. The volume
1 which is by Kittel Knight and Ruderman and
Purcell volume 2 which belongs to the same
series of Berkeley physics course. These 2
are excellent sources; Feynman lectures are
of course, great Griffiths book also gives
you all the essential tools and then there
is a nice book by Matveev mechanics and the
theory of relativity And this is a nice source
to read about current problems of current
interest; modern problems and classical electrodynamics.
Then, the reference to the software which
I mentioned the CAPE IIT M, the computer aided
physics education program that we have. it
is it is not a huge program but there are
few attempts made by some students who have
the enthusiasm to develop some educational
software, they contribute to this and with
that we pretty much conclude this unit.
If there are any questions, comments, I will
be happy to take. Otherwise, we will go over
to the unit 11 in the next class which will
be on chaotic dynamical systems.
