WEBVTT
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hello everybody i am rajdeep chatterjee from
the department of physics iit roorkee and
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i shall be talking on the special theory of
relativity
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well we plan for the next series of lectures
is something like this i shall talk on
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how relativity arose while reconciling the
laws of mechanics and electrodynamics but
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to be more precise reconciling the transformation
laws of mechanics and electrodynamics in that
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context i will be talking of galilean and
lorentz transformations moving over to be
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all important postulates of special relativity
okay
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and then i shall of course go over to the
consequences which are quite interesting we
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will be talking of length contraction time
dilatation mass energy equivalence and this
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is one thing perhaps many of you are quite
familiar with e = mc square let us try to
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see we will try to see at a certain point
of time how it all arose and in explaining
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all these things what i shall do is that as
and when necessary we will talk of certain
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problems
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we will try to do some problems so as to illustrate
the principles involved okay okay so let us
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start at the beginning mechanics
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this f = ma this perhaps is the most famous
equation in mechanics if i may say so you
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all know that if the force of is applied
on a particle of mass m it is going to accelerate
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with acceleration e you can even tell me that
this is actually newtons second law of motion
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okay but what is assumed here apart from of
course in this particular case that we treat
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m that is the mass of a particle that is that
is always constant okay
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well if you have any can argue that if you
have a variable mass you can talk of force
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as rate of change of momentum but let us stick
to this form of newtons second law for the
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motion okay yeah but what is important here
is that we always assume here that somehow
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we have a frame of reference where we are
able to measure this acceleration okay now
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frame of reference is actually a very fancy
name for for a simple coordinate system
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i mean coordinate system you know simplest
one of course it is the cartesian coordinate
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system i mean if you see where the walls of
of this room meet i mean the room you are
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in if it meets with the floor and then you
see the axis of the coordinate system so we
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have quite familiar with the word coordinate
system so that is a fancy name the frame of
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reference okay
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now another thing is we sometimes hear of
this word inertial frame so what is an inertial
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frame well a very solid definition is an inertial
frame is one in which newtons laws of motion
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are valid okay you know newtons laws of motion
are valid you know inertial frames so what
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does this explain where explain a person explain
and tell an engineer or a scientist where
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to look for this inertial frame we need to
be a little bit more precise than this idealistic
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definition and in doing that in defining that
we can say that it is a frame whose coordinate
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axis are fixed relative to the to the average
position of a fixed are fixed star in space
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of course
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or it is that frame is moving with an uniform
linear velocity that is a constant velocity
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relative to the star and of course there should
not be acceleration of this star otherwise
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the definition is not valid well once i have
said this and you realize that any frame the
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earth itself the earth itself is evolving
and it is there is day and night there is
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this rotation due to which you have of course
they are night and and and revolution around
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the sun that is the that gives the change
of season
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so there is some amount of acceleration yes
ok physically any coordinate system attached
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to its surface is known inertia but for many
purposes this acceleration is slight and then
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by many purposes i course do not mean for
all purposes if this acceleration can be considered
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slight here this particular frame that is
the frame on earth can be considered inertia
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a little better would be a non rotating frame
with with a origin fixed at the earths center
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and access appointed towards the fixed star
so that is we can say it is approximately
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interaction well even if you have this frame
to be fixed at the center of the sun for example
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that will be more inertia than compared to
the frame i just described ok but having said
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this we should be clear that non inertial
frames are actually quite common in in mechanics
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i mean must have heard of corioli forces okay
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so these are not inertial forces ok so let
us try to have a visual explanation of
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what we have been trying to say in more definitive
terms ok
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so here we have frame s ok so this is a three
dimensional cartesian the system a right handed
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cartesian system i have not written the
z-axis here but you can all guess and you
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can all you can all figure out that z-axis
here actually points outside the screen okay
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so this is frame s and then we have another
frame s prime let us say and then this frame
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s prime is moving with a constant velocity
or let us say uniform velocity v along the
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common x x prime axis okay
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now the coordinates here are the x prime y
prime and z prime and here is that prime again
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points outside the screen now at the beginning
at the very beginning general let us we
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have two observers who are let us say at the
origin so both these frames and at the beginning
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on both these flames coincide that is s and
s prime they have a common origin at the beginning
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at time t equal to and t and t prime = 0
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so i will talk of this chord at some time
a little bit later so how to do that i mean
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at simpler and you start with two observers
are there and they look at the watches and
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say that okay fine so our watches agree and
this is the time we set as t = our t prime
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=0 and then s prime frame starts moving with
uniform velocity v with respect along the
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common x x prime axis so the only line if
we have a point p which has coordinates x
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y z and then its measured at a certain time
t remember block is moving
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i mean time clock is working it is moving
i mean that time is flowing at a certain
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time t so at that same instant let us say
observer in s prime measures the coordinate
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as x prime y prime and z prime okay so how
are they related well you can say that it
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is simple actually so how are they related
x prime is related by this relation x prime
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= x - v of t v and t y = y prime and z
equal to z prime okay and it is important
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that both these time coordinates agree okay
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so so that is we this point is measured at
the same instant of time okay so they all
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started with synchronized watches so t prime
= t here okay now this transformation we
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call this transformation as galilean transformation
okay let us delve a little bit further
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how are the velocities if you measure the
velocities in in both these frames how are
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they related well that actually will be given
by the galilean velocity addition formula
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so let us see how it how it can be derived
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we have this we have this equation x prime
= x - vt see that now you differentiate
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x prime with respect to the time in its own
frame so that is dx prime dt prime and on
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the right hand side all you got to have dx
by dt prime - v dt dt prime okay and now you
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realize that on the right hand side you have
a dx by dt prime now x is the coordinate in
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the s frame but t prime that is the time that
is measured in the primed frame that is s
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prime frame
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so you need so if you need velocities so you
need coordinates of the same frame okay so
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a coordinate and the time of course in the
same frame i should say so the third step
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clarifies how to do that so you have dx dt
and then you take this dt dt prime - v of
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dt dt prime okay so then you realize that
dx prime dt prime that is the u prime that
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is the velocity velocity that is being measured
in the primed frame okay
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now you realize since t = t prime in galilean
transformation so dt by dt prime that is equal
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to 1 so you realize and then dx dt that
is u and so dt by dt prime that is 1 so - v
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of – v so you have u prime = u – v now
i mean there is a subtraction sign here so
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do not be too bothered about that when
i use the word addition formula because you
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can very well write the velocity which
is in the s frame in terms of the s prime
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frame by saying that u = u prime + v
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and what is v by the moment by the way so
it is just the velocity with which the s prime
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frame is moving uniform that is uniform velocity
with which the s prime frame is moving with
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respect to the s frame along the common x
prime axis ok so that goes for the velocity
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velocity addition formula what about the acceleration
mm-hmm
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well we start with what we have obtained for
the velocity so that is u prime = u – v
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so if you differentiate this once again so
you are going to have du prime by dt prime
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is du dt of course you know how to get this
now so this is du by dt you know how it would
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be so if du by dt frame take du by dt prime
and then you have dt prime dt by dt prime
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that is equal to 1 so the acceleration is
going to be the same in both frames
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so these frames are moving with uniform velocity
s with respect to one another so what we have
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is acceleration is being unaffected by if
you have frames which are moving with uniform
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relative velocities okay now on top of that
if you consider that mass is unaffected by
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motion of reference frames you come to
and we come to a very interesting conclusion
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we see that the form of newtons second law
is valid well actually newtons second law
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is valid in both these frames in both these
inertial frames okay
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so so what does this mean well this means
that by doing experiments entirely in one
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of these frames you cannot distinguish it
from the other okay so if you are doing extreme
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experiments entirely in one of the frames
so and this frame is moving with a uniform
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velocity v with respect to another frame you
will you cannot distinguish this particular
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frame from any other inertial frame okay so
by by mechanical experiments alone that is
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what im going to say here
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so what you can ask so so what happens is
that since in newtons laws of motion are being
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valid are valid in these two frames so our
equations of motion okay we charge right from
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from them and consequently the conservation
laws so going to have the conservation laws
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same in all these inertial frames okay so
if you do your mechanics in one of one of
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these frames and you derive a conservation
law can be rest assured that another inertial
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frame it is going to be the same it is going
to be valid okay
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so we could say that the laws of mechanics
are being invariant in in all inertial frames
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okay this is an this is an important conclusion
so next we move over to what is going to happen
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in electrodynamics okay
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so what is the electro dynamics so you see
it is an interesting thing so here you
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have what does it give you it gives you that
if you have a changing electric field you
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going to have a magnetic field and then if
you have a changing magnetic field you are
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going to have an electric field okay now we
ask this question that is electrodynamics
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or the laws of electrodynamics invariant under
galilean transformation
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remember laws of mechanics they were invariant
under galilean transformation so we asked
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another branch of physics electrodynamics
are the laws they are invariant under galilean
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transformation well for that let us see what
those laws are okay with the basic laws
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they are the encapsulated all in maxwells
equations
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well divergence of e that that that is equal
to rho by epsilon 0 and e as you know is the
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electric field a rho is the charge density
and epsilon epsilon 0 that is the permittivity
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of free space and then the curl of e that
is - del b del t b is the magnetic field and
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then the divergence of v that is zero the
curl of v that is mu0 j mu0 that is the permeability
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of free space and j that is the current density
that plus of course mu0 epsilon 0 del e del
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t
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now do not be bothered too much about this
mathematical details so what do they stand
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for i mean i mean that is what i have written
on the right hand side obvious equations the
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first one is actually gausss law well so it
is a very important law it actually allows
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you to calculate the electric field if you
have symmetries in the in the problem symmetry
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the charge distribution that is okay
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the second one curl of e that is equal to
- del v del t that is actually faradays law
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and i am surely aware of it because had this
law not been there i mean did not have motors
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you electric motors that is you must have
heard of faraday is important experiment in
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which you had this he was moving us a magnet
within a solenoid and then he detected an
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emf within the leads of the solenoid okay
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then the third thing i mean a third equation
that i have written here and should not be
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talked as a third law that is divergence of
b = zero which which well does not have a
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law it does not have an name as such that
it is physical implication is that there are
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no magnetic monopoles so like that like you
have a you have a positive and negative charges
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you do not have charges in a magnetic poles
in isolation okaydo not have magnetic monopoles
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the curl of b that is mu0 j + mu0 epsilon
0 del u del t that is actually a curl of b
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= mu0 j that is actually amperes law okay
and then added to that is maxwells correction
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well that maxwell or well maxwells corrections
are actually quite important you sre going
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to see later on because it had rather quite
rather very interesting implications in showing
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that these the maxwells equations well by
the way so what you see here is of all e and
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b are coupled
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these are coupled partial differential equations
so well this correction is very important
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because he was able to show that well he put
he introduced this concept of displacement
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current and then was able to show that these
equations when written in terms of only e
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or only b could be framed in terms of the
wave equation the more of that a little later
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okay okay
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so our maxwells equations invariant under
galilean transformations under the transformations
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mechanics is invariant on so what were these
transformations once again so so you have
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an s prime frame moving with an uniform velocity
v along the common x x prime axis okay again
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z and that prime this axis are moving are
actually out of the screen okay they are pointing
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out outside the screen and this transformations
x prime = x – vt y prime = y z prime = z
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and t prime = t
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so our maxwells equation invariant under this
the answer is no okay they are invariant under
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a different transformation maxwells equation
are not invariant on the galilean transformation
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but maxwells equations are invariant under
lorentz transformation okay so what is that
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so again we have the frames s prime moving
the uniform relative velocity v along the
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common x x prime axis but here we need x prime
to be given by not only x - vt divided by
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root over of 1 - v square by c square okay
and of course here a y prime = y z prime is
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equal to remember we are moving along common
xx prime axis
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what is interesting here is that see that
the times are not matching in galilean transformation
22:45.670 --> 22:54.260
we had t is equal to t prime but here t prime
= t - vx by c square divided by root over
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of 1 - v square by c square ok so this v is
actually the velocity with which the frame
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s prime is moving with respective s frame
ok what is this see here well if you have
23:15.700 --> 23:21.880
guessed that is the speed of light but all
of a sudden how come this speed of light is
23:21.880 --> 23:31.300
there so remember this is the transformation
under which maxwells equations are invariant
23:31.300 --> 23:32.430
ok
23:32.430 --> 23:43.300
so do we see do we see c and that is the velocity
of light im sorry the speed of light in vacuum
23:43.300 --> 23:47.830
explicitly in maxwells equation i mean
on the left hand side i have written that
23:47.830 --> 23:57.240
once again just for your convenience well
it is not present explicitly so we asked
23:57.240 --> 24:05.890
this question so where is this coming from
okay so is c ingrain somewhere within maxwells
24:05.890 --> 24:12.130
equation itself ok for that what you have
to do as i said is that maxwells equation
24:12.130 --> 24:17.390
these are posh coupled partial differential
equations
24:17.390 --> 24:23.890
now if you uncouple them ok we have if there
is a price to pay you see that you have a
24:23.890 --> 24:25.800
second order equation then ok
24:25.800 --> 24:36.700
so you have del square e = mu 0 epsilon 0
del 2 lt square of e and similarly for the
24:36.700 --> 24:45.960
magnetic field also you have the laplacian
i should say del square b or plus en of p
24:45.960 --> 24:54.790
that is mu 0 epsilon 0 is del2 by del t square
okay now this has an uncanny resemblance with
24:54.790 --> 25:03.540
the wave equation you know waves water waves
sound waves so it is wave equation here so
25:03.540 --> 25:13.490
the laplacian of f that is equal to 1 by v
square of del 2 f del t square okay
25:13.490 --> 25:22.890
so t is the time here and what is v v is the
velocity of the wave now you see in these
25:22.890 --> 25:29.640
two sets of equation if you compare maxwells
equation of b and e with the wave equation
25:29.640 --> 25:39.100
what you are going to see is that this term
mu 0 epsilon 0 can be compared with 1 by v
25:39.100 --> 25:46.950
square okay so which means that if it if it
i mean since it resembles the wave equation
25:46.950 --> 25:54.960
mu 0 epsilon 0 somehow has some sort of
relation with velocity okay
25:54.960 --> 26:01.630
it is actually you are going to see that 1
by mu 0 epsilon 0 is that does indeed turn
26:01.630 --> 26:10.850
out to be and 1 by root over of mu0 epsilon
0 does indeed have the dimension of velocity
26:10.850 --> 26:16.200
it is actually 3 to 10 to power 8 meter per
second okay so on that value later on when
26:16.200 --> 26:26.990
you put in values okay but also from physical
principles in hindsight you can also check
26:26.990 --> 26:36.320
that mu 0 epsilon 0 should have the dimensions
of 1 by velocity square well how to do that
26:36.320 --> 26:43.670
well well check any one of any one of maxwells
equations in e or b
26:43.670 --> 26:52.800
check the first one the laplacian of e = mu0
epsilon 0 del2 e del t square now this laplacian
26:52.800 --> 27:01.030
of e laplacian how does it look like del 2
del x square + del 2 del y square + del 2
27:01.030 --> 27:08.620
del z square that kind of a thing so it has
a dimension 1 by length squared okay so on
27:08.620 --> 27:13.830
the left hand side i mean for the moment look
at these operators that is then that is more
27:13.830 --> 27:19.690
important now because e and e so the ene they
have the same dimension
27:19.690 --> 27:25.230
so what we need to do is to balance the dimensions
of rest of the operators and rest of the things
27:25.230 --> 27:31.910
here but on the right hand side will be concerned
with mu 0 epsilon 0 del 2 del t squared okay
27:31.910 --> 27:38.400
now you have a t squared in a denominator
here so which means that that is time squared
27:38.400 --> 27:42.690
okay so on the left hand squared you have
on the left hand side you have 1 by length
27:42.690 --> 27:48.100
square and then on the right hand side you
have 1 by time square okay
27:48.100 --> 27:56.390
so what should be then the dimension of mu0
epsilon 0 so that you have this entire thing
27:56.390 --> 28:05.090
mu 0 epsilon 0 del 2 del t square to have
the dimension of length square okay well it
28:05.090 --> 28:13.750
has to have then the dimension of 1 by velocity
squared ok so then in hindsight we can actually
28:13.750 --> 28:21.270
we actually can figure out that mu0 epsilon
0 should have the dimension of 1 by velocity
28:21.270 --> 28:29.440
square that similarly has the same thing the
same conclusion you realize from the second
28:29.440 --> 28:36.980
equation laplacian of v is mu 0 epsilon 0
del to be del t square ok
28:36.980 --> 28:42.920
now on this value 3 into 10 to power 8 meter
per second okay and you might have already
28:42.920 --> 28:54.590
guessed that this number is actually the speed
of light okay so you see speed of light is
28:54.590 --> 29:02.260
actually ingrained within maxwells equation
itself and then del 2 laplacian of e is actually
29:02.260 --> 29:07.520
equal to a plus sign of the electric field
and the laplacian of the magnetic field you
29:07.520 --> 29:13.160
see that it is 1 by c square and then here
on top you have del v del t square and for
29:13.160 --> 29:17.390
the magnetic field l to be del t square okay
29:17.390 --> 29:24.090
now since it follows the pattern since it
follows the form of the wave equation okay
29:24.090 --> 29:32.510
so maxwell concluded that then light must
be an electromagnetic wave okay now this had
29:32.510 --> 29:44.290
a profound significance because light electromagnetic
wave and you see that i have written wave
29:44.290 --> 29:47.650
in in italics
29:47.650 --> 29:56.010
because in those days the bend in the 19th
century actually people thought that waves
29:56.010 --> 30:05.060
actually require a material medium to propagate
why was that the the reason that okay you
30:05.060 --> 30:11.070
have water waves which water to propagate
you have sound waves you need medium air we
30:11.070 --> 30:19.200
need air or even sound waves can travel through
another material for example to a metal but
30:19.200 --> 30:21.580
in any case you need a medium to propagate
30:21.580 --> 30:28.130
so so the reason that perhaps they also light
also should require a medium to propagate
30:28.130 --> 30:37.220
so and then they just name this medium as
ether or actually used to call it the luminiferous
30:37.220 --> 30:38.990
ether okay
30:38.990 --> 30:47.910
and then further reason that as light can
travel through vacuum then vacuum must contain
30:47.910 --> 30:55.390
this medium of light which is ether so vacuum
is full of ether a okay that is the medium
30:55.390 --> 31:02.260
of light okay now like every assertion in
physics and if you even if you make a theory
31:02.260 --> 31:09.320
it has to be proved it has to be validated
by experiments and so that is the challenge
31:09.320 --> 31:18.380
that confronted physicists at in those days
in the late 19th century is to detect ether
31:18.380 --> 31:22.620
and its properties okay
31:22.620 --> 31:30.330
so they were thinking of a possible experiment
in which to measure the speed of light in
31:30.330 --> 31:37.350
different inertia frames okay and and to see
if these speeds were different in in these
31:37.350 --> 31:39.730
different systems okay
31:39.730 --> 31:49.890
now in case they were different they will
look for evidence of a special frame where
31:49.890 --> 31:55.770
that is the ether frame and that that is going
to be a preferential frame where the speed
31:55.770 --> 32:01.250
of light is seated that c 3 into 10 to power
8 meter per second that is the speed of light
32:01.250 --> 32:10.440
in vacuum okay so they were they were looking
for an ether frame and this experiment remember
32:10.440 --> 32:16.160
was to be done on earth so sitting or not
they were supposed to detect ether
32:16.160 --> 32:23.740
now consider the fact that earth is in motion
okay so if an experiment has been done on
32:23.740 --> 32:31.080
earth and then earth is in motion so you should
be able to detect an ether wind in a sense
32:31.080 --> 32:36.790
quote unquote an ether wind okay and then
the magnitude and direction of ether of this
32:36.790 --> 32:45.280
heat oven would vary with season and of course
the time of the day because of rotation of
32:45.280 --> 32:52.341
the earth okay
so the point was to the suggested experiment
32:52.341 --> 32:57.880
was to measure the return speed of light okay
32:57.880 --> 33:07.170
so going and coming back okay since ether
was always gas in an ether frame in in different
33:07.170 --> 33:14.490
seasons and in various times of the day okay
33:14.490 --> 33:21.740
why because if earth is moving relative to
the ether frame the return speed of light
33:21.740 --> 33:26.570
would be different and this difference could
be detected then and that would be a test
33:26.570 --> 33:34.990
for the presence of heat remember that devising
such an experiment was indeed very difficult
33:34.990 --> 33:43.500
okay so but but there were smart people they
were wherever as they were michelson and morley
33:43.500 --> 33:48.460
who in the later part of 19th century
33:48.460 --> 33:56.920
they device then interesting instrument they
devised actually devised an interferometer
33:56.920 --> 34:00.650
which goes by their name
34:00.650 --> 34:09.050
so that had a light source okay so light is
emitted from the source it comes and hits
34:09.050 --> 34:14.609
the semi silvered better that is actually
a beam splitter okay so then you have mirrors
34:14.609 --> 34:21.399
on two sides perpendicular and parallel to
this light source and then and detector on
34:21.399 --> 34:29.560
the other side like it is shown here so what
happens is that light comes and hits this
34:29.560 --> 34:37.550
semi silvered mirror it splits into two parts
okay goes to the mirrors is reflected back
34:37.550 --> 34:38.550
okay
34:38.550 --> 34:44.779
so that you see the science a little bit different
symbols for this reflected rays and for this
34:44.779 --> 34:52.619
reflected ray then re combines and goes to
the detector and there will be constructive
34:52.619 --> 34:57.790
contract constructive and destructive interferences
due to which there will be a fringe pattern
34:57.790 --> 35:09.900
at this detector okay now remember this experiment
is being done on earth okay now as earth is
35:09.900 --> 35:12.869
moving in the ether frame okay
35:12.869 --> 35:22.380
and so if this flow of ether is parallel to
one of the one of these beam directions let
35:22.380 --> 35:31.190
us say parallel to going from if the direction
of ether is from light source towards the
35:31.190 --> 35:38.910
mirror on your right hand side and then what
will happen is that the returned speed of
35:38.910 --> 35:46.500
light will be different from the returned
speed on the perpendicular to the ether flow
35:46.500 --> 35:54.930
why because if you spiral to the flow of ether
and then once it goes parallel it is towards
35:54.930 --> 35:58.539
it is flowing with you know in the direction
of ether
35:58.539 --> 36:03.400
but when it is been reflected black its opposite
to the flow of it okay so there there is going
36:03.400 --> 36:15.019
to be a difference in time of the return speed
of of the return of light in both this axis
36:15.019 --> 36:21.869
and what you are going to have is that this
difference is going to cause a shift in the
36:21.869 --> 36:25.930
fringe pattern at the detector okay
36:25.930 --> 36:34.630
so the expected result was that there would
be a friend shift at the detector which would
36:34.630 --> 36:45.150
confirm the presence of ether okay but surprise
surprise the actual result was that although
36:45.150 --> 36:50.759
this was done on a different season different
times of the day no discernible friendship
36:50.759 --> 36:55.130
was observed i mean you could have argued
that maybe a more sophisticated instrument
36:55.130 --> 36:59.150
or later on they could have rechecked
36:59.150 --> 37:06.720
it was checked even by other people and also
by by more sophisticated equipments and there
37:06.720 --> 37:14.170
was no evidence of this ether frame well jokingly
of course sometimes people call and this is
37:14.170 --> 37:20.940
the most famous field experiment okay so there
was no ether
37:20.940 --> 37:31.450
now towards the end of the 19th century on
the other hand albert einstein was also very
37:31.450 --> 37:40.240
concerned and he was also concerned on a different
thing he was concerned that the laws of classical
37:40.240 --> 37:49.339
mechanics and electrodynamics we are not following
the same transformation laws they were following
37:49.339 --> 37:57.420
galilean and they all in transformation laws
okay so this was quite troublesome to him
37:57.420 --> 38:09.130
he being a theoretician so he is not that
does it mean that an inertial system which
38:09.130 --> 38:17.200
is actually indistinguishable by mechanical
experiments remember we saw earlier that with
38:17.200 --> 38:27.930
the help of mechanical experiments you are
not able to distinguish between inertial systems
38:27.930 --> 38:31.890
different inertial systems because newtons
law is going to be valid in each one of them
38:31.890 --> 38:38.279
in the same form ok so does it mean that okay
i mean with mechanical experiments it is not
38:38.279 --> 38:39.339
being possible
38:39.339 --> 38:48.400
but by other means by by other electromagnetic
means maybe optical methods can you then distinguish
38:48.400 --> 38:56.970
between inertial systems that to einstein
was a very why something because here you
38:56.970 --> 39:05.750
have then different branches of physics following
different transformation laws okay now he
39:05.750 --> 39:12.930
reason that this need not be so this that
there is there is somehow there is a there
39:12.930 --> 39:14.529
is a problem somewhere
39:14.529 --> 39:23.000
so he figured out that actually it is the
lorentz transformations which are more general
39:23.000 --> 39:28.569
than the galilean transformations we will
put the words more general in italics yes
39:28.569 --> 39:38.440
so explain that a little bit more later on
and he talked of the need to modify a mechanics
39:38.440 --> 39:45.269
the laws of mechanics accordingly so that
electrodynamics and mechanics follow the same
39:45.269 --> 39:55.690
transformation laws okay now to do this einstein
had to make two important assumptions okay
39:55.690 --> 40:06.630
they are actually the postulates of special
relativity so the first postulate that is
40:06.630 --> 40:13.910
the principle of relativity so which tells
us that the laws of physics are going to be
40:13.910 --> 40:20.500
the same in all inertial frames okay so there
is no that there should not be any preferred
40:20.500 --> 40:26.440
inertial frame okay there is no preferred
inertial and no preferred inertial frame axis
40:26.440 --> 40:36.319
okay and then the second postulate which says
which talks of the constancy of the speed
40:36.319 --> 40:38.670
of light
40:38.670 --> 40:45.220
the second assumption postulate and the speed
of light in free space it has the same value
40:45.220 --> 40:57.119
c in all inertial frames okay now with these
two postulates einstein started his calculations
40:57.119 --> 41:03.230
and let us go let us check a little bit more
on let us let us take this idea a little bit
41:03.230 --> 41:09.010
more on the second postulate
41:09.010 --> 41:18.170
so here we have the two frames s and s prime
moving with velocities going with a velocity
41:18.170 --> 41:26.599
v with respect this prime frame is moving
with a velocity v along the common x x prime
41:26.599 --> 41:38.059
axis then and then of course at t = t prime
they started so the it coincide and we considered
41:38.059 --> 41:45.839
a ray of light starting from a common origin
and reaching point p okay and then let us
41:45.839 --> 41:53.050
measure the distance op and o prime p in both
these frames
41:53.050 --> 42:00.599
so what would an observer in s frame measure
op as and what an observer in s prime frame
42:00.599 --> 42:08.690
measure o prime ps okay so the distance wise
that would be op that would be x square +
42:08.690 --> 42:15.779
y square + z square so that is equal to c
square t square the member c is the speed
42:15.779 --> 42:25.180
of light and then o prime p that is x prime
square + y prime square +z prime square that
42:25.180 --> 42:29.670
is equal to c square t prime square
42:29.670 --> 42:36.519
now i notice of course so get into the second
postulate we have again the speed of light
42:36.519 --> 42:42.119
to be the same in both these frames okay
42:42.119 --> 42:53.170
now you are assured that x square + y square
+ z square now you would you subtract out
42:53.170 --> 42:59.519
c square t square okay it is going to give
you 0 and a similar thing where going to happen
42:59.519 --> 43:08.049
if you subtract out c square and t prime square
from the primed from x prime square + y prime
43:08.049 --> 43:13.500
square + z prime square that is going to that
that that two is going to give you 0 so now
43:13.500 --> 43:17.190
the same thing is going to happen if you go
to another frame moving with certain other
43:17.190 --> 43:20.690
velocity v prime let us say or v double prime
let us say ok
43:20.690 --> 43:27.059
so the distance there could be x double prime
square + y double prime square + z double
43:27.059 --> 43:35.079
prime square and if the observer there has
measured time t prime - c square t prime squared
43:35.079 --> 43:42.920
remember that this speed of light is taken
the same in all inertial frames here but we
43:42.920 --> 43:51.569
point out that the quantity x square + y square
+ z square - c square t square is an invariant
43:51.569 --> 43:56.339
quantity ok so that is the thing that is not
changing
43:56.339 --> 44:03.599
now for this invariant quantity so which of
these transformations galilean or lorentz
44:03.599 --> 44:09.839
preserves this invariance ok the answer is
and you can actually check this out you can
44:09.839 --> 44:19.190
put x prime = x - vt x prime is y prime = y
is that z prime = z t prime = t and check
44:19.190 --> 44:27.940
if this invariance is preserved you are going
to see that it is not so it is only the lorentz
44:27.940 --> 44:32.920
transformation which is going to preserve
this invariance okay
44:32.920 --> 44:40.670
now let us see what that is so well we have
we have been introduced to lorentz transformation
44:40.670 --> 44:45.900
before well i have been talking about the
laws of electrodynamics so but let us write
44:45.900 --> 44:46.900
down once again
44:46.900 --> 44:56.501
so that is x prime = x - vt by root over of
1 - v square by c square y prime = y z prime
44:56.501 --> 45:05.190
= z and t prime = t - vx by c square root
over 1 - v square by c square of course i
45:05.190 --> 45:11.579
mean if this looks a little bit more complicated
sometimes people actually write it a more
45:11.579 --> 45:21.630
compact form by taking this ratio v by c as
beta and then writing gamma as 1 - bi by 1
45:21.630 --> 45:28.390
by root over of 1 - v square by c square which
is the same as 1 by root over of 1 - beta
45:28.390 --> 45:29.390
square
45:29.390 --> 45:37.220
so lorentz transformations can be very concisely
written in terms of in this fashion written
45:37.220 --> 45:49.529
in this white box x prime = gamma times x
- beta ct ok so why beta c because you see
45:49.529 --> 45:57.940
beta is equal to v by c ok and here we had
x – vt so we had to write beta c here okay
45:57.940 --> 46:04.359
so the ys and zs are the same here but it
is very interesting to write the time coordinates
46:04.359 --> 46:12.130
so if you multiply that by c so it has the
dimension of length again
46:12.130 --> 46:23.079
so ct prime = gamma of ct - beta x have you
noticed one thing it is that notice the thing
46:23.079 --> 46:28.890
for this x prime the transformation equation
for the x prime and the ct prime okay see
46:28.890 --> 46:37.119
that x prime and the last you have beta times
ct ok but when you have ct prime you have
46:37.119 --> 46:44.720
in the last you have beta times x okay and
then so you see it is it looks very symmetrical
46:44.720 --> 46:49.960
so when have the when you have the length
coordinate you have the time coordinate
46:49.960 --> 46:54.759
and when you have the time coordinate you
have the length coordinate now this is something
46:54.759 --> 47:00.450
new this is something new to us this is something
which is not natural to us i mean we are quite
47:00.450 --> 47:05.900
used to galilean transformation in our real
life okay but here what you see is that in
47:05.900 --> 47:14.430
this time coordinate you have some you have
you have the length coordinate as well okay
47:14.430 --> 47:19.390
so naturally this is going to have consequences
and we are going to check all these things
47:19.390 --> 47:22.299
in subsequent lectures okay
47:22.299 --> 47:32.140
just one more thing we we were actually talking
of all we were actually always talking of
47:32.140 --> 47:39.650
what is the quantity in the s prime frame
in terms of quantities in the s frame so we
47:39.650 --> 47:46.390
can also talk of the opposite thing so what
is the how can you say what what are the quantities
47:46.390 --> 47:53.839
in s frame in terms of quantities in s prime
frame for that it is very easy to concern
47:53.839 --> 48:00.540
it is it is it is the same situation if you
consider s frame to be moving with a velocity
48:00.540 --> 48:04.640
-v with respect to the s prime frame
48:04.640 --> 48:13.519
in that case you can simply write down x = gamma
times x prime + beta ct prime of course y
48:13.519 --> 48:21.369
and y prime are the same z and z prime of
the same and then ct = gamma times ct prime
48:21.369 --> 48:32.910
+ beta x prime now what i have done here as
i said was to express the quantities in in
48:32.910 --> 48:40.910
in s frame so you want to understand calculate
quantities in x frame and in the s frame in
48:40.910 --> 48:45.630
terms of quantities in the primed frame okay
48:45.630 --> 48:55.440
so like we can take a break here and in the
next talk we are going to focus on again on
48:55.440 --> 49:03.200
the postulates of relativity then the consequences
of the lorentz transformations where we are
49:03.200 --> 49:11.990
going to carry our discussions a little more
okay so to summarize what we have been doing
49:11.990 --> 49:22.380
today we have been looking at the laws of
mechanics and electrodynamics and we saw that
49:22.380 --> 49:27.720
actually they were not a transformation laws
of mechanics and electrodynamics
49:27.720 --> 49:34.359
they are not the same there were two galilean
and lawrence because einstein was so concerned
49:34.359 --> 49:45.609
and then he he he showed that you take if
he took he actually showed that the lorentz
49:45.609 --> 49:51.769
transformations were actually the more general
transformation laws and if you take that you
49:51.769 --> 49:57.400
want to you need to change mechanics okay
so these are certain things that we will be
49:57.400 --> 50:14.260
considering in our future talk’s okay thank
you