relativity1

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B. Tech Physics PHX(101)

By

Dr. Arvind Kumar

Physics Department

NIT Jalandhar

e.mail: [email protected]

Website: https://sites.google.com/site/karvindk2013/

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Contents of Course:

I. Theory of Relativity

II. Laser

III. Quantum Theory

IV. Electrostatics

V . Magneto statics and Electrodynamics

VI. Optics

Total Marks = 100 First Minor = 20 Marks 2nd Minor = 20 Marks Major = 50 Marks

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B.Tech Physics

Chapter 1 Theory of Relativity

Lecture 1.1

Books:

1. Concept of Modern Physics by Arthur Beise,

McGraw Hill Publications

2. Engineering Physics by S.L. Gupta, Dhanpat Rai

Publications

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What is relativity ?

_{Suppose Akbar draw a line AB (see fig)}

and order the people to make this line shorter without rubbing it. Everyone Surprised!

How it is possible to make line shorter without rubbing it?

Then Birbal came and draw a line CD which was longer than line AB.

He now ask Akbar, Sir, whose line is shorter ?

Akbar said: the line drawn by me is shorter as compared to your line.

So now Birbal smiled and said that I have shorted your line without rubbing it.

So the length is relative.

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Consider two friends A and B standing on opposite sides of river. There is

house on the bank of river.

Friend A says it is left to me.

Friend B says it is right to me

So who is right and who is wrong?

Both are right.

Actually

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 Consider a ball B and compare it with the

size of atom A. The size of ball is very large.

Now compare the size of the ball with the size of the Earth. In this case the size

of ball is very small.

We say that the size is relative.

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 _{Consider the train moving with} velocity 60 km/hour. The train is observed by three observers.

1st observer is at rest on earth.

2nd is moving with velocity 20 km/h

towards the train. The 3rd is moving

with velocity 30 km/h in opposite direction to train.

For 1st observe the train will appear to move

With velocity 60 km/h.

For 2nd observer the velocity of train will appear as

40 km/h.

For 3rd_{it will appear as 90 km/h.}

So the three observer different velocities of earth.

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The theory which deals with the relativity of motion and rest is called the theory of relativity.

Today we have two theories of relativity (i) Special theory of relativity

(ii) General theory of relativity

Before understanding about the above theories we need to know about few more terms

(iii) Frame of reference: Any system relative to which the motion of an object can be described is called a frame of reference.

There are two frames of references

(i) Inertial or unacclerated frame of reference.

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(i) Inertial or un-acclerated frame of reference:

An inertial frame of reference is one in which Newton’s first law of motion holds.

Now Newton’s first law says that a body not acted upon by an

external force is un-accelerated. In other words a body at rest remains at rest and if it is , it remains moving with constant speed.

The frame relative to which a body not acted upon by any external force, remains un-accelerated, are called inertial frames.

Any frame of reference that moves at constant velocity relative to an inertial frame is itself an inertial frame.

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The theory of relativity deals with the consequences of the lack of a universal frame of reference.

The special theory of relativity which is given by Einstein in 1905, treat the problems that involve the inertial frame of reference.

A frame of reference is said to be non-inertial in which the Newton’s Laws of motion are not valid. These are accelerated frame of references.

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Galilean Transformations:

These are the equations which relate the co-ordinates of a particle in two inertial frames. These two inertial frames have negligible speed in comparison with the speed of light.

We consider two inertial Frames say S and S’ such that the frame S’ is

moving with constant speed v w.r.t S.

Now we suppose that

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We choose the axes such that the X and X’ are

parallel to velocity v (see fig) . Let Y’and Z’are parallel to Y and Z respectively.

We count the time from the instant when the two origins O and O’ coincide.

From fig we can write,

Or ---(1)

Since there is no motion along Y and Z axes,

We have ---(2)

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Also, time is independent of space co-ordinates and we have

t’ = t ---(4)

Eqns (1) to (4) are known as Galilean transformations,

The inverse Galilean transformation are expressed as,

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Now we relate the velocity components in S and S’ frames. To get This we differentiate eq. (1) to (3) w.r.t. T and we get

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To find the acceleration we differentiate velocity eqns w.r.t. t, so we have,

note that velocity v is constant

---(7)

Above eqn shows that in all inertial frames a body will be observed to have same acceleration.

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Galilean Invariance: The quantities or laws that do not change under a transformation are called invariants.

According to Galilean transformation, some quantities like velocity are changed while mass , length, time and acceleration etc. are

not changed.

The basic laws of physics are same in all inertial frame of references.

Invariance of Newton’s Law:

According to Newton Law,

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Now Galilean transformation we have,

---(2) Differentiate w.r.t. t, we get

---(3)

In system S,

---(4) In system S’

where m’ = m ---(5)

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Exercise: Show that the law of conservation of momentum remains Invariant under Galilean transformations.

_{Shortcomings of Galilean transformations:}

1. The laws of electromagnetism are not invariant under above transformations

2. Velocity of light is not same in all frames. If in frame S the velocity of the light is c then in the frame S’ the velocity will

be c-v_x. But according to special theory of relativity the velocity of Light is same in all frame of references.