Chapter 26Lorentz TransformMichelson/Morley

Experiment

Hint: Be able to do the homework (both theproblems to turn in AND the recommended ones)you’ll do fine on the exam!

Monday, May 10, 1999 10:30am - 11:20amChs. 20, 26, and 27

You may bring one 3”X5” index card (hand-writtenon both sides), a pencil or pen, and a scientificcalculator with you. Same format!

Hint: Review notes from my review lectures! Tryto do some of the old homework recommendedhomework, and exam problems.

Monday, May 10, 1999 11:30am - 12:30pm

everything we’ve covered

You may bring one 8.5”X11” sheet (hand-writtenon both sides), a pencil or pen, and a scientificcalculator with you.

Monday, December 15, 1997 11:30am - 12:30pm

everything we’ve covered

Format: 5 problems, pick 4.

1 problem on each of the following topics:

Electrostatics Circuits Magnetism

Optics Modern

An electron is accelerated across an electrical potential of 100 million volts. What is the final speed of the electron?

PE = qV = (1.6 X 10-19 C)(108 V)

Initially, the electron’s energy is all potentialenergy...

= 1.6 X 10-11 J

electron

When the electron reaches the opposite plate,all the energy will be kinetic

electron

KE = 1/2 mv2 = 1.6 X 10-11 J

m = 9.1 X 10-31kg

v = 5.9 X 109 m/s = 19.7c !!!!

Classical theory has let us down again!

Which led to a flood of work trying toreconcile classical theory with observation!

At first, many thought that Maxwell’s equations must be wrong. After all they were but 20 years old at the time while Newton’s and Galileo’s had been around hundreds of years!

H.A. Lorentz found that Maxwell’sequations were invariant (i.e., of the sameform) under the following transformation:

xx ut

u c'

/

1 2 2t

t ux c

u c'

/

/

2

2 21

Provided a means by which to reconcile theelectromagnetic observations with Maxwell’sequations in a variety of reference frames.

xx ut

u c'

/

1 2 2t

t ux c

u c'

/

/

2

2 21

Well, shortly after Lorentz’s work, HenriPoincare suggested that the laws of mechanicsshould also remain invariant under a Lorentztransformation.

Einstein took on the task of rewriting theold laws of mechanics in a new form thatwould satisfy Poincare’s proposal.

In tackling this task, Einstein put forthtwo postulates, from which the entirety ofspecial relativity can be derived:

1) absolute, uniform motion cannot bedetected (your book says, “The laws of physics arethe same in all inertial reference frames,” which isn’t quiteas good of a statement…)

2) the speed of light is always measuredto be 3 X 108 m/s in vacuum independentof the motion of the source or of the observer.

And the correction to Newton’s lawsproceeded along the following lines:

Newton’s second law implicitly assumedthat the mass of a body is a constant,independent of the body’s velocity.

F m

t

v

If we dispose of this classical limitationand instead allow mass to vary with velocity,then Newton’s second law says instead

F

m

t

( v)

And, of course, Einstein derived therelationship between mass and velocity...

where mo is called the “rest mass” of the body and is defined to be the mass of the body in the frame in which it is at rest!

mm

v c

0

2 21 /

Notice that as the velocity v of the bodyincreases toward c, the mass increasestoward infinity!

Now we can understand why our experimentwith the electron and the capacitor couldnever produce an electron travelling at orgreater than the speed of light:

electron

If the mass of the electron increases toinfinity, then it’s kinetic energy must alsoincrease toward infinity: KE = 1/2 mv2

electron

And that kinetic energy must come from the initial potential energy the electron had before it started moving across the potential gap.

Potential Energy = e V

So we must increase V to infinity!

This “little” adjustment to the mass termthroughout Newtonian mechanics results in a consistent set of physical laws that remain invariant under Lorentz transformations.

And the work that went into proving these new laws of mechanics were correct created a legendary man known as….

What is the mass of anelectron moving 0.999c?

mm

v c

0

2 21 /

91 10

1

31

2 2

.

/

kg

c c(0.999 )

m = 2.0 X 10-29 kg

The greatest experiment to produce anull result in the history of physics!

We have to go back in time 18 years beforeEinstein’s theory of relativity…to 1887.

Scientists were trying to understandMaxwell’s theory that light propagatedlike a wave...

Well, sound waves travel in air. But they do not travel in vacuum (that’s why the astronauts need microphones when out on their space walks)!

Water waves also fall apart when theyget to shore, transferring some of theirmomentum and energy to objects on shore.

The waves they knew about had one thingin common:

They all propagated in somemedium or other!

So, light must travel in a medium as well,they concluded. The medium was called

A very strange medium that allowedlight waves to propagate, but had no massitself and no effect upon the planetsorbits or the motion of any other matter!

The Michelson-Morely Experiment wasdesigned to detect this ether.

bulb

mirrors

partiallysilveredmirror

detector

L

L

The distancesto the mirrorsfor the two pathsare identical.

L

L

So if the device is at rest relative to the ether,we will see constructive interference at ourdetector. On the contrary, if the device isactually passing through the ether...

L

L

…then we expect that light will requiredifferent amounts of time to travel the twodifferent paths. At some velocity relativeto the ether, we should observe destructive interference.

Let’s consider amore classicalexample to betterunderstand why...

Let’s look at a boat (it represents thelight in the Michelson-Morley Experiment)travelling across a river (that representsthe ether).

v

L

L

Both boats travelat the speed c.

v

L

L

How long does it take the brown boat to traveldown river a distance L and back again?

tL

c vdownstream

tL

c vupstream

t t tL

c v

L

c vtotal downstream upstream

c+vc+v c-vc-v

tL

c v

L

c v

L c v L c v

c v c vtotal

( ) ( )

( )( )

2 2

12 2 2 2 2

Lc

c v

Lc

c v c( / )

tL c

v ctotal 2

1 2 2

/

/

Brownboat

v

L

L

What about the red boat?

In order for the red boat to sail straightacross the river, it must aim its bow slightlyupriver.

c

vv c vtotal 2 2

v

L

L

c

v

v c vtotal 2 2

The same will be true as the red boat sailsback across the river...

v

L

L

So, the total time for the red boat to crossthe river both was is:

tL

c vacross

2 2t

L

c vback

2 2

t t tL

c vtotal across back

22 2

Redboat

tL

c v

L

c v ctotal

2 2

12 2 2 2 2( / )

Clearly these times are NOT the same!

tL c

v ctotal

2

1 2 2

/

( / )Red boat

tL c

v ctotal 2

1 2 2

/

/Brown boat

The Michelson-Morley Experiment iscompletely analogous to our boatingexpeditions! Instead of boats and ariver, we have light and ether.

Despite valiant efforts, interference ofthe two light beams was NEVER observed.

Conclusions:

1) there is no ether2) light travels the same speed in all

reference frames (EINSTEIN)

Swallowing the second of those twoconclusions requires a bit of faith...

Accepting it, however, completely changesthe Universe in which we live!

Not only does mass change with thevelocity at which we move, but so doapparent distances and times!