Slide 1

Praise to the mighty Sun, which brings us the day and the light and the air and clouds and all weather, by which all life is upheld in all creatures.

MAXWELLS EQUATIONS AN EXPRESSION OF NATURES

HARMONY !!

OUR GOAL IN THIS LECTURE IS TO DEVELOP THE ELECTROMAGNETIC MODEL FOR RADIATION .

THE MODEL WAS DEVELOPED BY JAMES CLARKE MAXWELL.

AT THE END OF THE LECTURE WE SHOULD BE ABLE TO

APPLY IT TO UNDERSTAND THE PROPAGATION OF EM WAVESTHROUGH A MICROWAVE WAVEGUIDE AND AN OPTICAL FIBER. MOST IMPORTANTLY WE REALISE THE RELEVANCEOF THIS MODEL BEING THE UNDERLYING PRINCIPLE UNIFYING EVERYTHING IN NATUREINCLUDINGTHE PRINCIPLE OF HOW WE SEE !!!

Environment around charges that are

stationary.

moving with uniform velocity.

accelerating.

COULOMBS LAW UNIVERSAL IN BEHAVIOUR

EVERY CHARGED PARTICLE HAS ITS INFLUENCE ON EVERY OTHER CHARGED PARTICLE IN THE UNIVERSE.

COULOMBS LAW UNIVERSAL IN BEHAVIOUR

EVERY CHARGED PARTICLE HAS ITS INFLUENCE ON EVERY OTHER CHARGED PARTICLE IN THE UNIVERSE.

CHARACTERISATION OF SPACEAROUND A CHARGE.ELECTRIC FIELD

Total Flux through any closed surface Total charge enclosed by the surface. In Point form or differential form .

DS LET US NOW SEE WHAT HAPPENS TO SPACE WHEN THE CHARGES ARE MOVING WITH UNIFORM VELOCITY !Magnetic field arises due charges in motion. A steady magnetic field is due to charges moving with uniform velocity [steady current ]. The Steady Magnetic field is related to the steady current through the relation and in point form is Both are Amperes Law in different forms.

B = m H

Since the dot product is positive when It enters the surface and negative when it leaves.It could also be interpreted as due to non existence of monopoles .

In point form it is A CLOSED SURFACEMaxwells Equations for stationary charges and for charges moving with uniform velocity.

D= eE

B =mH D -- Coulombs per sq.meter E -- Volts per meter H Amperes per meter B -- Weber per sq. meter or Tesla

Now the most exiting aspect of Maxwells work !!Now the most exiting aspect of Maxwells work !!ENVIRONMENTAROUNDACCELERATINGCHARGES.

Two new concepts Electric field produced by changing Magnetic field. concept from the experimental work of Michael Faraday. Magnetic field produced by changing electric field resulting from the experimentalwork of Ampere andtheoretical work of James Clarke Maxwell.

The First Concept

Electric field produced by changing Magnetic field.

Faradays law states that emf around the closed path is equal to (-ve) of time rate of change of magnetic flux enclosed by the path. i.e. ButUse Stokes theorem

.

.

.

Negative sign indicates later value of B is less than the Initial value. TIME RATE OF VARIATION OF MAGNETIC FIELD AT A POINT PRODUCES AN ELECTRIC FIELD AT THAT POINT.The Second Concept

Magnetic field produced by changing Electric field.

AMPERES CIRCUITAL LAW

THE STATE OF AFFAIRS ---THEN

THE STATE OF AFFAIRS THENMAXWELLS THINKING.

NATURE LOVES SYMMETRY ! SOMETHING IS WRONG ?

Point form of Amperes Circuital Law for steady magnetic fields, It is inadequate for time varying fields. Take divergence of both sides.

Since divergence of a curl is always zero, we get For time varying current [ or time varying charge density] and is actually, as we shall see in the next slide.

Since the current is electric charge in motion, the total current flowing out of some volume must be equal to the rate of decrease of charge within the volume since charge can neither be created nor can be destroyed. The conservation of charge can be written as I =

By Gauss divergence theorem, =

Point form Continuity Equation.

NEGATIVE TAKES CARE OF DECREASE We now modify this Law for time varying situations. We start with , and substitute for from Gausss law, . Thus,

from which, we get CONTINUITY EQUATION FOR TIME VARYING FIELDS is the total current density for time varying fields. --- Conduction Current Density --- Displacement Current Density.

The current density in Amperes Circuital Law for steady currents, should be replaced by for Time Varying Fields.

Amperes circuital law in point form for time varying field becomes

has dimensions of current density, amperes per square meter.

"The insertion of Displacement Current

in Amperes law was Maxwells

greatest contribution to Humanity."

Richard Feynman

BASICALLY, THESE EQUATIONS DESCRIBE THE FOLLOWING FEATURES OF NATURE.

* STATIONARY CHARGES PRODUCE STATIC ELECTRIC FIELD

* CHARGES MOVING WITH UNIFORM VELOCITY PRODUCE STATIC MAGNETIC FIELD.

* ACCELERATING CHARGES PRODUCE RADIATION.DIRECTION OF VIBRATIONOF ELECTRON

NO RADIATION IN THIS DIRECTIONMAXIMUM RADIATION IN THIS DIRECTION

CROSS SECTIONAL VIEW

The Immense potentialities of these equationsmade Ludwig Boltzmann comment,

Was it a God who wrote these lines ?

Let us solve these equations !

[ ] = [ ] = - [ ] Free Space = - [ ] Permeabllity = 4p X 10- 7 H / m = Permittivity = 8.854X 10- 12 F / m

SimilarlyFOR FREE SPACE

where

We learn about the unknown from the known !

KNOWN

33

In Free SpaceKNOWNUNKNOWN

E and H propagate with a velocity

34

E and H propagate with a velocity

In Free SpaceALL FREQUENCIES TRAVEL WITH SAME VELOCITY IN FREE SPACE AND WITHOUT ATTENUATION.

KNOWNUNKNOWNONE OF THE POSSIBLE SOLUTIONS

35

=

PROPAGATION IN A MATERIAL MEDIUM

=Ohms Law in Point Form

Since , the phase velocity in the medium decreases , since

v=

is frequency dependant.The phase velocity of the wave in the medium will depend on the frequency the dispersion phenomenon.

Since , the phase velocity in the medium is altered.

Only Electric Field is shown

is frequency dependant.The phase velocity of the wave in the medium will depend on the frequency the dispersion phenomenon.

Since , the phase velocity in the medium decreases.

s involves a random process the first terms represent a loss. This term causes an exponential decrease in the amplitude of the wave.Only Electric Field is shown

A phase delay occurs at the boundary , the delay depends on the permittivity of the medium. This is the origin of Refractive Index.Permeability m = m0 mr , mr = 1 Permittivity e Conductivity sDIELECTRIC MEDIUM

FREE SPACESUMMARISINGTHE GRAND CONCLUSIONS

The spatial variation of the charge gets impressed on the Electric andMagnetic fields and they propagate with velocity of light. EVERY POINT IN THE UNIVERSE

[ VACUUM OR MATERIALS ]

KNOWS

THE MOVEMENT OF EVERY CHARGE !!!!!

THE SPACE TIME FLUCTUATIONS OF CHARGES

RESULTS IN AN

IDENTICAL FLUCTUATIONS IN

THE ELECTRIC AND MAGNETIC INTENSITIES

IN THE SPACE AROUND IT..

THE FLUCTUATIONS TRAVEL WITH THE VELOCITY OF LIGHT THE TRAVELLING FLUCTUATIONS CONTAIN

INFORMATION ABOUT THE CHARGE MOVEMENT.

INFORMATION COULD BE

VOICE, PICTURE, DATA OR WHATEVER.

MAXWELLS EQUATIONS PREDICT THE MECHANISM OF

INFORMATION TRANSFER TO

REST OF THE UNIVERSE !

JAMES CLERK MAXWELL13 June 1831 5 November 1879

As of now, if we are able to establish very superior communication systems or to develop extremely accurate sensors or to get a glimpse of the Universe, it is through the use of these equations,carried out during 1860 and 1871.

SEEN TEN THOUSAND YEARS FROM NOW,THERE CAN BE LITTLE DOUBT THAT MAXWELLSTHEORY OF ELECTROMAGNETISM WOULD BE ONE OF THE MOST SIGNIFICANT EVENTS. RICHARD FEYNMAN

THUS THE UNIVERSE IS KNIT TOGETHER. THE ATOMIC MOTIONS

OF DISTANT STARS STILL HAVE SUFFICIENT INFLUENCE AT THE

GREAT DISTANCE TO SET THE ELECTRONS IN OUR EYE IN

MOTION.

IF THIS LAW DID NOT EXIST, WE WOULD ALL BE IN

THE DARK ABOUT THE EXTERIOR WORLD !

RICHARD FEYNMAN

MAXWELLS EQUATIONS AN EXPRESSION OF NATURES HARMONY !!