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When isaac newton formurated the law of gravity.                            
                                                                           
He had to postulate forces acting at a distance.
                                                                           
That is to say, the sun could exert a force                                                                                         
directry on the earth even though they were                                
separated by a vast distance with nothing in between.                       

That was an idea that satisfied absolutery no one.

Even isaac Newton's most ardent admirer, that is 
isaac Newton himself, was embarrassed by it.

then, in the 19th century, Michael Faraday imagined
his lines of force, and in the next generation,
Maxwell turned that idea into the field of force,
and everyone was happy.

At first glance, it's hard to see what had changed.

Masses and charges still applied the same forces to
each other with nothing in between.

And yet, the idea of the field of force seemed to 
make all the difference in the world.

why?

The answer is because the field had objective reality.

It had properties that could be predicted, that could 
be calculated manipurated,�@envisioned.

The field existed. 

It became itself the proper subject of scientific inquiry.

For example, we know what happens when the field is 
disturbed.

The disturbance propagates away at the speed of light,
a speed that was inherent in the force laws themselves.

In retrospect, we know that was the first great triumph
of field theory.

And so, we're going to study the properties of those fields.    

But the study of the properties of fields didn't begin 
with electricity and magnetism.

It began in another science altogether, the science of 
hydrodynamics, the flow of water.

Today, we're going to study the properties of all of those
fields--

Electric fields, magnetic fields, and the fields of flowing
water.

Since the creation of the univers, or, in scientific terms,
the big bang, nature and everything in it has gone with the
flow.

Out there, magnetic fields stream though the mostly uncharted
oceans of outer space.

Out there, electric fields bind molecules and atoms into the 
stuff of the universe.

Closer to home port, in the chain of planetary islands known
as the solar system, magnetism and electricity also make their
presense known.

But down here, on the third planet from the sun, there are a 
few differences and a few connections between them.

One difference is that, to the best of anyone's knowledge,
only here on planet earth is there any liquid water.

The second difference, life, took shape on the bottom of the
sea, and the bond between all life and all water remains
intact.

During the courese of evolution, some life changed, thought
it had better things to do on land, and made a crawl for it.

But despite form or function, destiny or direction, all
living things retain that link to the water from whence
they come.

Perhaps that's why, when a form of life called homo sapiens
got around to contemplating the mysteries of electrucity and 
magnetism, it turned for inspiration to the field of flowing
water.

While the flow of electricity and magnetism may seem harder
to get into than a flowing river, the fields of all three have
very much in common.

And even in the field of flowing water, there may be more going
on than meets the eye.

For example, the apparent calm of this bowl of water is soon
pierced by a growing whirlpool.

A bit of ink injected into the water reveals the swirling 
pattern of flow around the vortex.

Not that, even without the ink, it's impossible to see evidence
of the flowing motion of water.

But then, it's mostly the effects rather than the flow that one
actually sees.

In the natural cource of events, air bubbles, leaves and twigs,
and above all the agitated shape of the surface reveals the
motion of water.

 But looking more deeply into the subject, something very interesting
emerges. 

It may seen to be a small point at first, but when it comes to
the fields of water, electricity, and magnetism alike,
it still happens to be the naked truth.

Inside any body of water, there's definite velocity at each
point in space and at each instant in time.

And since velocity is a vector quantity, it follows that the flow
of water, no matter where it flows, can be represented by a pattern
of vectors.

In exactry the same way, electric forces on a test charge can be represented
by a pattern of vectors.

In exactly the same way, electric forces on a test charge can be 
represented by a pattern of vectors in space.

And so can the pattern of directions taken by a magazined needle 
anywhere in space.

Of cource, in real life, it's impossible to see electric and magnetic fields.

However, by observing their effects, it's possible to study their forms
and to discover that, like water, electric fields and magnetic fields
are vector fields, which means that all three are patterns of vectors
in space and that all three can be described with the beautiful flow
of mathematics.

All vector fields obey equation s describing their forms.

the electric flux through any closed surface is proportial to the 
net electric charge inside.

But the magnetic flux through any closed surface is always equal to zerd.

The line integral of the electric field around any closed path is
equal to zerd.

But the line integral of the magnetic field around a closed path.

These equations are the modern expressins of the enigmatic forces
of electricity and magnetism.

But what can be said about the field of flowing water?

Consider, for example, this swift mountain stream. 

The flow is rapid, even turbulent in spcaces.  But for all the
sound and fury, in the larger scheme of things, this is a 
modest flow, a minor tributary in the earth's great network
of rivers and waterways.

Its floi pales in comparison to a major river like the Mississippi
or the Amazon, which can discharge over 180,000 cubic meters
per second.

But the length and the volume of water deliverd aren't  the only
factors in the flow of water.

For example, even through these pipes from the Colorado river
may carry less water than the Amazon, they may carry their water
with a much faster flow.

The the diffference between thise two ideas, total flow versus
velocity at a point, is best expressed by the concept of flux.

The word flux is latin for "water flow,"  but in the language of
mathematics , here's how it works.

Flux is the totalvamount of water passing through an element of area
in a given amount of time.   In time, t,  that's all the hater in a box of
width, vt,  area, a,  and volume, vta.

So the flux, or flow rate, is equal to the velocity, v, times the area, a.
However, if the area element is tilted with respect to the direction
of flow, less of the water passes through it.

So flux depends on the angle betweeen the flow field and the area element.

Defining an area vector perpendicular to its plane, the flux is 
conveniently expressed by the vector dot product.

The same idea is applied to other fields as well.

For example, the flux is found by adding, or integrating, the vector
dot product of the field and the area through which it passes.

That's also the area through which it passes.

That's also the way to express the magnetic flux.

The idea is to find the total number of lines of force through
the surface.

So, whether counting total lines of force or total flow of water,

The mathematical expression of flux is the same for all vector fields.

But all fields are not created equal when flux out of a closed surface
is considerd.

Imagine a surface that completely encloses a volume inside a body of
water.

No matter the flow fields,


 
WHATEVER QUANTITY OF WATER 
FLOWS OUT

MUST ALSO FLOW IN 
AT THE SAME TIME.

THAT'S THE ONLY WAY
IT CAN BE
 
BECAUSE WATER ISN'T CREATED 
OR DESTROYED INSIDE.

SO, IN WATER, THE FLUX OUT
OF A CLOSED SURFACE

IS ALWAYS ZERO.

THE SAME IS TRUE 
IN THE CASE OF MAGNETISM.

THE SAME IS TRUE
IN THE CASE OF MAGNETISM.

THE NET MAGNETIC FLUX 
OUT OF ANY CLOSED SURFACE

IS ALWAYS EZUAL TO ZERO.

BUT THE ELECTRIC FLUX

IS DIFFERENT FROM THAT
OF WATER AND MAGNETISM.

THERE CAN BE AN ELECTRIC FLUX
OUT OF CLOSED SURFACE.

ACCORDING TO GAUSS' LAW,

IT'S PROPORTIONAL TO
THE NET ELECTRIC CHARGE INSIDE.

THE FLUX 
OUT OF A CLOSED SURFACE

EXPRESSES
AN ESSENTIAL DIFFERENCE

AMONG THE VECTOR FIELDS.

THE FLUX OF MAGNETIC FIELD 
AND OF WATER

OUT OF ANY CLOSED VOLYME 
IS ALWAYS ZERO

BECAUSE NEITHER
CAN RADIATE OUTWARD

FROM A POINT IN SPACE.

ON THE OTHER HAND,

THAT'S PRECISELY 
WHAT ELECTRIC FLUX DOES--

RADIATE OUTWARD
FROM A POINT IN SPACE--

WHICH MEANS THAT RATHER THAN 
BEING LIKE WATER AND MAGNETISM,

ELECTRIC FLUX 
IS LIKE THE FLUX OF SUNLIGHT.

THE TOTAL FLUX OF LIGHT 
THROUGH A CLOSED SURFACE

IS PROPORTIONAL TO THE STRENGTH 
OF THE LIGHT SOURCE INSIDE.

THAT'S TRUE WHETHER THE SOURCE
IS A STAR IN THE HEAVENS

OR A BULB 
IN A FLASHLIGHT

AND EQUALLY TRUE NO MATTER
THE SIZE OF THE SURFACE.

SO THE SUN CREATES
A FLUX OF LIGHT

MUCH THE SAME WAY
AN ELECTRIC CHARGE

CREATES A FLUX 
IN THE ELECTRIC FIELD.

BUT WHAT ABOUT WATER FLOW
AND MAGNETIC FLUX?

IF THESE FIELDS 
AREN'T GENERATED 

BY SOURCES THAT CAN CREATE FLUX 
OUT OF A CLOSED SURFACE,

HOW CAN ANY FLOW 
BE CREATED AT ALL?

ONE ANSWER IS THAT WATER

CAN BE SET IN MOTION
BY STIRRING.

FOR EXAMPLE,

LOOK AT A BODY OF FLUID 
IN CIRCULAR MOTION.

ITS FLOW, MADE VISIBLE 
BY ALUMINUM PARTICLES,

STARTS AT THE WALLS 
AND MOVES INWARD.

EVENTUALLY, ALL THE MOTION
SETTLES INTO A RIGID STRUCTURE

WITH THE FLUID ROTATING 
  

AS IF IT WERE FROZEN 
INTO A SOLID BODY.

THAT'S ONE WAY 
TO GET THINGS MOVING.

HERE'S ANOTHER.

WHEN SOMEONE 
PULLS THE PLUG,

ALL MOTION BREAKS LOOSE,

AND SOONER OR LATER,
A NEW PATTERN ALWAYS DEVELOPS,
AND IT'S CALLED VORTEX FLOW.

IT'S AN EXTREMELY STABLE
KIND OF MOTION,

ALMOST IMPOSSIBLE TO STOP 
ONCE IT GETS STARTED.

THIS SAME PRINCIPLE 
IS AT WORK

IN THE GENTLE WHIRLPOOL 
IN THE FAMILY BATH 

AND IN THE DESTRUCTION
OF AN ENORMOUS HURRICAME.

IT OCCURS BECAUSE
EVEN BEFORE THE STORM BEGAN,

THE MASS OF AIR 
HAD A SPECIAL KIND OF MOTION.

IT'S CALLED CIRCULATION.

AS IN THE CASE OF A TUB 
OR A HURRICANE,

A FIRE IN THE FOREST
OR A TWISTER ON THE PLAIN,

CIRCULATION LEADS 
TO A VORTEX.

WHEN THE PLUG
IS FIRST PULLED,

WATER NEAR THE CENTER
OF THE BOWL,

WHERE THE INTERNAL MOTION
IS LEAST,

FLOWS DIRECTLY DOWN,

TO BE REPLACED BY WATER
FROM FARTHER OUT.

BUT THAT WATER,
CIRCULATING MORE RAFIDLY,

CONSERVES ANGULAR MOMENTUM
AS IT MOVES INWARD.

MASS TIMES DISTANCE 
FROM THE CENTER TIMES SPEED

IS CONSERVED
FOR EACH BIT OF WATER.

SO AS IT GETS CLOSER
TO THE CENTRAL AXIS OF THE BOWL,

EACH BIT OF WATER
SWIRLS FASTER AND FASTER

UNTIL THE WATER
RUPTURES INTO THE HOLE

AT THE CENTER
OF THE WHIRLPOOL.

BECAUSE EACH BIT OF WATER
CONSERVES ANGULAR MOMENTUM

AS IT SPIRALS INWARD
TOWARD THE VORTEX,

THE BASIC PATTERN
OF VORTEX FLOW

IS CIRCLES OF CONSTANT VELOCITY 
PROPORTIONAL TO 1/R,

THE DISTANCE
FROM THE VORTEX CORE,

OR HOLE IN THE FLOW.

THE TRAPPED ANGULAR MOMENTUM

ACCOUNTS FOR THE REMARKABLE
STABILITY OF THE VORTEX FLOW,

AND THE VECTOR FIELD 
HAS THE SAME FORM

AS THE MAGNETIC FIELD

OF COURSE,
IN THE VORTEX FLOW,

THE VECTORS REPRESENT 
VELOCITY OF FLUID,

WHILE IN MAGNETISM,

THE VECTORS REPRESENT 
STRENGTH OF FIELD,

NOT THE SPEED OF ANYTHING.

BUT THE SHAPE AND 
THE MATHEMATICS ARE THE SAME,

AND TOGETHER
THEY ILLUSTRATE THE FORCE

OERSTEAD FOUND
CIRCLING A WIRE.

THOUGH HIS DISCOVERY

CREATED AN INTELLECTUAL STORM
IN THE SCIENTIFIC COMMUNITY,

NO ONE REALIZED IT ACTUALLY 
TOOK THE SHAPE OF ONE.

BUT THAT IT DID,

AND THE NEW SCIENCE
OF MAGNETISM

WOULD REVOLVE AROUND
THAT VERY FACT.

SINCE THE FIELD 
OF CIRCULATING FLUID

HAS THE SAME SHAPE
AS A MAGNETIC FIELD,

IT SHOULD COME 
AS NO SURPRISE

THAT ITS MATHEMATICAL
DESCRIPTION HAS THE SAME FORM.




THE LINE INTEGRAL 
OF THE VELOCITY FIELD OF WATER

IS CALLED THE CIRCULATION.

SINCE THE WATER
FLOWS NEARLY IN CIRCLES,

WITH VELOCITY 
INVERSELY PROPORTIONAL

TO DISTANCE FROM THE CORE,

THE CIRCULATION 
AROUND A VORTEX CORE

DOESN'T DEPEND ON THE DISTANCE
FROM THE CORE.

IT'S A CONSTANT,
PROPORTIONAL
TO THE STRENGTH OF THE VORTEX.

IN THE SAME WAY,

THE LINE INTEGRAL
OF A MAGNETIC FIELD

IS PROPORTIONAL
TO THE ELECTRIC CURRENT

THAT CREATES THE FIELD.

MATHEMATICALLY SPEAKING,

THE VORTEX IS THE SOURCE 
OF FLUID FLOW

IN EXACTLY THE SAME SENSE

THAT ELECTRIC CURRENT IS
THE SOURCE OF MAGNETIC FIELDS.

JUST AS AN ELECTRIC CURRENT
CAN FORM A CLOSED LOOP,

SO CAN A VORTEX CORE

FORM A LOOP 
CALLED A VORTEX RING.

THE FLOW FIELD
OF A VORTEX RING

IS THE SAME 
AS THE MAGNETIC FIELD

OF A CURRENT LOOP.

VORTEX RINGS, LIKE ALL VORTICES,
ARE AMAZINGLY STABLE STRUCTURES.

WHETHER ELECTRIC, MAGNETIC,
OR THE FLOW OF WATER,

ANY VECTOR FIELD CAN BE
DESCRIBED BY TWO QUANTITIES--

ITS LINE INTEGRAL
AND ITS FLUX.

SO FOR EACH TYPE 
OF VECTOR FIELD,

THE QUESTIONS ARE THE SAME.

FIRST, CAN IT CIRCULATE?

AND SECOND, CAN IT FLOW RADIALLY
TO OR FROM A POINT?

FOR MAGNETIC 
AND ELECTRIC FIELDS

THAT ARE CONSTANT IN TIME,

THE ANSWERS ARE SIMPLE.

MAGNETIC FIELDS CIRCULATE,

BUT NEVER CONVERGE
TO A POINT.

ON THE OTHER HAND,

ELECTRIC FIELDS 
RADIATE FROM POINT CHARGES,

BUT NEVER CIRCULATE.

THESE ARE 
THE FUNDAMENTAL NATURES

OF STATIONARY ELECTRIC
AND MAGNETIC FIELDS.

ONE WAY OR ANOTHER,
IN ANY NUMBER OF SETTINGS,

IT'S COMPARATIVELY EASY TO WATCH 
WATER CIRCULATE AND FLOW.

HOWEVER, THE FLOW 
OF ELECTRIC AND MAGNETIC FIELDS

IS ANOTHER MATTER ENTIRELY.

ELECTRIC AND MAGNETIC FIELDS,
THOUGH THEY'RE JUST AS REAL,

ARE HARDER
TO GET ONE'S HANDS ON.

NEVERTHELESS,
IN PHYSICAL TERMS,

ELECTRIC AND MAGNETIC FIELDS
EXIST IN SPACE

FAR FROM THE CHARGES OR CURRENTS
THAT CREATE THEM.

AND ALL THREE VECTOR FIELDS

HAVE MORE THAN 
JUST SHAPE AND FORM.

THEY ALSO HAVE ENERGY.

FOR EXAMPLE,

IN AN ELECTRIC FIELD
SUCH AS THE ONE IN A CAPACITOR,

ENERGY MAY COME
FROM A BATTERY 

DOING THE WORK OF MOVING CHARGES 
AGAINST THE FORCE

OF THEIR MUTUAL
ATTRACTIONS AND REPULSIONS.

IN A MAGNETIC FIELD,

SUCH AS THE ONE 
IN A SOLENOID,

SUCH ENERGY MAY ALSO BE PRODUCED
BY A BATTERY,

WHICH MAKES THE CURRENT FLOW
TO CREATE THE MAGNETIC FIELD.

IN A FLOW FIELD, 

WORK IS DONE
SETTING THE FLUID INTO MOTION,

AND THE ENERGY RESIDES
IN THE MOTION OF THE FLUID.

THE ENERGY
IS KINETIC IN NATURE,

1/2 mv-SQUARED
FOR EACH BIT OF FLOWING MATTER.

SINCE THE FLUID'S COTINUOUS,

IT'S BEST TO EXPRESS ENERGY
PER UNIT VOLUME

AS 1/2 THE MASS
PER UNIT VOLUME OR DENSITY

TIMES THE VELOCITY SQUARED.

BUT IN ANY CASE,

ENERGY RESIDES
AT EACH POINT IN SPACE

PROPORTIONAL TO THE SQUARE
OF THE VELOCITY FIELD

AT THAT POINT.

BY THE SAME TOKEN,

THE ENERGY DENSITY 
IN AN ELECTRIC FIELD

IS PROPORTIONAL TO E-SQUARED
AT EACH POINT.

WHETHER MAGNETIC, ELECTRIC,
OR FLOWING LIQUID,

THE ENERGY PER UNIT VOLUME

STORED IN EACH 
OF THESE VECTOR FIELDS

IS PROPORTIONAL
TO THE SQUARE OF THE FIELD

AT EACH POINT IN SPACE.

SO WHETHER IT'S FLUX,
ENERGY, OR CIRCULATION,

SOME IMPORTANT PRINCIPLES
OF THE PHYSICAL WORLD

CAN BE SEEN AT WORK

IN ONE OF ITS MOST
PRIMITIVE ENVIRONMENTS,
THE FREE-FLOWING RIVER.

OK. I KNOW 
WHAT YOU(RE THINKING.

THERE ARE TOO MANY CANDLES
FOR IT TO BE MY BIRTHDAY.

BUT I'M GOING TO 
BLOW SOME OF THEM OUT.

I'LL DO IT THE HARD WAY,

USING VORTEX RINGS
FROM ALL THE WAY OVER THERE.

THOSE SMELLY CHEMICALS
I PUT INTO THIS CANISTER

MAKE SOME SMOKE.

THAT WILL MAKE THE VORTEX RINGS
INTO SMOKE RINGS.

THE SMOKE RINGS
ARE VERY STABLE STRUCTURES.

THEY'LL LAST LONG ENOUGH 
TO GET OVER THERE

AND BLOW OUT THE CANDLES
WHEN I TAP ON THIS MEMBRANE.

HOW MANY THINK
I CAN BLOW OUT CANDLES THIS WAY?

YOU CAN LEAVE.
THE REST STAY AND WATCH.

WATCH THIS.

[APPLAUSE]

THIS DEMONSTRATION, 

WHEN IN WAS FIRST DONE
IN THE 19th CENTURY,

MADE SUCH A BIG INPRESSION 
THAT FOR HALF A CENTURY,

SOME OF THE BEST MINDS
IN EUROPE

THOUGHT THAT ATOMS
WERE VORTEX RINGS IN THE ETHER.

THAT'S A STORY 
WE'LL GET TO LATER.

RIGHT NOW, LET'S GO GET 
SOME FRESH AIR.

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