Some years ago I posted to various discussion groups
which were taken over by Google who are now ruining
the layout (thank heavens for Vortex - though I do
wish Vorts would stick to the text only rule).
There was one post on Dimensional Analysis (as
relating to permeability and permittivity) that I
wanted to dig up. Unfortunately, I couldn't for
then life of me remember when I wrote it or which
of the many physics groups I'd sent it to. All I
could remember was that I had used the pseudonym,
Theresa [my daughter's name and an anagram of
aethers (grin)] and that it would have the word
dimension in it. Thanks to googling I was able
to find it in 10 seconds flat.
I am pasting it below because of its relevance to
another post I am preparing on the relation of the
Beta-atmosphere/aether to EM potentials. It will
be more convenient to refer to it in the Vortex
Archive rather than to have to give a URL for the
google site where it is accompanied by distracting
stuff in both margins.
Cheers
Frank
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Subject: The Mother and Father of all Lights
f.grimer Feb 26 1999, 12:00 am
Newsgroups: sci.physics.relativity
From: [EMAIL PROTECTED]
I have a problem that may interest the readers of this newsgroup since
it relates to the speed of electromagnetic radiation. For simplicity,
I will use the word light to represent all wavelengths of
electromagnetic radiation both shorter and longer than those of
visible radiation.
>From a practical point of view the speed of light is a variable.
By this I mean that its effective speed varies in different media.
In this respect it is not unlike the speed of sound.
Sound’s speed also varies from one media to another.
Paradoxically, light has its highest speeds in what to us is
relatively intangible stuff. Its speed in air is hardly less than its
speed in the vacuum of outer space. It would be difficult to imagine
anything more intangible than the atmospheric vacuum of outer space.
With sound on the other hand it's the other way around. Roughly
speaking the more tangible things are, the faster sound travels.
I once read of someone who calculated how fast sound would be
transmitted across an atomic nucleus. I think the speed turned out to
be faster than the speed of light. Not really surprising considering
the fantastic density of a nucleus. No doubt some reader will supply
the actual speed if he knows it.
It is rather interesting that sound and light are the inverse, so to
speak, of each other.
It is as though,
John Sound lives on the 13th floor with all his walls painted black,
and all his furniture painted white, whilst
Janet Light lives on the 12th floor with all her walls painted white,
and all her furniture painted black.
It is as though,
like Janet and John themselves, their rooms are complementary.
It is as though light and sound
are the yin and the yang of information communication.
Another interesting similarity between sound and light is the equation
that relates speed to other physical properties. This equation is of
the form
X = (Y.Z)^(0.5)
Or in words, X is the geometric mean of Y and Z.
One could hardly think of a much simpler relationship than that.
I suppose it could be argued that,
X = ( Y + Z ) /2
is even simpler since it’s easier to add than to multiply, and though
division is harder than either, finding a square root is harder still.
How can we clad the naked symbols of this simple Son et Lumiere
relationship with garments to stimulate new views?
How can we flesh out this skeletal equation to give our imaginations
something to get hold of, something to generate new ideas?
If we are going to flesh out things then it seems to me that the best
flesh to use is our own. We will let our hair down, loosen up and have
a go. Let’s view the equation as an organic process,
the X as generated by the Y and the Z,
as the offspring of the marrying of the Y and Z by the multiplication
sign. Everyone loves a wedding.
Also, since roots are organic things, that helps the imagery along.
I must confess though, I’ve never seen a square one.
Let’s imagine that Y and Z are the Mother and Father of X. Perhaps Z
could be the father snoozing in his armchair while Y is the mother
holding up her hands in horror because their offspring, X, is putting
the cat in the microwave..
Armed with this whimsy let’s consider the equation for the speed of
light, an equation with which all readers of this newsgroup will
certainly be familiar.
light speed = sqr.rt (permeability x permittivity)
or c = (u.e)^0.5
Now at this point I get a hunch that something’s not quite right. Oh
the equation works all right. I know that. But somehow I get the
feeling that the magnetic mother, u, and the electric father, e, are
not getting on very well. They seem to be at odds with each other.
And their child, c, doesn’t look at all happy about it.
Let's see if we can find out what the trouble is using the technique
of dimensional analysis. Let’s stump up the money for a trip to the
dimen-analyst.
In the technique of dimensional analysis Mass, Length and Time are
conventionally take as the fundamental dimensions. Other qualities
such as Force, Energy, Power and so on are expressed in terms of these
three.
It must be emphasized that Mass, Length and Time are only
conventionally fundamental, not fundamentally fundamental,
if you get my meaning.
Other choices of fundamental dimensions are possible;
and in some cases more convenient.
The choice one makes relates to the particular problem under study.
I suppose one could say that it’s all a question of relativity.
Not Special Relativity though;
or General Relativity either;
just ordinary commonal garden relativity,
with a small r ;-).
I recommend any reader who does not understand the previous paragraph
to read a book on dimensional analysis if he wants to fully appreciate
the next section. As for readers who believe in the fundamental nature
of mass, length and time as firmly as they believe in their God (or
gods), then I suggest they delete this post now. I wouldn’t want to
endanger their simple faith.
When one expresses the dimensions of permeability and permittivity in
terms of [M]ass , [L]ength and [T]ime one finds that they are very
different creatures.
Not the type of people likely to get on with each other in fact.
No wonder their kid is out of sorts;
all screwed up as you Americans might put it.
Now let's lead c's mother and father gently into the dimen-analyst’s
consulting rooms for some marriage guidance counseling.
Combining the two words, permeability and permittivity with no space
in between them gives permeabilitypermittivity . Since this is a bit
of a mouthful we can take the permeab- bit of permeability and the
-ivity part of permittivity to give us Permeabivity, the name for the
Mother and Father united, the name for the couple as one.
To emphasize that it is a group name, and because it’s easy to get it
confused with the two other names, I’ve given Permeabivity a capital
letter, as you can see.
To recap, we now have, the equation
light speed = the square root of Permeabivity,
I’ve not bothered to put this equation in symbols as well as words.
To do so would mean introducing a new symbol for Permeabivity
and that could lead to confusion.
We can now write down the dimensions for the property,
the quality, of Permeabivity,
in terms of the conventionally fundamental dimensions,
the properties, the qualities, of mass, length and time.
When we do this we notice that some of the dimensions are redundant.
In other words they appear in both the denominator and the numerator.
If we treat these dimensions in the same way as we treat letters in
algebra we can simplify the expression by cancellation,
just as we would in algebra.
If there is an [L] in the numerator and another [L] in the denominator
we can cancel them.
Likewise with the [M]s; and likewise with the [T]s.
After all the cancellations are completed the dimensions of
Permeabivity simplify down to [L].[L] in the numerator
and [T].[T] in the denominator.
So the simplified conventional fundamental dimensions of the
Permeabivity dimension are:- [L]^2 / [T]^2
In purified dimensional terms then, the speed of light is
sqr.rt ([L]^2 / [T]^2).
Having aggregated the dimensions of permeability and permittivity to
give the dimension of Permeabivity and purified those dimensions by
cancellation of redundancy, we can now reverse the process.
We can now dis-aggregate the dimension of Permeabivity to give us...
To give us what?
To give us ([L]/[T]) . ([L]/[T])
To give us two speeds, a Mother speed and a Father speed.
So now we can write,
C = sqr.rt ( Cm . Ce )
(I’ve switched to upper case letters for speeds so that I can us lower
case letters as suffices.)
C is the speed of light
Cm is a speed associated with magnetism
Ce is a speed associated with electricity
Now the idea that the interaction of two speeds should give rise to
another speed seems eminently reasonable. After all, the father and
mother of a rabbit are both rabbits. You don’t get a rabbit by mating
a cat with a mouse, do you?
Exactly the same procedure can be carried out on the equation for the
speed of sound. It’s somewhat simpler since there are fewer dimensions
to deal with.
I like to think of the process as a kind of Dimensional Chemistry
comprising three stages
combination - purification - fractionation.
A bit like making whiskey really.
Are there examples of an overall speed being the combination of two
partial speeds?
There are plenty.
The speed of the Pony Express Mail Service is a combination of the
speed of the pony and the speed of the rider handing over the mail at
the staging post. Even in a relay race the runners presumably slow
down somewhat when handing over the baton.
The question I would like readers to consider is this.
Are there any physical (as opposed to mathematical) examples where an
overall speed is a geometric mean of two other speeds.
I realize that those of you who regard the speed of light in a vacuum
as sacrosanct might not want to even consider such a question because
of its obvious implications.
However, since exactly the same question arises for sound you may be
prepared to lend your manifest intellectual prowess to the lesser
problem. ;-)
--
Theresa
--
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