Physics of Lenses and Idiots (part I)

Abstract:

Found a magnificent Lens, Got the story.

and thoughts about Idiocy in organizations

This time I didn't lie!

How's about me keeping y'all on your toes ha?

Well, as promised this post will deal with lenses, and idiots.
Following our recent brush with hypoxia, my lovely wife and myself proceeded to tour the coastline of California, we drove through San Francisco, met with a distant relative, and all in all had a great time.

At some point we drove through Cambria, which is a quaint little town somewhere on the coastal road between Carmel-by-the-sea and L.A.

WARNING: next is a tacky description fitting of a less-than-mediocre dungeon master:

"It was a cool morning. the air was still. thick fog covered the valley and wayward sounds made their eery way to our ears. the cold ate at our bones, and it was difficult to see anything further then the tip of your nose. 

Somewhere to the left you hear the crack of dried bones, and you think you might have felt something brushing against your boot, or maybe it was your imagination...."

Ok, sorry, got carried away there just a tad....
Anyway, it was cold that morning so  the town was shrouded in mist. we took a little stroll down to the waterfront, where it was rather stinky as the sea washed ludicrous amounts of kelp to the shore and it just lay there rotting.

Well we needed to refuel so we pulled over, and while we were in the station, after grabbing a cup of coffee something caught my eye.

by that time the sun was about half way between horizon and zenith, and the glint of a glass house - the kind you should not throw stones from - was pulling at me.

We went to investigate only to find something incredible.
there is was, encased in a glass gazebo, welded shut and otherwise unapproachable, like a queen enthroned in a crystalline palace - The most wonderful Fresnel Lens I have ever laid eyes upon....

Piedras Blancas lighthouse Fresnel Lens at Cambria

I was psyched.
We went to find someone who could maybe open the locked carousel, so I could take a closer look at it, and found an elderly fellow who was kind enough to tell us the wonderful story of this lense.

In his crackled yet fiery voice he told the story of the lens....  

It seems that ever since a couple of decades ago, as the US coastguard decommissioned several lighthouses along the US coastline, this lens was all but forgotten by all, laying embedded in a non-active lighthouse's torch.

Imagine this untouched treasure trove, laying partly in the sand, covered by rotting kelp, the sand and salt slowly eating away at the tempered glass, scratching it, scarring it....
oh the humanity!!! 

The good people of Cambria led by local coastguard veterans secured the consent of the US coastguard, took the lens, fixed it, re-polished it, and enshrined it in the glass carousel in town,
by the veterans center.

Now as it turns out, the US coastguard, found out that this kind of lens is not manufactured anymore. kind of like the precise German lenses and mirrors used in WWII to blind British air-fighters on their runs over bombarded Berlin. it just costs too damn much to manufacture these kind of lenses especially when there are "better" alternatives to be found, more on that later...

That no one manufacture these anymore is a nicer way of saying these babies cost a small fortune each!!! and now, the US coastguard found out about that and wanted to put their hands on this lens.... snatch it away from the land lubbers' keep.

After a long and epic struggle, almost as epic as Gandalf's struggle with the Balrog at the bridge of Khazad-dum (and later falling, and in the waters, and on mountains... sheesh he must've been really tired that night), or the no-less epic struggle of scraping myself out of bed each morning....
the land lubbers won the lens and it can still be seen lit on special events and festivals of Cambria

An epic struggle

An equally epic struggle

Another nice story about this lens is this:

When first they brought the lens to town, they put a regular old incandescent light bulb at the focus of this lens, only to find out that the housing for the light bulb melted completely!!!

it took them a while (and several light bulbs and housings) to understand that at the focal point things get heated. so much so in fact that the whole thing simply melted.


Warning: physics ahead-

Let's try to understand how hot it gets at the focal point shall we?

Let's say the incident area of the lens is about 1 square meter.
assuming the light bulb is approximately round with a radius of say 5 cm the incident area it presents is about \(\frac{\pi}{4}\cdot 10^{-3}\,m^2 \).

Now the light flux that goes through the lens goes to the focal point with a fractional portion accounting for flux losses due to diffraction, interference and other energy losses (crepuscular rays for instance) so basically:
\[\Phi_{bulb}=\varepsilon\Phi_{lens}\]

Now the flux per area averaged over all earth's incident area, over all emission wave-lengths, and over night-day at the mean earth-sun distance is called the earth's solar constant, and the observed value is \(S_{\oplus}\approx1360\, \frac{J}{sec\cdot m^2} \) , so:
\[J_{bulb}=\frac{\Phi_{bulb}}{A_{bulb}}=\varepsilon\frac{\Phi_{lens}}{A_{bulb}}=\varepsilon S_{\oplus}\frac{A_{lens}}{A_{bulb}}\]

Where \(J\) denotes the energy flux per area unit, which is proportional to the temperature in the 4th power by Stefan-Boltzman's law:
\[J=\sigma T^4\]
In which \(\sigma\) is the Stefan-Boltzman constant which is given in SI units by \(\sigma=5.67\cdot 10^{-8} \frac{Jouls}{sec\times m^{2}K^{4}}\)
And so if we substitute all the constant and data we get that:
\[T_{bulb}=\sqrt[4]{\varepsilon 3.05\cdot 10^{12}}\]

Where providing no energy loss whatsoever (\(\varepsilon=100\%\)) we get the temperature there to be about 1300K or about \(1000^{\circ}c\).
Let's say the situation is pretty bad and we have \(50\%\) energy loss we then get "only" 1100K
which is about \(800^{\circ}c\)... you get the picture.

Just for proper comparison purposes:
Aluminum melts at about 900K, Copper at about 1350K, Lead and Zinc  at 600K and 700K respectively and Iron at about 1800K.
So obviously if the bulb housing was made with aluminum wiring it is fairly easy to see the whole thing would simply melt even in relatively cold days, not to mention the plastic in the housing...
after a fairly short research it seems most plastics melt at a temperature no higher than about \(200^{\circ}c\), while even temperature and fire resistant brands do not fair well above \(400^{\circ}c\) to say the least. so really the lens could be in a horrendous state or simply our assumption for \(\varepsilon\) could be way off, since even at \(\varepsilon=5\%\) it turns out the plastic housing would have melted...

Ok, so what they did in the end is lower the bulb housing by maybe ten centimeters, so now, while the lighthouse does not yield such illumination as it used to, it is now operable.

So my lovely wife went through this post and had several remarks:
1. This post is too long.
2. I am a complete and utter nerd, but she loves me still.
3. it is late and I should go to bed if I prefer hell not to be risen....

Thus I am at this point apologizing, and cutting this post in two.

Let's conclude for now by deriving an interim equation for the ambient stupidity in an organization...
 
It is obvious that  the bigger an amount of money is involved in a project, the more foreign consideration (i.e. bribes, conflicting interests etc. )  will come into play.
Also there is a pretty known principle where a chain or for our purposes a net is only strongest as it's weakest link, this also applies to teams, where a team is only strongest as its weakest member.
and for our purposes a project team is only as capable as it's dumbest member.
Also we want to make an observation that not only the above is true, but as team grows larger it becomes more sluggish and bureaucracy bound, now the only question is what are the relations, so , obviously when a lot of people work on the same project with no proper guidance (i.e. tyranny or otherwise simple dictatorship) the average work tend to scatter in all directions equally thus yielding zero work.On the other hand it is a well known fact that given a strong and small team, work is done in greater efficiency.
so I propose the next relations:
\[eff \propto Ne^{-\frac{N}{N_0}}\]
Where N is the number of people working on the same project, and \(N_0\) is there for normalization purposes, and \(eff\) denotes the efficiency.

also it is a well known fact that you need a team of people to fix one persons stupid mistake, and if I am VERY lenient I will only take a quadratic decrease in efficiency in respect to the amount of money, thus I propose a refinement:

\[eff\propto \frac{B}{\beta B_s}\frac{N}{$^2}e^{-\frac{N}{N_0}} \]

Where \(B\) is the average brains of the team, \(B_s\) is the brains possessed by the dumbest person on team, \(\beta\) is the coefficient that represents how many other team members it takes to fix one dumb mistake...

Keeping in mind that idiocy is the reciprocal of efficiency we get the ambient idiocy of a team or organization to be:

\[I_{diocy}\propto \frac{\beta B_s}{B}\frac{$^2}{N}e^{\frac{N}{N_0}}\]

This could very well account for the coastguard behavior in that instance, albeit in military organizations this dynamic is really not enough to account for all the stupid s@#% one encounters there, and I have rich experience of such encounters...  

Also - basically whichever organization it was that hired me, probably raised it's idiocy bar by several orders of magnitude... :p

That's all for now folks, part II of this discussion will be very interesting, with an explanation of why, for instance,it's highly unreasonable that the Klingon ship captained by Kurn son of mogh, with Worf on board out maneuvered two Klingon birds of prey by flying into a star's corona, and while Kurn's ship was able to escape and break orbit, the pursuing ships fell into the star's sea of lava and consequently exploded...