Summary:
+ Introduction
+ + Refraction Reflection
+ + telescope diffraction
and spherical aberration
+ Dispersion and Telesopio Newton
+ + luminiferous ether
Birefringence and Polarization
- Interference
; A strange hail
a strange stretch of sea
;
Thomas Young's double slit experiment
Read Fresnel
Details on the simulation of hail
+ Rainbow
A strange hail
The other day I had to leave the house while the time promised a good time ... and I had un buco nel tetto. Per non rischiare danni maggiori, ho sistemato un bel catino proprio sotto al buco; ma quando sono tornato la situazione era questa:
C'era stata una grandinata eccezionale, e io ho trovato sì il catino pieno di ghiaccio, ma molti chicchi erano sparsi sul pavimento. Una spiegazione potrebbe essere che il catino si sia riempito troppo, o che qualche chicco ne sia rimbalzato fuori; nonostante queste ipotesi, ho voluto appurare se per caso non succeda qualcosa durante il viaggio : siamo sicuri che tutti i chicchi di grandine arrivino sempre nel catino?
Nell'animazione qui sopra mostro 4 casi diversi:
A - the hailstone falls away from the hole, and then never comes on my floor.
B - exactly the grain passes from the center of the hole, and it falls into the basin.
C, D - these two hailstones bounce off the edge of the hole, deflect their trajectory and fall on the floor. New
curiosity: how likely is it that the hailstones fall out of the bowl?
With a computer simulation, I dropped the hailstones from random locations, recording for each the point of arrival on the floor. Accumulating a million tests I could build the following graph: the variable height of the white bell-shaped graph indicates the amount of grain that has fallen into every spot on the floor.
The point marked A on the floor is just below the hole in the roof, and the red line that indicates the height of the bell at that point, shows that the probability that a hailstone falls right there are greatest. Point B is in an area not is located underneath the hole, yet the probability that the hailstones fall there (indicated by the length of the green line) is far from negligible. Finally point C, far away from the hole, also has a good chance of being reached by hail!
Yesterday I discovered a second hole in the roof, and another storm was coming. Certainly do not hope to get potermela placing a simple pair of basins, on the other hand I got another curiosity: now that there are two holes, what are the chances that the hailstones fall at various points of the floor?
Figures 1 and 2 show the two holes in the roof. The green line above the point A is the total probability that the hailstones falling on the floor at that point: in fact the area in red indicates the probability who have to go to the various points of the ground beans that pass through the hole 1, while the white area refers to the hailstones that pass through the hole 2, the sum of these two curves of probability says how much ice will fall on every point the floor. ▲
a strange stretch of sea
After clearing everything (and plugged the holes in the roof), I went to relax at sea. As if by magic, my thoughts have fled the hail ... to deal with waves: for I have seen live the phenomenon of diffraction which we have already parlato qui . Mi sono chiesto come variasse l'altezza dell'acqua momento per momento, e ho ottenuto quanto segue:
La linea rossa indica in ogni istante l'altezza dell'acqua che si trova in corrispondenza della linea grigia. Ciò che però mi interessa in modo particolare è l'ampiezza delle onde, non l'altezza istantanea dell'acqua: nel disegno che segue, il grafico rosso indica l'altezza massima raggiunta dall'acqua in ogni punto, e quindi l'ampiezza delle relative onde.
Mi sono stupito nel vedere la netta somiglianza fra il comportamento di queste onde e quello della grandine che abbiamo visto prima. Nell'immagine qui sotto, a sinistra mi sono limitato ad "appiattire" un po' la campana della grandine: la somiglianza è davvero notevole!
Voglio ricordare che, per quanto i grafici relativi sia alla grandine che alle onde siano ottenuti grazie simulazioni fatte al computer, i procedimenti di calcolo (che riproducono la realtà dei fenomeni fisici) sono totalmente diversi: per la grandine si tratta di una statistica di eventi casuali governati dalle leggi della meccanica, mentre per le onde si tratta della simulazione di una matrice di punti materiali legati da molle, come spiegato qui . Proprio il fatto che i risultati siano così simili spiega perché it was so difficult to understand what it was made light, when waves or corpuscles!
Continuing my walk along the seafront I came across another phenomenon: the waves pass simultaneously from two different slots.
Initially the waves behave normally, but when it spread to the point of overlap is generated interference between the two groups of waves, the result can be seen particularly well in the final frame of the animation above.
In the following we can see the amplitude of the waves, as we did for the waves that were generated by a single slit:
compare it with the curve of the hail that passes through two holes:
We see immediately that there is a notable difference: the left curve (hail) proceeds sweet, very constant; the right one shows a trend of maximum (indicated by the yellow line) is very similar, but also exhibits rapid oscillations, the maxima and minima very frayed: in fact this kind of curve, referring to the wave phenomena, it is even called "fringe interference ".
What accounts for the interference fringes? The waves generated at the two cracks have the same amplitude and depart with the same phase (ie, there are always at their peak at the same time). The waves, however, bring us some time to spread, and this causes "lag" variables in the waves:
1 - Top left you see two waves using exactly the same time to reach the point intersection on the purple line. At that point the waves arrive exactly in phase, their peaks always coincide, so their effect is added, the red wave obtained by the union of the two waves have twice the amplitude of the waves each green and yellow.
2 - Top right the length of the path taken by the two waves differs for exactly half a wavelength. This means that when one is at the peak, the other is the peak opposite: there is practically a continuous full-load compensation, and the wave amplitude has generated nothing.
3 - At the bottom left the two waves arrive in phase again, but given the different length of the paths trodden, the wave turns yellow swing more than the green. The result is a wave of amplitude equal to that generated in case 1.
4 - Bottom right the confluence of the two waves occurs at a point where they are almost out of phase, ie the positive and negative peaks are offset but not completely vanish: the wave that emerges has a width very small.
Note: All points of confluence of the waves that depart from the cracks are aligned on the same purple line, it is precisely with this type of process are calculated interference fringes.
In summary, we can say that waves and particles exhibit many similar behaviors, but not interference. ▲
Thomas Young
In the history of optics we've seen in a number of significant characters, such as Fermat, Huygens, Newton. What I present to you now, for some versi, è forse il più incredibile di tutti: ecco a voi... Thomas Young!
Nasce nel 1773 in Inghilterra. A due anni sa leggere e scrivere; a sei conosce già il latino, a quattordici anche greco, francese, italiano, ebraico, caldeo, siriaco, samaritano, arabo, persiano, turco ed etiope!
Nel 1792 inizia il corso di studi in medicina a Cambridge, infatti agli inizi del XIX secolo Young è (almeno in teoria) un medico: ma è molto più interessato alle malattie, che ai malati. Per fortuna un vecchio zio vede bene di lasciargli una ricca eredità, per cui gradualmente lascerà ad altri la cura dei pazienti per dedicarsi liberamente ai suoi studi.
His research interests in many fields: bears the name of "Young's modulus" a coefficient used in the study of deformation of the material, making major studies on capillarity and surface tension of liquids, in particular, is the first to use The term energy in the sense that we give them today, but ... we had not said he was a doctor? Well, actually he graduated in physics at Göttingen, who knows if it will have the opportunity to learn German too!
About medicine and physiology in particular, already in 1793 can explain, following which he made original experiments, which focus the eye depends by the change of curvature of the inner surface of the lens. In 1801 he published his findings under which the sensing element would be the retina of the eye, not the lens as many believed, that color perception would be due to "only" three different types of nerve endings sensitive to red, green and blue ... and all that, even before the age of thirty.
In those same years is also involved in research on light ... but first I must explain what we're doing over here, along with his portrait, that text written in ancient characters: This is a section of the famous Rosetta Stone, the stone "trilingual" which allowed to decipher, after enormous efforts, the two Egyptian scripts (hieroglyphic and demotic) from a greek text (the greek does not appear in the photo).
Well, the first to figure it out ... it was our Thomas Young, contributing to the fundamental discovery that it is based on two scriptures phonograms (each sign is a sound, as in modern alphabets), instead of semagrammi (each symbol corresponds to a concept) , as claimed by those who had studied the Egyptian writing until then. Finally
with biographical anecdotes: in his article "Language," written for the Encyclopaedia Britannica, Young compares grammar and vocabulary of four languages \u200b\u200b... yes, I think that the German he had learned! ▲
double slit experiment
During his studies on the physiology of the eye, Young can not help but wonder about the nature of light, and goes believing that it is a wave phenomenon. Apparently familiar with the phenomenon of interference, it try to verify if the light produces interference phenomena.
As light has a wavelength very small (less than one thousandth of a millimeter), e quindi apparentemente sia molto difficile da maneggiare, l'esperimento è più facile da realizzare di quanto sembri. Infatti l'interferenza delle onde ha la proprietà per cui, allontanandosi dalla doppia fenditura, le zone di rafforzamento o di elisione delle onde diventano sempre più ampie; le linee nella figura che segue sono tracciate in corrispondenza dei punti sui quali non si generano onde:
Ho provato a realizzare l'esperimento per conto mio usando un piccolo puntatore laser ( ATTENZIONE : se decideste di replicare l'esperimento, non dirigete mai la luce del puntatore direttamente negli occhi, né vostri né altrui):
Top left you see a card that acts as a shield, which has two small holes with a pin. The laser pointer illuminates the two holes for a double beam of light projected on a sheet into rectangles positioned about five feet away (even if you do not see well, the squares have more or less the same size in all four shots) . Here are the results of the experiment:
- Top right one sees the projection of the laser beam as it is, without a screen interposed.
- In the lower left is the projection of the laser beam passing through one of the holes in cardboard: the disk of light is wider but ha un'intensità ridotta, ed intorno ad esso è visibile qualche anello di diffrazione.
— In basso a destra il raggio laser passa attraverso i due forellini: si vede comparire chiaramente la frangia d'interferenza. Insomma: è proprio vero che nelle condizioni giuste luce più luce fa buio!
Ovviamente Young non dispone di un raggio laser, ma riesce a notare qualcosa di sorprendente anche con la semplice luce solare. La quale, come sappiamo, è una mescolanza di colori, ciascuno associato ad una diversa lunghezza d'onda: Young infatti non vede zone di chiaro-scuro (quelle si vedono solo con una sorgente di luce monocromatica), ma frange variamente colorate.
Nel 1802 pubblica the Philosophical Transactions of the Royal Society a paper entitled "Report of some cases of production of color not yet described" in describing this and other experiences, that accounted for by the wave theory. But immediately this article does not get his big awards: one is written in a manner not very clear, using a mathematical approach rather elementary (and hence is a bit 'snubbed by other scientists), but most of all ...
... Young is guilty of the crime of "treason"! In fact, supporting the wave theory of light goes against the views of Newton, and this is something that the scientific English absolutely can not tolerate, especially if it is just an Englishman! ▲
Read Fresnel
The witness then passes directly through the Channel ... and is taken over by the French, such as Malus we talked about the polarization . The most important person in this stage was Augustin-Jean Fresnel (one of "Fresnel lenses", 1788-1827): over many years develop formulas that explain all the phenomena of reflection, refraction, interference, diffraction. the study of polarized light after many uncertainties him to accept a light composed of transverse waves, supporting the hypothesis more difficult to digest for the propagation of light. In fact, when Fresnel
presents his studies at the Academy of Sciences of Paris, between 1815 and 1819, raises the immediate protests from the likes of Pierre Simon Laplace and Siméon-Denis Poisson, supporters of the corpuscular theory. In particular, Poisson
studied the works of Fresnel in the hope of finding some weak point, and at one point is sure to have found it: according to the Fresnel formulas, if a light beam involves a circular screen, beyond the obstacle along the axis that connects the light source and the center of the screen, you should see a ray of light, which according to Poisson is clearly impossible.
The verification is done immediately, and the result is that this ray of light "materializes" indeed! The explanation lies in the fact that at the time that the light generated in the screen A reaches B, B will radiate from the edge of the waves that generate the normal diffraction. Then all the points that lie along the axis that extends beyond the center of the screen will be equidistant from the edge of the screen itself, and receive the waves diffracted in phase, adding. I
created a simulation of this phenomenon, modifying my program wave to do the calculations on a three-dimensional matrix. After about six hours of calculations, the computer gave this result:
In fact, this phenomenon was already described in 1723 by Giacomo Filippo Maraldi (1665-1729), but his observation was largely ignored until it was rediscovered nell'occasioe I told above. Ironically, today this phenomenon is known as Poisson's Spot. "
On the other hand is called "Fresnel diffraction" the complementary phenomenon: if you di investire un ostacolo circolare il fascio di luce viene fatto passare in mezzo ad un'apertura rotonda, poco al di là dell'apertura si viene a creare un "punto nero", una zona in cui la luce proveniente direttamente dalla sorgente luminosa e le onde diffratte dal bordo dello schermo si elidono completamente. Ecco qui un'altra simulazione (altre sei ore di calcolo per il mio povero computer); il "punto nero" è evidenziato dal rettangolo rosso alla fine dell'animazione:
Con questa storia abbiamo assistito a uno dei casi più straordinari di applicazione del "metodo scientifico": una massa di indizi (i fenomeni of light), allowing you to develop mathematical laws (the laws of Fresnel), which promise to be a phenomenon never seen before (Poisson's spot), which exceeds the experimental evidence: it seems to me something really amazing! And Fresnel had to be of the same opinion, in fact writes:
All the compliments I received [...] I have never done so much pleasure as the discovery of a theoretic truth, or confirmation of a calculation experiment
▲
Details on the simulation of hail
All texts that deal with interference make the comparison between the behavior of particles or waves that pass through holes or cracks, but very often replace the word "particles" in the word "bullets".
Initially I thought of speaking of bullets, then it occurred to me that the hail could be a better plot device. The substance, however, did not change.
The affair of the bullet (or hail) led me astray in writing the simulation program for at least two reasons: first, that the hole in the roof must have exactly the same diameter of hailstones.
Infatti provando con un buco più grande, le probabilità che i chicchi passino in mezzo al buco, senza toccarne il bordo, sono nettamente superiori a qualsiasi altro percorso che comprenda un urto con il bordo stesso. Con un buco più grande il risultato sarebbe questo:
Il secondo problema riguarda il calcolo dell'angolo di deviazione del chicco di grandine dopo l'urto. Infatti ho scritto un programma che riproduceva le normali leggi fisiche relative agli urti elastici fra corpi rigidi:
Il chicco di grandine scende in direzione verticale, e un punto della sua circonferenza (il cui moto è indicated by the green arrow) collide with the edge of the hole in the roof (red lines). At the moment of the grain gets a boost oriented according to the range shown in black (the radius that connects the point of impact with the center of the grain), and is in practice as if it is a rebound of grain on the yellow line, perpendicular within the black. As a result of impact the trajectory of the hailstone is diverted so that the angle β is equal to the angle α.
running the simulation program I was very surprised to see results like this:
The side sections of the curve are very similar to those we have seen above, in the explanation of interference, but looking at the central area turns out that the probability of finding hailstones just below the hole in the roof are virtually nil!
This is a phenomenon that I would have to wait ... Instead I lost several hours searching for the "bug" in the program. The origin of this unexpected behavior of hailstones can be understood with the following scheme:
For the angle β is very close, the horizontal distance d between the left edge of the grain and the edge of the barrier (see also the magnification on the right) must be very small, so the odds che il chicco di grandine sfiori il bordo del buco così di striscio da non essere quasi deviato nella sua traiettoria sono praticamente nulle: per chi mastica un po' di trigonometria, il problema risiede nell'elevatissima sensibilità della funzione arcoseno per angoli vicini ai 90°.
Alla fine ho optato per un altro genere di simulazione: non un urto elastico fra due corpi rigidi, ma una repulsione di tipo "quadratico inverso", analoga a quella che si ha fra cariche elettriche o poli magnetici dello stesso segno.
Durante la sua caduta, l'oggetto c viene respinto da entrambi i poli a e b che are ideally positioned to the sides of the hole in the roof. The forces depend on the inverse of the square of the distance since the distance is less than the ac bc , the object c went right from the pole to more than it is to the left of the pole b , and then deviates to the right (purple arrow).
The results, like those I have shown talking about interference, are exactly what you see in scientific texts. My simulations can not reflect the behavior of the hail, as I said that was chosen as a narrative device to make it easy to understand the phenomena, on the other hand, the method of calculation che ho adottato rispecchia fedelmente il comportamento di altri fenomeni fisici, maggiormente appropriati allo studio dell'interferenza, ma che sarebbero stati più complicati da descrivere. ▲
Prossimo capitolo: L'arcobaleno
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