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October 6

Feynman Lectures. Lecture 27. Ch. 27–7 Resolving power [1]

Quote

It is necessary that the difference in time between the top ray and the bottom ray to the wrong focus shall exceed a certain amount, namely, approximately the period of oscillation of the light: t2−t1>1/ν,(27.17) where ν is the frequency of the light (number of oscillations per second; also speed divided by wavelength). If the distance of separation of the two points is called D, and if the opening angle of the lens is called θ, then one can demonstrate that (27.17) is exactly equivalent to the statement that D must exceed λ/nsinθ, where n is the index of refraction at P and λ is the wavelength.

I tried to derive $D>{\tfrac {\lambda }{n\sin {\theta }}}$ , but I got another answer. According to image PNG dwg we can write:
$c't_{0}-c't_{1}\approx PP_{1}=D\sin {\theta };$
$c't_{2}-c't_{0}\approx PP_{2}=D\sin {\theta };$
Where $c'=c/n$ - speed of light in medium. Adding both equations:
$c't_{0}-c't_{1}+c't_{2}-c't_{0}=2D\sin {\theta };$

$t_{2}-t_{1}={\tfrac {2D\sin {\theta }}{c'}}={\tfrac {2Dn\sin {\theta }}{c}};$
But $t_{2}-t_{1}>1/\nu ={\tfrac {\lambda }{c}}$ .

${\tfrac {2Dn\sin {\theta }}{c}}>{\tfrac {\lambda }{c}};$

$D>{\tfrac {\lambda }{2n\sin {\theta }}}.$
Where is my mistake?

I'm not sure what angle does Feynman mean by "opening angle of the lens"... May it be the angle SPR (Fig. 27–9)? Username160611000000 (talk) 03:03, 7 October 2016 (UTC)[reply]

How old are beach pebbles?

Or to put it another way, how long does it take the sea to grind pieces of rock into sand? SpinningSpark 18:17, 6 October 2016 (UTC)[reply]

Per articles like Sand and Rock there are a bewildering array of materials and methods by which rock gets turned into sand. There's no reasonable way to give even the most general answer. --Jayron 32 18:27, 6 October 2016 (UTC)[reply]

Could you have provided an answer if I had asked a more specific question? Let's say the pebbles on Brighton beach. SpinningSpark 20:39, 6 October 2016 (UTC)[reply]

Some beaches, Brighton and the UK south coast generally are good examples, have a range of ages for pebbles, sorted along the length of the beach. As the coast behaves as one long straight beach with a single Eastwards direction of motion along it (although groynes try to control this), "pebbles" begin anew as boulders from cliff erosion at one end and travel along the beach, eroding as they go, right down to cobbles, pebbles and eventually sandy beaches. By tracking their age and material, a correlation of age and size can be made.

For the South coast, the chalk cliffs erode in West Sussex and the soft chalk erodes quickly to sand. Pebbles that survive as such (and travel further East) are from the harder flint found within the chalk. This still goes on slowly today, but the bulk of them still date from the rapid erosion of the area (and so accelerated boulder deposition) at the end of the ice age and glacial meltwater erosion, maybe 10,000 years ago. The production of loose flint pebbles (flint doesn't really appear as large boulders) was so rapid in this period of great erosion that the beaches are still chewing on the remnants.

Much older though, and fascinating, are the Brighton Raised Beach structures. These are archaic pebble beaches, formed about 1/4 million years ago and now some 10m above modern sea level. They can be traced the length of the coast and for some distance inland.

Another UK pebble beach of much study is of course Chesil Beach.[2]

The classic book source (I remember it from school geology) would be Ellis, Clarence (1965). Pebbles on the Beach. Faber. ISBN 0 571 06814 6. and a more modern comparable text might be Zalasiewicz, Jan (2012). The Planet in a Pebble. OUP. ISBN 978-0199645695.

Some South coast pebbles are only a few decades old. After WWII, a lot of brick and concrete appeared on beaches, from coast defences broken up in situ (accidentally or deliberately). These were often easily trackable and helped to illustrate some beach erosion processes. Andy Dingley (talk) 13:50, 7 October 2016 (UTC)[reply]

@Andy Dingley: thanks for the informative answer. I had often wondered where the pieces of brick come from. SpinningSpark 14:59, 9 October 2016 (UTC)[reply]

This [3] paper presents what looks to be an interesting model of sand bank formation, and give some information on characteristic time scales in terms of other parameters. This [4] has tons of information about weathering of silicate minerals, and discusses an apparent disconnect between laboratory and field-derived estimates. This [5] work uses everything from cosmic ray mechanics to spall mechanics to discuss erosion and denudation at large scales, and has some interesting results for different region (Figs 3,5). The final one is pretty close to addressing the question "How many years must a mountain exist, before it is washed to the sea" :) SemanticMantis (talk) 18:42, 6 October 2016 (UTC)[reply]

Thanks, but those look like they are concerned with weathering and erosion. I'm interested in the action of waves on beach pebbles. SpinningSpark 21:48, 6 October 2016 (UTC)[reply]

It's not really guaranteed it will happen at all. For example, all along the Appalachian Mountains there is a layer of Pottsville Conglomerate that retains the pebbles of the ancient beach to this day. Wnt (talk) 12:24, 7 October 2016 (UTC)[reply]

They have only survived in that stratum because the action of the sea had stopped at some point. I believe that it is the case that pebbles that are continuosly rolled by water will always have a finite life. SpinningSpark 13:29, 7 October 2016 (UTC)[reply]

Maybe you aren't interested in those papers, but erosion is a large source of beach gravel/sand, and weathering absolutely includes wave action on beaches. I will repeat for clarity that my first ref is 100% about formation of sand on beaches. SemanticMantis (talk) 13:55, 7 October 2016 (UTC)[reply]

Thanks for explaining that. I have no access to that site so could only read the abstract and was not able to tell from that whether it would answer my question. SpinningSpark 17:45, 7 October 2016 (UTC)[reply]

Note that the title question seems to be different from what you are asking:

A) Title question: "How old are beach pebbles ?" would mean how long ago did the rock which has since been ground down to pebbles first form ?

B) Your question seems to be about how long they have been "pebbles", in which case we would need to define the range of sizes classified as pebbles. StuRat (talk) 12:43, 7 October 2016 (UTC)[reply]

According to our article pebble, "A pebble is a class of rock with a particle size of 2 to 64 millimetres based on the Krumbein phi scale of sedimentology. Pebbles are generally considered larger than granules (2 to 4 millimetres diameter) and smaller than cobbles (64 to 256 millimetres diameter)." DuncanHill (talk) 13:32, 7 October 2016 (UTC)[reply]

You have overlap on the size ranges you listed for granules and pebbles, as does the article you linked to. StuRat (talk) 19:14, 7 October 2016 (UTC)[reply]

According to the Wentworth Grain Size Scale granules are a subset of pebbles. DuncanHill (talk) 19:23, 7 October 2016 (UTC)[reply]

Looks like our article needs fixing then. Care to volunteer ? StuRat (talk) 20:16, 7 October 2016 (UTC) [reply]

Yes, that is exactly the question I am asking. Are you now able to provide an answer? For the purpose of this question, pebbles are small enough for the current to be able to roll them, but large enough that the current cannot lift them off the bed altogether. SpinningSpark 14:29, 7 October 2016 (UTC)[reply]

No, I am not qualified to answer, but by clarifying the Q, hopefully others can. StuRat (talk) 19:16, 7 October 2016 (UTC)[reply]

As an amusing semi-relevant side track, one Christopher Ball from Brighton has written a book called Reverse Theory in which he propounds that pebbles are not ground smaller, but instead built up from sand by "the process of tidemark"; claims that the whole of geological dating (including fossil evidence, plate tectonics and radioactive decay measurements) is fundamentally derived from a (therefore) illusory rate of erosion from rock to sand; and that therefore (I gather) Creationism is true and humanity has devolved from a higher to an animal state. I myself do not intend to spend either time or money reading the book, and I doubt any of this is sufficiently notable fruitloopery as to merit coverage in Wikipedia. {The poster formerly known as 87.81.230.195} 90.197.27.88 (talk) 19:33, 7 October 2016 (UTC)[reply]

Trying to identify the name of a physical scaling constant

There's a certain constant usually associated with the permeability of free space having a value of $10^{-7}$ . I've never actually seen it referred to alone anywhere and yet I've encountered it enough working with physics equations that it seems deserving of its own designation. For example, its presence (or rather its inverse) in the equation $q_{\text{P}}={\sqrt {10^{7}m_{\text{P}}\ell _{\mathrm {P} }}}$ , where $q_{\text{P}}$ is the Planck charge, $m_{\text{P}}$ is the Planck mass, and $\ell _{\mathrm {P} }$ is the Planck length. Anyway, I generally think of it as a "charge scaling constant" relating to the distance between charges used to define the coulomb (being a single meter, that is) but I'd really like to know if there is a formal definition for this value. Earl of Arundel (talk) 22:46, 6 October 2016 (UTC)[reply]

The permeability of free space or "magnetic constant" is concerned with production of a magnetic field by electric current

µ₀ = 4π×10⁻⁷ N / A² ≈ 1.2566370614...×10⁻⁶ H / m or T·m / A or Wb / (A·m) or V·s / (A·m)

It appears in the constituitive form of Ampère's circuital law

\mathbf {B} =\mu _{0}\mathbf {H} \,\!

. But I don't recognize the decimal multiplier

10^{-7}

as a physical constant. Have you read anything in the article about Planck units that leads to the equation you quote for

q_{\text{P}}

? AllBestFaith (talk) 02:25, 7 October 2016 (UTC)[reply]

Yes, and it appears in other equations as well. But to answer your question, no, I arrived at that equation myself. At some point the thought just crossed my mind that the Planck charge might not be fundamental. Consider that the gravitation constant has dimensions consisting of nothing more than length, mass, and time. Moreover, the dimensions of the resulting force (the newton) in both gravitational and electric-charge calculations also involve these sole three dimensions as well. Then it occurred to me that charges in equations always seem to appear as powers which are multiples of two. And so I just squared the Planck charge and divided that by the Planck length. The result agreed well enough with the Planck mass (after multiplying by the aforementioned scaling factor of

10^{-7}

) to convince me, at least, that the formula was correct. Wouldn't be surprised if this is already a well-known fact somewhere in the scientific community, though. If not, it probably deserves further consideration. Earl of Arundel (talk) 18:25, 7 October 2016 (UTC)[reply]

It comes from the definition of the Ampere. Greglocock (talk) 08:16, 7 October 2016 (UTC)[reply]

Basically the Ampere was defined in cgs units, which have this kind of bizarre scaling. (The odd 10^2 comes from the centimeters) There is no SI unit more unwanted and chronically confusing to me than this two-wire thing - I don't understand why we can't just use an Avogadro number of electrons instead of the Coulomb, etc. i.e. Faraday constant. Wnt (talk) 14:28, 7 October 2016 (UTC)[reply]

I imagine ^{[citation needed]} that the effort to cleanse SI units from anthropocentric concepts such as "one forty-millionth of the Earth's perimeter" was the occasion to pick definitions without weird constants in them, but the ampere was allowed to stand because the force between two wires in the vacuum was comparatively "clean". Tigraan^{Click here to contact me} 15:06, 7 October 2016 (UTC)[reply]

That's not the reason for retainng the ampere, although the fact that it had an acceptable definition certainly helped. The definitions of the amp, volt and ohm were established by international conference of electrical engineers in 1881. The international nature of the telegraph had made it essential to have a common set of units. By the time the MKS and SI sytems came along this was so well established it would have been very difficult to get the industry to change. Especially so as a set of MKS electrical units without the 10⁷ factor would be entirely impratical, either much too large (the amp and ohm) or much too small (the volt) for everyday use. Pretty much the same objection would apply to using the Faraday constant. To answer the original question, I've never heard this factor given a name, I'm pretty sure it doesn't have one, it's just a number. SpinningSpark 17:27, 7 October 2016 (UTC)[reply]

Well, instead of saying a current of 20 A you might say 207 umole/s; instead of a 110 V outlet you'd have a 10.6 MJ/mole outlet; a resistance to make 110 V flow at 20 A would not be 5.5 ohms but 51.2 GJ s/mole^2, I think; the power of 110 V at 20 A would not be 2200 W but ... oh, yeah, 2200 W. That's not an electrical unit! :) A picofarad capacitor (1E-12 C/V) would give way to 1E-12 * 1.03E-5 mole / (96485 J/mole) = 1.074e-22 mole^2/J ... I think. (I really ought to clear some cobwebs from my head and recheck that math) So OK, maybe you'd need to invent a prefix or two. But on the other hand... you'd immediately know how many actual electrons you can put into the capacitor for a certain amount of actual energy. ~~(Somehow I doubt it's actually ten, like I said, I should debug this, but can't right now)~~ oh wait, those are mole^2 so if this corresponds to a physical number of electrons it's the root of that or 1e-11 mole, which is a bit bigger. Wnt (talk) 10:52, 8 October 2016 (UTC)[reply]

Odd, I was just looking at Planck units. Taking the Planck unit of charge = 1.875 545 956(41) × 10−18 C divided by 96485.33289(59) C mol−1 gets 1.943866E-23 mol, which is 11.70624 / Avogadro's number. But why is it not 1 per electron but 11.7 per electron? Oh wait, that's the square root of the fine structure constant. I feel like describing charge in moles (or numerical bundles of any other size found more convenient) is more fundamental than Planck units, oddly enough, but alpha is something mysterious that only a genuine communicant of physics truly understands. Wnt (talk) 15:58, 10 October 2016 (UTC)[reply]

Well you did set up the equation to relate planck units to moles though (and thus Avagadro's number), which is why the fine structure constant (FSC) popped up like that. What makes planck units so special is that they form a dimensional rationale for physical constants. The speed of light is planck length divided by planck time, planck's constant is planck length squared times planck mass divided by planck time, and so forth. And of course you could just multiply the ratio of feet to meters with the planck length and then you'd have an equally valid definition of the planck length with respect to the USCS system of units. A truly fundamental planck scale would simply set all units to unity, but then that would just make day-to-day calculations more tedious. Which is why the current convention of describing planck units in terms of the SI system is really the best compromise. That said, dimensionless contants like the FSC are in a sense more pure, so perhaps they could be used to develop an even more fundamental basis still. But until that happens, planck units are going to reign supreme. There just isn't anything else that comes close to being as intrinsic. Earl of Arundel (talk) 01:22, 11 October 2016 (UTC)[reply]

Right, so there is an incongruence of sorts with the units chosen to represent electrical phenomena. And while it would be nice if we could have a more homogeneous system, it probably isn't worth the trouble. That said, in lieu of that it does seem reasonable to formally define this thing. Otherwise, we're just stuck with using an ad hoc constant to define things like Coulomb's constant (

k_{\text{e}}={\text{c}}^{2}10^{-7}

). At best, it's arcane..at the very worst, confusing. Earl of Arundel (talk) 18:25, 7 October 2016 (UTC)[reply]

Appalachian endorheic basins

As far as I can tell, there aren't any endorheic basins in the Appalachians. Why not? Or am I wrong, in which case, where are these Appalachian basins located? Nyttend (talk) 23:33, 6 October 2016 (UTC)[reply]

As can be told by looking at maps, or reading the Appalachian Mountains article, the Appalchians are low, old, and narrow. There does not appear to be any major geologic barriers to water landing in the Appalachians from reaching the ocean, the same way that there is at, say, the Basin and Range Province, which has numerous such basins due to the specific type of faulting in the region. Such faults do not exist in the Appalachians, which are mostly low rounded hills without dramatic elevation changes or significant barriers to water flow; water always has a way out. --Jayron 32 01:28, 7 October 2016 (UTC)[reply]

The thing I wonder about, in particular, is the Ridge and Valley Province. Here, you have lots of long mountains with low valleys between them, and in most cases, each mountain and each valley will be only a few miles wide. Some mountains even curve significantly, as can be seen in File:Bedford-co-air.jpg. But every valley has an outlet: very few of them lack a "normal" connection to other valleys (there don't seem to be any valleys that are blocked up at both ends), and the few exceptions, e.g. 37°5′53″N 81°20′32″W / 37.09806°N 81.34222°W all seem to have water gaps to adjacent valleys. It's not as if I'm looking for something on the scale of the Great Basin or even Devils Lake (North Dakota); something a mile wide and a few miles long would suffice. Nyttend (talk) 20:07, 7 October 2016 (UTC)[reply]

The water gap is one form of erosion I described below. That is, some water will almost always flow out, even if only drips. But, over millions of years, those drips erode a larger channel and eventually you have a water gap. StuRat (talk) 20:14, 7 October 2016 (UTC)[reply]

Two reasons, both dealing with age of the land:

1) Sedimentation, filling in any basins. Windblown particles may also settle out in the bottom of the basin.

2) Land erosion, wearing down the walls of the basins (some of which becomes sediment within the basins).

So, to have basins, you need active geological processes, such as volcanism, glaciation and tectonic uplift, to counter the above forces. StuRat (talk) 20:01, 7 October 2016 (UTC)[reply]

Appalachian geology is remarkably regular, but not perfectly so. There are weird formations like Fort Valley, Maryland where you see a ring of hills around a central region. But there are ways the water goes out. This is illustrated by Delaware Water Gap, Pennsylvania where a stream has been flowing steadily since, I assume, before the hills were first eroded out of the earth. (in other cases, like the nearby Wind Gap, Pennsylvania, you see that the stream stopped at some point and the pass gradually went up into the air as the earth eroded all around it) I don't know, but I suspect that in such a moist environment the water was always flowing and so every basin that came into being has a built-in back door for the water to get out. I always think of endorheic basins being in dry areas and I wonder if this is a reason for it? Wnt (talk) 18:58, 8 October 2016 (UTC)[reply]