Feeds:
Posts

## The photoelectric troll (all that and a bag of chips!)

Somewhere in the wilderness of Toelek is a store.  Inside the store are clerks.  The clerks sell bags of Wavy Lay’s potato chips:

Now, the thing is, if you give money to clerks to buy some chips, the clerks never give any money back: they’re greedy.  What’s more, they leave the store immediately with whatever money they have left over from the transaction.  However, the clerks don’t always get very far, because out behind the store is a bridge that the clerks have to cross, and the bridge is guarded by a troll named Voltar.  The troll is greedy, too.  He demands a toll, and if a clerk can’t pay up then he can’t cross the bridge.

Toelek is a weird place.  Its citizens have a very rigid society, and people are required to wear differently colored clothing depending upon how much money they have in their pocket.  The money is always in multiple of 50 cents.  For example, if you carry \$0.50 then you must wear red; if you carry \$1.00 you must wear orange; if you carry \$1.50 you must wear yellow.

Look, over there—I see a line of people approaching the store.  They’re all wearing red.  They enter the store…but I don’t see any clerks exiting out the back, and consequently no clerks cross the bridge.  More and more reds go into the store, and a faster and faster rate, but it doesn’t matter.  There are no clerks coming out.  The troll doesn’t get any business.

I conclude that a bag of chips costs more than \$0.50.

Later, I see a line of oranges go into the store.  I now observe some clerks coming out, but none of these can cross the bridge.  I conclude that chips cost \$1.00, but since the clerks have no money left over, they can’t pay the troll’s toll.

Even later, I see a line of yellows go into the store.  Clerks are coming out, and these can cross the bridge.  The troll must be demanding a toll of \$0.50 or less.

There are many quantities which are important in analyzing this situation: the amount of money a person has before entering the store (use E for entering), the price of a bag of Wavy Lay’s potato chips (let’s call this price W), the toll that Voltar the troll demands (let’s call this V), and the amount of money a clerk has, K (they speak Dutch in Toelek, so the clerks are called klerks) upon exiting the store.  It should be obvious that

K = E – W,

since the amount of money a clerk has upon leaving the store is just the amount of money a person has upon entering the store minus the cost of some chips.  Additionally, for a clerk to cross the bridge, it must be true that

E – W ≥ V

so that the clerk has enough to pay the toll.  If the clerk barely makes it, this inequality is an equality and

E – W = V.

The V at which this happens (for a given E and W) is called the cutoff toll Vo;  if Voltar were to increase the toll by any amount at all, the clerks wouldn’t get to cross the bridge.

It’s interesting to graph the cutoff toll Vo vs. money that customers have upon entering the store E.  You get something like this:

Notice that the cutoff value of E is \$1.00, which is the price of a bag of chips.  At or below this value the troll need not charge any toll at all, since no clerk will have any money to pay him.  That is, when V = Vo = 0,  then E = W.

WHAT’S GOING ON?

The physicists reading this blog have already guessed the game I’m playing: I have presented an analogy for Einstein’s explanation of the photoelectric effect (hence Toelek, from fotoelektrisch).  Make the following transformations:

Customers = photons;

Color of customer = frequency of photon;

Money = energy;

Store = photoelectric material;

Price of chips = work function;

Clerks = electrons;

Clerk’s money = kinetic energy;

Bridge = potential difference;

Voltar’s toll = kinetic energy required to jump the gap.

With these transformations, you can re-write the story as follows:

There is a photoelectric material, a metal such as platinum.  Inside the metal are electrons.  The electrons can be liberated if enough energy is added.

Now, the thing is, if you give energy to electrons to liberate them, the electrons don’t give any energy back: they’re greedy.  What’s more, they leave the metal immediately with whatever energy they have left over from the transaction.  However, the electrons don’t always get very far, because out behind the metal is a potential difference that the electrons have to cross in order for current to be observed.  Jumping this gap requires a certain amount of kinetic energy, without which the electrons don’t produce current.

How do we add energy to the metal?  Well, by shining light on it.  Light energy is quantized, in chunks called photons.  The energy of a single photon is proportional to its frequency, by Einstein’s formula E = hf.

Look, over there—I see some red light (E = 1.9 eV) approaching the metal.  Unfortunately, no electrons exit out the back, and consequently there is no current.  More and more red photons hit the metal, and a faster and faster rate, but it doesn’t matter.  There are no electrons coming out.

I conclude that the amount of energy need to liberate an electron (called the work function) is greater than 1.9 eV.

Later, I see orange light (E = 2.1 eV) go into the metal.  I now observe some electrons coming out, but none of these produce current.  I conclude that the work function is W = 2.1 eV.

Even later, I see yellow light (E = 2.18 eV) go in.  Electrons are coming out, and these can cross the potential difference.  The potential difference must be 0.08 volts or less.

There are many quantities which are important in analyzing this situation: the amount of energy a photon has before entering the metal (E), the work function (W), the voltage that electrons have to jump (V), and the amount of energy an electron has upon exiting the metal, K.  It should be obvious that

K = hf – W,

since the amount of energy an electron has upon exiting the metal is just the amount of energy a photon has upon entering the metal minus the cost of liberating an electron.  Additionally, for an electron (with charge e) to jump the gap, it must be true that

hf – W ≥ eV

so that an electron has enough kinetic energy to overcome the potential difference.  If the electron barely makes it, this inequality is an equality and

hf – W = eV.

The V at which this happens (for a given f and W) is called the stopping potential Vo.

It’s interesting to graph stopping potential Vo vs. energy of incoming photons E.  You get something like this:

Notice that the cutoff value of E is 2.1 eV, which is W.  At or below this value there need not be any potential difference at all, since no electron will be liberated.  That is, when V = Vo = 0,  then E = hf= W.

I hope you find this analogy useful.  As for me, I need to go to the store: all this talk of potato chips has made me hungry.

### 12 Responses

1. Brilliant! Please tell me you’re a physics teacher…

• Assistant professor of physics, Western Carolina University…

• May you proceed to full Professorship! You have an excellent way of explaining it, Mathew. Animating electrons as “greedy” worked brilliantly for my non-mathematical mind.

2. Sure…

• You’ve looked through my blog so you know I’m an atheist. I see no evidence for the gods or for the supernatural. Cosmology is gorgeous to me. I find meaning in it and am genuinely baffled why anyone wouldn’t. However, in one of my posts I teased the idea of what it would mean (for our species and, subsequently, spiritualism) if we discovered that atoms could store miniscule beads of information; an echo of the things it’d been… from a gas cloud to a bridge to part of a human hand. I’m not suggesting this information could change or alter an atom, but if such a thing were discovered it would (I’d imagine) re-cast our going notions of death. I apologise in advance if this question is utterly absurd, but in your learned opinion might atoms have such a capacity? Is it at all feasible?

• Was it that stupid? It’s ok, you can say so. You won’t hurt my feelings 🙂

• I’m not sure I follow the question. An atom only has a very small number of changeable parameters (quantum numbers) and therefore can’t hold a particularly large amount of information. However, in the spirit of your hypothetical, *if* atoms could store information then I still think it wouldn’t do us much good, since information is useless if it cannot be extracted by some physical process. For example, all the knowledge of human history can be encoded by a single cut on a 1-meter-long iron bar. To encode all of human knowledge (I am going to make an educated guess and say 100 billion terabytes of data, or 10^23 bytes) first take the knowledge and write it out as a single number between 0 and 1 (it would obviously have to have many, many decimal places, more than 10^23). Then make a cut at *exactly* that location on the iron bar. For example, if the “human knowledge number” starts 0.658399344901… then the cut is made at about the 65.8-cm mark. Now there are several problems with this; for one thing, we can’t measure such a location that precisely; it would require us to be able to measure things immensely smaller than a quark. Secondly, who knows if space itself isn’t quantized? If so then such a cut would be impossible. Thirdly, extracting the information back from the cut would run us up against the wall of Heisenberg’s uncertainty principle. So such a cut is basically impossible, and our dream of limitless knowledge is but a dream…

• Thanks Matthew. I guess what I was hitting at was seeing if there could be any tangible/physical evidence for something akin to reincarnation. Here I don’t mean the elaborate transfer of consciousness or even thoughts, rather the persistence of something as simple as a positive or a negative record. For example, a carbon atom plays a part in, say, a human body for a few months before continuing on its merry way. Our understanding is that it has not changed in any way in that time: it’s still just an atom of carbon and will remain that way until (if the conditions ever occur) it fused into oxygen. But is this absolutely correct? Can we ever be entirely confident that nothing of those few months (being part of a human, or anything else for that matter) is not somehow etched into the atom?

Now your point about information retrieval was interesting. If information can’t be accessed it’s useless, and that would pretty much answer my question. You did however mention that there is a capacity (quantum numbers) for something to be stored, and if this is at least ‘possible’ (albeit highly unlikely) then we can’t necessarily rule out retrieval at plank lengths which we can’t consciously ever hope to perceive.

I apologise for taking your time up on my mental ramblings. Religion fascinates and infuriates me at the same time. I feel genuine sympathy for the average theist because in the end they are mostly simply frightened individuals, and it’d be nice to do something to alleviate that fear. The current drug of choice (dogma) isn’t healthy, hence my interest in this subject.

Thanks so much. You have a beautiful mind.

3. Awesome metaphor as usual!!
Yes, I guessed – except from: What the hell is Toelek? I thought this is a character from Lord of the Rings I might have missed 😉

• … guessed it … (Why doesn’t WP let us edit typos?)

4. Toelek, from fo-toelek-trisch

5. Excellent article. Hard to find a post of
the quality. Thank you