Built on Facts

Here’s a final scheduled post before I get back home later today. Spammers haven’t overrun my comments, I trust? Either way, consider this an open post to discuss whatever you wish if you’d like!

Until tomorrow, let’s contemplate the probability density for the Schroedinger solution of a bouncing ball in a fairly low excited state and compare it to the probability density for a classical bouncing ball, both normalized. Pretty, no?

I’m out of town for the weekend, and though I am not around to update this site it’s smart enough to update itself. Well, anyway it’s smart enough to schedule this post for Saturday morning when I’m actually typing this on Thursday night. I won’t exactly be in an internet-free wilderness, but I’ll probably be too busy to get online. It’s my significant-other’s sister’s wedding in Atlanta! Fun times will be had by all, I’m sure. I’m not exactly a wedding connoisseur, but I am a wedding reception connoisseur. Free food and free cake are something I don’t usually turn down.

So while the site auto-posts in zombie mode as it waits for me to get back, here’s a physics experiment at the University of Queensland which has been people have been waiting on for a long time. It’s the longest running of any physics experiment. It goes like this: different liquids have different thicknesses. Viscosity, we call it. Water isn’t very viscous; maple syrup is much more so. Pitch is much more so. It’s so thick it seems solid at room temperature - but it isn’t quite. In 1930 some scientists put some pitch in a funnel and let it start dripping. In the last 70 years, it’s dripped a total of 8 times. The most recent of which happened in the year 2000. At the current pace, we’re not too many years away from the next drop falling.

Have a great weekend!

See that equation there? It’s the unsung hero of Maxwell’s equations. It says in English, “This is where I would tell you how magnetic fields are generated from magnetic charges, but there aren’t any such thing as magnetic charges.”

In its integral form as above, it’s usually the second of Maxwell’s equations that my students see in their intro E&M class, after Gauss’ Law and before Ampere’s Law. It doesn’t get a lot of press. In fact it doesn’t really even have a name, unlike the other four equations. In a way that’s understandable. Rarely is a problem solved by suddenly remembering that there are no magnetic charges. But it is a fact, and it’s an important one, and it’s one that I make sure my students know.

The professor I’m teaching for this semester takes a different tack - one that’s legitimate, but not one I’m entirely comfortable with. He begins the subject of magnetism by talking about permanent magnets. Obvious enough, but his mathematical treatment of them involves a fictitious magnetic charge on the pole ends of the magnet. Which is to say, he treats the dipole field of a bar magnet as literally produced by “north charge” and “south charge” distributed on the surfaces of the ends of the magnets.

It works, and it’s mathematically justifiable as a calculation device. You want to figure out how magnetic forces from a permanent magnet scale with distance, and this method will serve you well. Everyone is happy. The problem is that it’s wrong. Magnetic monopoles: there ain’t no such animal. This is probably why our textbook treats magnetism from the other way around. It spends a chapter talking about magnetic fields and what they do to moving charged particles. Then it spends another chapter on what makes magnetic fields, which focuses almost exclusively on current-carrying wires. Finally it spends a brief few pages with a purely qualitative discussion of paramagnetism, diamagnetism, and ferromagnetism. That’s the way I’d prefer to do it. If you’re going to include the artificial monopole approximation, do it only after the basic concepts are straight. Certainly don’t lead off with it.

Oh well.  I guess this is why we have recitation instructors, so the professors’ lectures can be translated into something that doesn’t seem quite so arcane to new students.

I gave this quiz to my Physics 208 class this Monday, modified slightly from one of the textbook homework problems. I was going to work it out here, but I think I’ll leave the solution as a challenge for you.

A straight conducting wire of mass M and length L is placed on a frictionless inclined plane tilted at an angle θ from the horizontal. There is a uniform, vertical magnetic field B at all points. To keep the wire from sliding down the incline, a voltage source is attached to the ends of the wire. When just the right amount of current flows through the wire, the wire remains at rest. Determine the magnitude and direction of the current in the wire that will cause the wire to remain at rest.

Force on a straight current-carrying wire in a magnetic field:

Where L is the length of the wire in the direction of current flow. I realize “direction” is a little ambiguous since I haven’t included a diagram, so don’t worry too much about that part of the problem. Hints below:

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Inside Higher Ed has an article about graduate students at the University of South Carolina.  Seems they don’t make very much money for their work.  By not very much, I mean it’s about the same as the average income… in Brazil.

Graduate students at South Carolina make an average annual stipend of $9,590, a sum that students say is insufficient to meet the rising costs of tuition (for those without waivers), living costs, and university-mandated health-care coverage.

Nine thousand dollars, and some have to pay their own tuition?  Good grief, that’s pretty much just working for free.  You can’t even live in a cardboard box on that salary.  Maybe there’s more to the story.

University officials say they want to be competitive with regional peers, including Virginia Tech. But Virginia Tech’s current average stipend of $18,000 nearly doubles that paid by South Carolina. The discrepancy, however, could be in part attributable to higher stipends typically paid to graduate students in the heavily-represented science fields at Virginia Tech.

Ah, there’s a little bit more to the story.  The difficulty of living as a graduate student varies heavily on what you’re studying.  Take at the law school model, for instance: you don’t get paid at all, and tuition is very expensive and not waived.  But the upside to that is that you’re not in school very long, you can live comfortably on loans, and once out you can probably get a high-paying job which can pay down your debt fairly quickly.  So lack of pay is not in and of itself the problem.

Where things can get trickier is in some areas (like physics!) where you can be in school for a pretty long time and your job prospects are not necessarily very high right out of the gate.  Six years for a physics Ph.D. is not at all atypical, and while professors usually make pretty good money, the job market is quite tight and a newly minted Ph.D. can easily spend even more years in a low-paying postdoc job without even the guarantee of a tenure-track position at the end.  There’s always the national labs and industry, so a physics Ph.D. can usually bail out of the academic world and find a well-paying job elsewhere in need be.  But a lot of physicists find that unappealing and so they’re willing to take the risk and low pay of the hunt for a tenured position.

In total though, the long duration of science (and physics especially) graduate education combined with the uncertainty of employment afterwards translates into salaries a lot better than the nine thousand dollars above.  The graduate schools that I received offers from generally offered stipends a little north of $20,000 a year, with tuition waived.  I think this is almost universal among physics schools.  It’s probably fairly usual in other majors as well:

Several stipend levels at South Carolina certainly exceed the average cited by graduate students. In chemical engineering and biomedical science, for instance, the university pays graduate students $22,000 a year, more than twice the university average.

You won’t be rich, but at least you won’t be living under an overpass somewhere.

Are there any majors out there with the low $9k average salary above that don’t have quick graduations and good post-graduation jobs (like law)?  I would not be surprised.  I hope the University of South Carolina and other places can find the funds to make sure that doesn’t happen as much.

Lots of times we might have a complicated function which we wish were less complicated. Take the differential equation describing the simple pendulum, for instance:

I can’t solve that one explicitly. Nobody can - the sin term prevents a closed-form solution and so nearly the only practical way to work with it is to use numerical methods. There’s nothing wrong with that, but it’s not necessarily what we’d prefer. A stack of floating point numbers is great but it doesn’t necessarily breed understanding. An approximation might be helpful. For small angles,

Take a look at their graphs. Near the origin they get closer and closer to being the same thing:

So we can just put θ in the differential equation where we used to have sin(θ) and solve it easily. The result will only be valid for small displacements, but most of time a pendulum isn’t swinging at very high angles anyway.

This is just one example of a power series approximation. Basically you assume the function can be written as

And you do some calculus voodoo to find the constants. Then you just keep the first few terms depending on how accurate you want your approximation to be. Talking about this in more detail will be a good topic for a future post, but for now I just want to show off one of my favorite pathological functions.

It’s continuous everywhere except the origin, and even there it’s only a point discontinuity so we can define it to be equal to its limit, which is 0 at that point. That means the function is continuous everywhere, and differentiable everywhere. Its derivatives of all orders are continuous as well.

You can do the calculations to figure out the power series yourself, but here’s the result: its first derivative at the origin is 0. So is its second derivative. So is its third derivative. In fact, all of its derivatives of all orders are 0 at the origin. It’s power series thus seems to be 0, though the function itself is clearly not the zero function.

Why is this? Well, it turns out that power series are intimately connected to the behavior of the function in the complex plane. Though our function is perfectly well-behaved at 0 on the real line, it is horribly behaved at 0 in the complex plane. It has an essential singularity at that point; the limit as you approach 0 depends on how you do the approaching. For example, f(0.1) = 3.7 x 10^-44, but f(0.1 i) = 2.7 x 10⁴³. Clearly they’re not approaching the same thing. In fact, Picard’s great theorem guarantees that the limit will approach any given value depending on which direction you approach from. As you might expect, this wrecks the possibility of expanding the function in a power series about the point of singularity.

When you next do a power series approximation, spend a little time being thankful that most functions in physics aren’t so badly behaved.

The Large Hadron Collider is scheduled to come online in a matter of months.  For all the thousands of theorists working in high-energy physics, SUSY, string theory, and related topics, no one really knows what kinds of new particles are going to be coming out of the collision point.  This, of course, is the entire point of doing the experiment!  It’s the entire point of every physics experiment - we don’t know for sure what nature is going to do in a given situation and so we find out.  If it does what we expect then we have an additional measure of confidence in the theory describing that situation, and if not then we have new data to begin working on new improvements or replacements to current theory.  There are some cases - like the LHC - where theory isn’t well-developed enough to even make solid predictions as to what we expect and so the outcome will be at least a partial suprise regardless of what happens.

But we have some ideas.  The big one is the Higgs boson.  The standard model solidly predicts that it exists, and sets a range of possible mass values that’s well within the energy range testable by the LHC.  If it exists in the way we expect it to, the LHC should find it without trouble.  Beyond that, whatever else is observed in the particle detectors at the collision point is expected to include a tremendous number of unexpected and interesting things.  Some of my friends working on SUSY models are hopeful that evidence of supersymmetric particles will be found.  String theorists (and the string theory popular press!) are hopeful that evidence of strings will be found.

So what’s actually going to happen?  I’m a physics grad student with zero experience in particle physics, and furthermore I’ve never even taken a high-energy physics class and so everything I know about it I’ve picked up from osmosis.  But that’s not going to stop me from making some wild guesses!  In fact, my friends and I have something of an unofficial pool as to what mass the Higgs is and what else will be found.  I’m angling for the long-shot dramatic win: the Higgs boson won’t be found at all, and neither will evidence for SUSY or string theory.  But what we will find will be amazing, exciting, and totally unexpected.

I’m probably totally wrong (except for the amazing and exciting part!), but I always did support the underdogs.

Any guesses from the readers?

How tall are you? You might give an answer in feet and inches if you’re an American, or in meters most other places. And if you give than answer, I’ll have a good understanding of how tall you are - all because we’re using the same units of measurement. In practice your ruler might be a little different from mine because of slight manufacturing defects, but your meter is probably essentially the same as mine.

Scientists have to be more precise sometimes. Precision tests of (for instance) relativity and QED require many decimal places of accuracy and so well-defined units have to be available. For instance, the meter used to be the distance between two fine scratches on a particular bar in a government vault. This wasn’t very practical because the scratches weren’t infinitely thin, and a bar can’t be kept at a perfectly constant length. So the meter was redefined as the distance that light travels in a particular tiny fraction of a second. The second was defined as some large number times the period of a particular type of light emitted by cesium atoms. So on and so forth for the other units - instead of physical objects, they were defined in terms of well-understood and mostly easily-reproducible physical phenomena.

The kilogram is the one exception. Right now, one kilogram is defined as the mass of a particular lump of a platinum-iridium alloy in Sèvres, France. Here’s its mug shot. It’s the metal cylinder at the center of the nested jars.

It’s not the ideal. Making comparisons with it is very difficult because it must be kept absolutely undamaged and clean, and furthermore the mass of it or its copies doesn’t seem to stay completely put. Making a real-world physical object behave with the required precision is simply very difficult. Scientists are looking to fix this by defining a kilogram that’s not tied to any specific physical object. They’d prefer a more reproducible and fixed definition such as that enjoyed by the meter. There’s a few approaches. One that’s been in the media recently is an atom-counting approach that works like this.

Make a perfect silicon sphere of a very precisely specified size. The mass of that ideal sphere is defined as 1 kilogram. That way any sphere made to the specification can be used as a kilogram. The difficulty with this is in the making of a sphere perfectly and accurately enough to be more reliable than the current kilogram in the bottle in France. The Avogadro Project is working on this, and they have developed silicon spheres which are perfectly spherical to within unbelievable tolerances - just a few dozen nanometers in global error, with local surface irregularities smaller than a nanometer.

Of all the approaches to a new definition of a kilogram, I like this one the best. Unlike the Watt balance which is intrinsically delineated in terms of electrical properties, the silicon sphere technique actually gives you a physical object which has a mass of one kilogram while still retaining an abstract definition based on pure physical constants.

You know, I wonder if there’s a correlation in preference of method between experimentalists and theorists. Theorists might prefer the Watt balance since there’s fewer steps between the fundamental constants and the definition than there are with the atom-counting approach. Experimentalists (like me) might prefer the atom-counting approach since it actually produces a one-kilogram object directly.

Either way, it will be nice to have a kilogram not based on a single object. The International Prototype Kilogram is a relic of a bygone age in science, and it really should be in a museum by now.

A few days ago I wrote about how to move in space, you needed to bring along something to push against. Dr. Pion objected. “It’s not what you push against that makes you move, it’s what pushes against you.”, if I can paraphrase.

I thought that was a silly objection. It’s true, but doesn’t everyone know that something you push against necessarily pushes back with the same force? It’s Newton’s Third Law, and equivalent to conservation of momentum besides.

But thinking about it some more and remembering teaching intro mechanics, no it’s not necessarily so obvious to a lot of people starting out in physics. And physicists ought to be precise with their language to avoid conceptual mistakes. Consider the following scenario: you have a truck pulling a trailer. No matter how hard the truck pulls on the trailer, the trailer pulls equally hard back on the truck due to Newton’s Third Law. How is it that the truck ever gets anywhere? This is a suprisingly confusing question to many students in their first mechanics class.

Look at the truck. There’s two forces acting on it. The road is pushing the truck tires and the trailer is pulling backwards on the truck. The first force is larger and thus the truck accelerates forward. Same thing for the trailer; there’s one force acting on it - the truck pulling it forward, and so the trailer accelerates forward as well. To actually calculate the forces requires knowing the masses of each vehicle, but if you know those the calculation is very simple because of the constraint that both vehicles have to have the same acceleration to remain connected.

In summary, when thinking about “equal and opposite reactions” it doesn’t matter in the slightest what you’re pushing on. What matters is what’s pushing on you. The fact that those forces happen to be the same is important but not the whole story.

Apollo 11 launch
July, 1969

Today is the 4th of July, the day the United States of America celebrates its independence. It’s a unique and beautiful country, and I’m glad I had the good fortune to be part of it.

For much of its history the country was a bit of a scientific backwater, contributing modestly to the progress of human knowledge but rarely in groundbreaking ways. The first few decades of the Nobel Prize in physics include only a few Americans. The Second World War shattered the preexisting system, and due largely to massive research efforts during the war and the emigration of scientists from Europe to America, physics in the US began its rise. The post-war economic boom and the looming threat of Soviet scientific advances kept both interest and funding for physics at a fever pitch.

The results began building on each other and the Nobel Prizes kept piling up. Feynman, Gell-Mann, Weinberg, Lamb, Mather and Smoot, Bardeen, Cooper, and Schrieffer… Today the US and its unparalleled university system keeps growing its own brilliant scientists while also serving as a place for scientists and graduate students of every nationality to study and research their own fields.

The world is a big place, and love of country in no way implies lack of respect and admiration for the others. So on this, my country’s birthday, I think it’s worth looking back with pride and hopefulness that the land of the free and the home of the brave will always continue to be thus.

Might I suggest two ways to celebrate the 4th with physics? If it’s legal in your area, the dry ice bomb is a very cool, very cheap, and VERY loud demonstration of both phase transitions, gas pressure, and the propagation of compression waves through air. Be careful! It’s less dangerous than many seemingly mundane fireworks, but there’s still substantial risk involved. A less dangerous and less noisy but prettier effect involves powdered non-dairy creamer. Get a big bottle (about $2 at Wal-Mart or similar), gently shake a thin cloud of the powder over a flame, and enjoy the fireball. Done outside away from flammable things, this is actually a lot safer than it looks. In my experience at worst you’ll singe arm hair, but I make no guarantees and gently remind you that you are responsible for your own safety and well being. It’s an interesting exercise in chemical thermodynamics, and the vast difference a large surface area to volume ratio can make in the behavior of a substance. Here’s a friend of mine (I hope he doesn’t mind!) doing this trick a few years ago. Don’t worry, the fireball only lasts about one second.

Have a happy 4th!