Charging the Earth

Category: Physical Science

Teaching Physics 201 has me digging out some of my old favorite concept-y problems. Nothing dramatic in the mathematics, but at the 201 level you can't even assume knowledge of derivatives. But you can try to catch their minds with interesting examples. Here's a classic one:

You've got the earth and the moon. They have mass and so they attract each other with gravity. Both the earth and the moon are pretty large, and so the attraction is considerable. On the scale of earthbound undergraduate lab equipment however, gravity from anything but the earth is pretty hard to measure. Pretty much impossible, in fact. We can do a little better with electrical forces. You can charge a glass rod by rubbing it with fur, and observe the effect of the charge on little sheets of foil or similar. These electrical forces are very, very tiny. But even so we're able to see them, which is more than we can say for gravity.

So, if you sprinkled some extra electrons on the earth and the same number of electrons on the moon, how much charge would you have to add in order to completely cancel the gravitational attraction? Let's figure it out. Call the charge q, and we'll put the gravitational force on the right and the electrical force on the left

Of course M_e of the mass of the earth, and similarly for the moon. The q² really is the product of the charge on the earth and the charge on the moon, but we'll assume they're equal for conceptual convenience. Nicely, the distance r will cancel and solving for q we get:

Plug in the appropriate numbers to find the required charge on each body. I find that the answer is 5.71 x 10¹³ coulombs. That's a pretty huge amount of charge in total. But for the earth it's only about 10^-11 coulombs per kilogram. And that itself is only sixty million or so excess electrons per kilogram. And considering each kilogram of earth contains trillions of trillions of electrons, only an absolutely tiny percentage of the total atoms in the earth would actually need to be ionized.

We often think about gravity being the strongest force in our daily experience. It isn't. Electrical forces just have the decency to cancel themselves out to very high precision. Gravity builds up on itself without limit. Which in the final analysis is a pretty lucky thing for us - it's nice having these solid planets which hold themselves together and solar systems which stay mostly stable over quite long time periods.

Still, next time lint clings to your shirt think about how amazing it is that electrical charge can produce forces strong enough to have visible effects from something so small. Gravity couldn't hold lint to your shirt in a million years.

Posted by Matt Springer at 10:00 AM • 5 Comments

Two Happenings in Physics

Category: Physical Science

Physics is a continuous thing, progressing steadily forward with only rare dramatic leaps. This is not the kind of style that makes for flashy news stories in the popular press. When there are interesting things being reported, they're usually wrong. "Faster than light" laser pulses, quantum teleportation, invisibility cloaks... if it's in the popular press it's probably not anything remotely resembling what they tell you it is. It's like asking me to report on avant-garde fashion.

But every once in a while some interesting things pop up. Today two things did. The first is the sun.

The sun is currently in one of the minimums of its 11-year sunspot cycle. Very unusually it's gone a month with no sunspots at all, and the last few months have had a much smaller number of sunspots than normal even for the low end of the cycle. Why? I have no idea. I'm not an astronomer and I'm pretty sure even the astronomers aren't sure in this case. The sun is very complicated. It's a fairly constant star as far as stars go (and it has to be in order to support life), but it's interesting to remember that in fact we live around a variable star. Not a very variable one, but a variable one nonetheless.

As far as I know no one anticipates this quiet period causing any problems. Past minimums have been associated with freezes and famines, but from looking at the graph this minimum doesn't seem to be nearly as small or as long as the old damaging minimums. Either way, unlike in science fiction even if it did cause problems humanity is not quite capable of adjusting the output of the sun just yet. It's interesting to watch nonetheless.

The other story is the demise of Bell Labs. Bell Labs was of course the research arm of the Bell telephone monopoly, and was responsible for an unbelievable number of advances and discoveries. It has something of a legendary reputation in the golden age of physics; they collected Nobel prizes by the fistfull.

Long story short, the breakup of the Bell monopoly meant that Bell had increasingly little money to spend on fundamental research. Bell Labs was split off the company into Lucent and lasted for another decade or so before finally calling it quits. Overall the increased competition in the telecommunications industry is probably a good thing for the country at large, but the loss of Bell Labs is a bitter price to pay. Cross your fingers that someday another huge company will take an interest in fundamental physics and create their own new institutes. Gates? Jobs? Heck, Carlos Slim? I've got a proposal for y'all...

Posted by Matt Springer at 10:00 AM • 9 Comments

Walking the Plank

Category: Physical Science

In honor of Physics 201 which I'm teaching this semester, I present a very elementary statics problem.

Here we have a board of uniform composition and weight W. It has length l and the supports are separated by a distance s. What are the two forces (call them A and B) on the boards?

The board isn't moving. It's just sitting there, and so if there's no acceleration there's no net forces. That gives us

Hmm. That's two unknown quantities with one equation. Not enough. Fortunately there's another equation we can use involving the torque. Torque is the angular equivalent of force. Get a wrench and turn a bolt, and you're applying a force at a distance from the bolt. That distance times the force is the torque about the bolt. If there's no angular acceleration (it's not spinning at all here), there's no net torque. There's no bolt either, but fortunately it turns out that you can measure the toque about any convenient point - some might make the problem easier than others. Here's let's pick point A as the point to measure the torque about, and we'll write down each force times its distance from A. They'll have to add to zero since there's no angular acceleration and thus no torque.

That factor of (l/2) represents the torque from the weight of the board. Since the board is uniform we can just pretend all the mass is at the center point. So now we can solve the second equation for B, and use that to find A in the first equation. I get

and

Easy, huh? Everyone has to start somewhere though, and this is a pretty good place to start for statics.

Exit question (easy): If s is too small, the board will fall over. What is the smallest value of s that will allow the board to balance in the picture and how do we know this from the equations?

Exit question (medium): If the board is not uniform but instead has a weight per length (x increasing toward the right) given by

The constants are of course picked so that the total weight is still W. What is the minimum s that allows for stability? Sadly there are no prizes other than fame in the comments, but I'm sure the thrill of success will be reward in itself!

Posted by Matt Springer at 10:00 AM • 11 Comments

The Ladder

Category: Physical Science

Fundamentally, you can start off with the Standard Model. It's not perfect, but it's a pretty good description of the particles and forces of nature especially at the mostly low energies of our soar system. Using quantum mechanics you can built up those particles into distinct nuclei, and calculate how they can fuse to form new nuclei. Tack on even a rudimentary Newtonian understanding of gravity and you're already able to describe stars from scratch.

Keep going with atoms and eventually things move out of the physics building into the chemistry building as those atoms combine to form compounds. If there's carbon involved you can keep progressing up the ladder of interaction complexity to organic chemistry, and if there's not complicated carbon molecules you can describe inorganic reactions such as those that eventually lead to geophysics, geology, meteorology, you name it.

But those complicated carbon molecules get very complicated and they form proteins. You usually have to move buildings again to this point until you find the biology building. Proteins form organelles of cells, and those cells form plants and animals. Those animals form groups and suddenly you move departments again and you end up in the ecology building. But if those animals happen to be people, they get even more complicated and the tools of analytical science begin to overload with the sheer volume of variables and you have to settle for a phenomenological description, which has its own buildings with names like psychology and sociology.

In theory though, the chain of steps from the Standard Model all the way up is unbroken. Not every step is understood, but for any given science, the one "below" and the one "above" are usually pretty well connected.

So in theory you can start with the quantum mechanics and derive the logistics of a hurricane evacuation. It might take a while.

Regardless of the chain of steps, I'm looking at the end result. Gustav is pointed right at south Louisiana where a lot of my friends are, and its end point actually passes pretty much right over College Station. My apartment's going to be occupied with some evacuee friends for a day or two and this weekend has been a little scrambled with me getting prepared. Here's what the track looks as of this writing (via Weather Underground), though of course as time passes the actual and the predicted may not look the same.

So tomorrow and the next day could conceivably not have updates to this site due to the busyness. If that happens, don't worry because I'm unlikely to be dead or anything.

Happy Labor Day (Dirty Jobs marathon on Discovery!), stay safe, and if you're not in the hurricane path now's a good time to think about planning ahead for whatever natural disaster might be in your area's future. Once it's on its way you're usually too late.

Posted by Matt Springer at 10:00 AM • 2 Comments

Sunday Function

Category: Physical Science

We can't graph here, this is bat country! Complex bat country.

...well ok, let's stop and take a look anyway. But no graph.

You've seen this jewel of mathematics. It's Euler's identity.

It comes from the more general expression right below this paragraph, which is today's Sunday Function. You might wonder where this expression comes from. It's a long story, but if you want to plow through it, I commend you.

Set θ = π, and Euler's identity pops right out. But let's set θ = π/2 instead and see what happens:

In other words, we've just figured out a new way to write the imaginary number i:

Now let's raise that to the ith power.

"Wait, what? What does it even mean to raise a complex number to a complex power? Something like 2³ is easy, it means 2*2*2 = 8. Fractional and negative exponents are a little weirder since they involve roots and division, but conceptually they're not that much of a stretch. But an imaginary number to an imaginary power? That's crazy talk!"

Don't worry. Just follow the rules and be consistent, and even this problem will fall to your skill. Write iⁱ, but for the lower i use that expression we just derived:

Now invoke one or two high school exponent rules:

And as the last step we can rewrite that term on the right:

And that, dear friends, amazes me even more than Euler's identity. It is almost too numinous for words.

I can tell that you mathematicians are about to fire up your keyboards and note that my exponential expression for i is not unique due to the fact that I have failed to pin down the possibility of a 2πn phase in the exponent. True, but we can worry about that later. For the moment it's enough just to be awed by the power of writing a complex number as an exponential.

Posted by Matt Springer at 10:00 AM • 8 Comments

Real Science

Category: Physical Science

First, a Public Service Announcement: As a decade-long former south Louisiana resident who was in Baton Rouge for Katrina, I have some advice. If you're in Louisiana anywhere south of about Alexandria, now's the time to start packing. You might be ok sticking around till Saturday night or possibly Sunday morning to see if it turns, but that's really pushing it. If you're actually in New Orleans you should leave now regardless. Now to our regularly scheduled post.

Science. On TV it goes something like this: Scientist gets brilliant idea. Scientist goes to lab, puts serious expression on and conducts voodoo-looking ritual involving the lab equipment du jour in a clean and gorgeous lab. Ten-minute experiment provides expected result, confirming brilliant idea. Accolades all around. It's the CSI model of experimental science.

More typical is what we did last Thursday in my professor's lab. The first step starts off the same - you need a clever idea. They don't generally come from flashes of brilliance as much as they do hard work and a lot of thought, but scientists are in the thinking of ideas business and they are pretty good at it. Long story short, this idea involved putting laser pulses through a fluorescent Rhodamine 6G solution. So we set up our optical fish tank (no kidding), poured some solution in it, and fired the laser through it.

One thing I'll say about laser physics is that the experiments themselves are often very strikingly pretty. I'm going to have to invest in a good camera so I can take some nice pictures and post them here. In this case you have a bright green line running through the solution. Self-focusing effects, thermal defocusing effects, and the motion of the fluid combine to produce an undulating filamentary structure of light which is quite otherworldly.

"It's very pretty, but I'm not sure we can do anything with it", said the professor. But the point of the experiment is to find out. So as per the clever idea, we fiddled with some characteristics of the pulsed beam and looked for the solution to behave as we thought it should. One of the two effects predicted worked. The other didn't. We changed the pulse characteristics. We changed the path length through the dye. We agitated the dye. We got out the spectrometer and measured the characteristics of the beam in the dye - there were two distinct peaks when there should have been one. Or at least there were sometimes - they appeared and disappeared in a way that didn't make a lot of sense.

We fought with the process for hours, and arguably ended up knowing less about the physics of the situation than when we started. And later on we'll probably do the whole thing again. And we'll keep on doing it and thinking about it until we understand it. That's real science.

Now it's Saturday, so let's have a few random items to start the weekend.

A reader emails about a new site called The Graduate Junction, which aims to connect graduate students in collaborative research efforts across many disciplines. Take a look, see what you think.

You probably all read Cocktail Party Physics regularly anyway (and you should!), but this post about Witchblade and the physics of pool is especially neat. The slow-motion video at the end is fantastic.

T-Rex on the wonders of modern science. That pretty awesome "The Future" that all those people in the 50s were talking about? We live in it. We just usually don't notice.

Have a great weekend, ladies and gentlemen! Tomorrow, a Sunday Function with a result even more bind-bending than the Euler identity.

Posted by Matt Springer at 10:00 AM • 6 Comments

Sarah Palin and Parity Invariance

Category: Politics

I really loathe politics, and have mostly tried to avoid writing about it. But finally something interesting has happened, and it's worth a brief comment.

In physics we like to talk about symmetry. Conservation laws and symmetry are intimately related, and you can learn a lot about one by studying the other. One of the more interesting examples of this is parity. Parity basically means reversing left and right - if you do an experiment and then rebuild the entire thing backwards, everything happens as you'd expect. If you take a picture of a physical process and flip the picture in Photoshop, the resulting picture is still a picture of a perfectly valid physical situation. But it wasn't too long ago that it was discovered that certain subatomic interactions in fact are not symmetric under parity. This was a huge deal, and resulted in a small revolution in our understanding of particle physics.

There's sort of an equivalent symmetry breaking in politics. Take two otherwise precisely identical candidates, except make one a man and one a woman. Their political fortunes will not be the same - politics is not invariant under sex-reversal symmetry. And the consequences of this symmetry breaking are why McCain chose Sarah Palin.

Here's my reasoning on why it will probably help him much more than any of the standard choices like Romney, Pawlenty, Ridge, etc.

1. It will appeal to women - especially to the all-important undecided/moderate women who supported Hillary out of solidarity.
2. It will appeal to men - let's face it, being attractive doesn't hurt. Could a man shaped like Taft ever be nominated today regardless of his political views? I doubt it and it's a shame. But political reality is political reality.
3. It will appeal to disaffected conservatives - McCain was never popular with the conservative base. Sarah Palin is considerably more popular among movement conservatives.
4. It reduces the "Let's make history" appeal of Obama - now electing the Republican ticket will produce a demographic first as well.

Does she have disadvantages, like lack of experience? Absolutely. Do those disadvantages outweigh the above? Not a chance. We have to face the facts: everyone who cares about issues and qualifications has already made up their minds one way or another. VP picks are about pulling in the rest, and Palin will do that in spades.

I'm Matt Springer, and I approve you thinking that my analysis is completely lacking in data. It's pure guesswork. We'll find out how accurate it is soon enough.

Posted by Matt Springer at 1:42 PM • 28 Comments

Greatest Physicists Nominations!

Category: Physical Science

Top 10 lists are silly. But they're fun, which is why there's so many of them. In a week or two, I'm going to start a brief biographical series with a little bit of information on the lives and works of the great physicists.

The top 3 are obvious (Well, to me anyway). The top 5 - I think I have a decent idea what my opinions are. The top 10? Things start to get kind of fuzzy. There's a few dozen people who could make a pretty good case for being considered among the truly great physicists. But these lists are as much about who's not in them as who is. And heck, maybe we'll even make it a top 20 list for the educational/entertainment value of having a few more cool guys and gals of physics to talk about.

I'm not ranking according to raw intelligence, but mainly by the importance of their ideas and discoveries to the advancement of our understanding of the universe. And while this won't make much difference, I'm disqualifying those people who are better known as pure mathematicians - so someone like Gauss won't be on the list even though he was one of the smartest people of all time and made several notable contributions to physics. Mathematical physicists are fine.

So, time for your suggestions! Let's have your top 10 physicists, or even just unordered suggestions for who should be on my list.

Posted by Matt Springer at 10:00 AM • 41 Comments

Falling from Heaven

Category: Physical Science

Nine dayes they fell; confounded Chaos roard,
And felt tenfold confusion in thir fall
Through his wilde Anarchie, so huge a rout
Incumberd him with ruin: Hell at last
Yawning receavd them whole, and on them clos'd,
Hell thir fit habitation fraught with fire
Unquenchable, the house of woe and paine.
- Milton, Paradise Lost

The biblical description of Satan's fall is sparse. Like lightning, says Luke's gospel. Milton's poetry fleshes this out with his typically beautiful and dramatic imagery. A fall of nine days into Hell.

Maybe it's just me, but the physicist in me is immediately curious at this kind of particularly physical literary license. If the fall took nine days, how far into space was he when he started falling?

A long way. Way too long to just assume he's accelerating at 9.8 m/s^2 the whole way. After all, the earth isn't all that big in the grand scheme of things, and you don't have to go too far before the gravity and thus the acceleration gets much weaker. Satan is going to spend a good part of those nine days accelerating very slowly at the beginning of his fall before quickly picking up speed toward the end.

Falling bodies obey Newton's laws, and so Satan's mass m times his acceleration will be equal to the gravitational force at his distance x from earth which itself has mass M:

Now here's where we hit trouble. The mass m cancels, but after that we're faced with a second order nonlinear differential equation. It's autonomous, which means we have a definite chain rule solution method that's available. But the resulting integrals are ugly. Really ugly. My guess would be they're not solvable in closed form, or at least not usefully so. Typical, for a problem involving the devil.

But it wasn't so long after Milton's time that Newton, Euler, and others developed numerical methods for handling these sorts of equations approximately. They were still a pain in the neck to do by hand, but today we automate those procedures by computer. I've done this for our particular equation, assuming an initial velocity of 0, and adjusting the starting position so that the falling object has dropped down to the radius of the earth after nine days - which is 777,600 seconds. Here's a plot, with the upper curve representing the distance from the center of the earth in meters as a function of time, and the line just above the x-axis is the radius of the earth for reference. When they touch, the fall is complete.

How high up did he start? By my math (I encourage you to recheck it!), the initial distance was 5.801 x 10⁸ meters. That's somewhat higher than the orbit of the moon. This makes sense: Apollo 11 took around 3 days to reach the moon. Since fall time increases very rapidly at larger distances, this seems fairly reasonable as a rough estimate. You can't tell at this scale, but if you zoom in on the edge of the graph right as the falling object approaches the earth's surface and measure the slope, you'll see that it's about 11,000 m/s. This is very nearly the escape velocity of earth. No matter how far away from earth you start your fall, you can't hit the ground faster than that under the influence of gravity alone.

I doubt Milton had any of this in mind. Poetry is not built to be beaten over the head with a differential equation. If my opinion is worth anything though, I think classic literature is improved by a little physics.

Posted by Matt Springer at 10:00 AM • 29 Comments