The Physics of Finding Osama bin Laden (As Mis-Reported on NPR)

Over in Scientopia, Janet notes an interesting mis-statement from NPR, where Dina Temple-Raston said of the now-dead terrorist:

[O]ne intelligence officials told us that nothing with an electron actually passed close to him, which in a way is one of the ways they actually caught him.

As Janet notes, this would be quite a feat, given that electrons are a key component of ordinary matter. But for the sake of silly physics blogging, let's take this seriously for a moment. Suppose that Osama bin Laden really could make himself utterly devoid of electrons: would that be a good way to hide?

To answer this, let's think about some of the physics involved. If bin Laden were totally electron-free, that give him a large positive charge, of one electron charge unit per proton in his body. These positive charges would attract the negative charges in nearby matter, and repel the positive charges. The resulting polarization turns nearby objects into electric dipoles, which leads to a force drawing those objects toward him. This is the process whereby you can stick balloons to the ceiling by rubbing them on your head first-- the rubbing transfers some electrons from one object to the other, and the now-charge balloon polarizes atoms and molecules in the ceiling, creating a force that keeps the balloon in place.

Given that, how much force would an electron-free Osama exert on his surroundings?

For the purposes of this silly calculation, let's say Osama has a mass of 100 kg-- that's almost certainly too big, but not by more than a factor of two, so the different doesn't matter. What would his positive charge be after he got rid of all his electrons?

Well, humans are mostly water, so as an easy approximation, let's ask how many molecules there are in 100 kg of water. Water consists of one oxygen, with a mass of 16 atomic mass units (on average), and two hydrogens with a mass of one atomic unit each. An atomic mass unit is 1.7x10^-27kg, so electron-free Osama would contain roughly 3.3x10²⁷ electron-less water molecules (What's holding them together, you ask? Pure fanatical nastiness. Just roll with it, OK?). Each of these molecules contains ten protons (eight for the oxygen, one for each hydrogen) for a total charge of 5.3x10⁹ Coulombs.

So, how much force does this exert on a nearby object, say a Navy SEAL sneaking around the compound? Well, the equation for the force between a charge q and a polarizable particle, lifted from my notes for first-year E&M, looks like this:

The force depends on the distance between the centers of the objects, r, and the polarizability of the second object α, plus a bunch of constants. Plugging in the numbers, you find that for the charge calculated above, and the polarizability of water (from this page that I Googled up), the force is 0.73 N for a separation of 1m.

"That's less than one Newton," you say, "and a Newton is the weight of a 100-gram object (an apple, say), which is pretty insignificant."

True enough. Except that's the force on a single water molecule. To estimate the total force, you'd need to multiply by the number of water molecules in a Navy SEAL, which would be roughly the same as the number of water molecules in Osama, namely 3.3x10²⁷. Which means the total force would be around 2.4x10²⁷N, or something like a hundred thousand times greater than the gravitational force keeping the Earth in orbit around the Sun.

But, of course, that's the force between electron-free Osama and a SEAL a meter away, and if they get that close, he's dead, anyway. But the force drops off very rapidly with distance-- as 1/r⁵-- so maybe a more distant SEAL wouldn't notice it so readily. So, how far away would the searchers need to be for the force to be an all-but-undetectable 1N?

You can get the answer by setting F=1 and solving the above equation for r, and it comes out to roughly 300,000m. So, as long as nobody looking for him came within 300km, this might not be a problem. Other than, you know, all the objects that started out 1m away crashing inwards at relativistic speeds due to the huge electric polarization force.

So, you know, probably not the most effective strategy for an international fugitive. The best bet would be to keep anything containing electronics away, though even that obviously wasn't foolproof...