In fact, this method is so simple that I can tell you what to do right now. Just tweak the charge on the electron so it is a bit out of balance with the charge on the proton. Just a little bit. How little a bit, you might ask? A very little bit. Really, this doesn't sound hard. I mean, sure, you have to do it for all of the electrons in the earth, but we're talking about a very very small percentage change. Not convinced? Let me show you just how small a change we're talking.
If there is a charge imbalance in the electron and the proton, this will give the earth a net charge, throughout it's volume. I've got to make a few assumptions about the earth here, so hold on. I'm going to assume that the earth is a uniform density everywhere, and I'm going to assume that the earth is made entirely of iron.* Now, the net charge of any iron atom will be
\[ (q_e-q_p)Z=(q_e-q_p)26\]
where Z is the atomic number of iron, the number of protons (and electrons) the atom has. The net charge of the earth, Q, is the number of iron atoms, N, times this charge,
\[Q=(q_e-q_p)ZN\]
I've previously estimated that N is about 3*10^50 atoms. Now, the electric potential energy of a sphere of radius r with charge q uniformly distributed throughout it's volume is
\[U_e=\frac{3kq^2}{5r}\]
where k is the coulomb constant.
Dissolution of the earth will occur when the electrostatic energy of the earth equals the gravitational potential energy of the earth. The gravitational bound energy of the earth is given by
\[U_g=\frac{3GM^2}{5R}\]
Where M is the mass of the earth, G is Newton's gravitational constant, and R is the radius of the earth. Setting this equal to the electrostatic energy of the earth,
\[\frac{3GM^2}{5R}=\frac{3kQ^2}{5R}\]
\[Q^2=\frac{G}{k}M^2\]
so
\[(q_e-q_p)ZN=\left(\frac{G}{k}\right)^{1/2}M\]
Now, N is given, in our approximations, by
\[N=\frac{M_{earth}}{m_{iron}}\]
so
\[q_e-q_p=\left(\frac{G}{k}\right)^{1/2}\frac{m_{iron}}{Z_{iron}}\]
Now we can plug in some numbers. G=6.7*10^-11 m^3*kg^-1*s^-2, k= 9*10^9 m^3*kg*s^-2*C^-2, m_iron=9*10^-26 kg, Z=26. Thus,
\[q_e-q_p=3*10^{-37} C\]
To put this in perspective, the charge on the electron is 1.6*10^-19 C, so this is roughly 10^18 times less than that charge. Put another way, if the charge on the electron was imbalanced from that of the proton by roughly 1 part in 10^18, the earth would cease to exist due to electrostatic repulsion.**
As I told you at the beginning, you only have to change the charge by a very small amount! So get working. There are only about 1000000000000000000000000000000000000000000000000000 electrons you need to modify!
*According to the internet, the density of the earth, on average, is roughly 5.5 g/cm^3. The density of iron is 7.9 g/cm^3 at room temperature, and the density of water is 1 g/cm^3 at room temperature. So, while the earth is not entirely iron (of course), it is a better approximation to assume the earth is iron than the earth is water. And those, of course, were really our only two choices.
** It turns out that this is a good argument for the charge balance of the electron and the proton.
Of course, if you subscribe to the one-electron universe hypothesis, then you only need to change the charge on one electron. Much easier!
ReplyDeleteAssuming atoms could remain stable, how does an ensemble of 'neutral' atoms with a slightly different proton/electron charge balance look any different to an ensemble of normal atoms with a specific net charge? Would it not therefore be the case that the earth would simply gain/lose charge from free space until it became electrically neutral, even if we introduce an electron/proton charge imbalance?
ReplyDeleteAlso, given that the earth is likely to be relatively conductive, one would expect any net charge to eventually move to the surface of the earth where it could be carried away by a relatively small mass fraction, thus avoiding complete destruction :)
George,
ReplyDeleteThe main way in which it looks different is the way in which the charge is distributed. If it was an ensemble of atoms and ions, we might well expect the earth to accumulate a net surface charge (which it does in fact have). If we introduce a charge imbalance as above then the charges are stuck in the volume of the earth. This makes it much harder to balance the electrostatic forces, (and much easier to calculate).
Certainly if we changed things gradually, we might expect that gradual change to be negated, one way or another. However, if we switched things suddenly, there would be no time for any kind of compensation, and the earth would just blow appart. (also, a 'gradual' change that slowly changes things by 1 part in 10^18 would be impressively slow).
Re: Jesse's reply to George
ReplyDeleteChanging the charge suddenly would require, of course, an enormous amount of energy. Dumping that energy into the earth by pretty much any mechanism would blow up the earth.
Changing the charge should require, if you're perfectly efficient, exactly as much energy as I said you would gain in potential energy if the charges were imbalanced. Just conservation of energy. Which, by construction, is exactly as much energy as it would take to overcome the gravitational energy bound in the earth. So yes, any time you put in that much energy the earth should come apart (more or less). What I think is interesting about the calculation is how little you would have to change the charge, by 1 part in 10^18, to achieve that.
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