Skip to content

In case you hadn’t heard, the universe is governed by four fundamental forces.  But when it comes to understanding nature at almost any level larger than a nucleus and smaller than a planet, only one of them really matters: the Coulomb interaction.

The Coulomb interaction — the pushing and pulling force between electric charges — is almost incomprehensibly strong.  One common way to express this strength is by considering the forces that exist between two electrons.  Two electrons in an otherwise empty space will feel pulled together by their mutual gravitational attraction and pushed apart by the Coulomb repulsion.  The Coulomb repulsion, however, is stronger than gravity by 4,000,000,000,000,000,000,000,000,000,000,000,000,000,000 times.  (For two protons, this ratio is a more pedestrian $10^{36}$ times.)

When I was a TA, I enjoyed demonstrating this point in the following way.  Take a balloon, and rub it against the top of your head until your hair starts to stand on end.  Then stick the balloon to the ceiling, where it stays without falling due to static electricity.  Now consider the forces acting on the balloon.  Pulling up on the balloon are electric forces between the relatively few electrons I just rubbed off from my hair and the opposite charge that they induce in the ceiling.  Pulling down on the balloon are gravitational forces coming from the pull of the entire mass of the Earth.  Apparently the electric force created by those few (something like $10^{10}$) electrons is more than enough to counterbalance the gravitational pull coming from every proton, neutron and electron in the planet below it (something like $10^{51}$).

So electric forces are strong.  Why is it, then, that we can go about our daily lives without worrying about them buffeting us back and forth?

The short answer is that they do buffet us back and forth.  Pretty much any time you feel yourself being pushed or pulled by something (say, the ground beneath your feet or the muscles tied to your skeleton), the electric repulsion between microscopic charges is ultimately to blame.

But a better answer is that the very strength of electric forces is responsible for their seeming quietude.  Electric forces are so tremendously strong that nature will not abide having a large amount of electric charge collect in one place.  And so electric forces, at the scale of people-sized objects, are largely neutralized.

But what if they weren’t?

$\hspace{1mm}$

When I was a TA I got to walk my students through the following morbid little problem, which helped them see why it is that electric forces don’t really appear on the human scale.  Perhaps you will enjoy it.  Like most good physics problems, it is thoroughly contrived and, for a new student of physics, at least, its message is completely memorable.

The problem goes like this:

What would happen if your body suddenly lost 1% of its electrons?

$\hspace{1mm}$

Now, 1% may not sound like a big deal.  After all, there is almost no reason for excitement or concern when you lose 1% of your total mass.  But losing 1% of your electrons, without at the same time losing an equal number of protons, means that suddenly, within your body, there is an enormous amount of positive, unneutralized electric charge.  And nature will not abide its strongest force being so unrequited.

I’ll use my own body as an example.  My body has a mass of about 80 kg, which means that it contains something like $2 \times 10^{28}$ protons, and an almost exactly equal number of electrons.  Losing 1% of those electrons would mean that my body acquires an electric charge of $2 \times 10^{26}$ electron charges, or about $4 \times 10^7$ Coulombs.

Now, 4 billion 40 million Coulombs is a silly amount of charge.  It is about 3 million times more than what gets discharged by a lightning bolt, for example.  So, in some sense, losing 1% of your electrons would be like getting hit by 3 million lightning bolts at the same time.

Things get even more dramatic if you start to think about the forces involved.

Suppose, for example, that in their rush to escape my body, those 40 million Coulombs split in half and flowed to opposite extremities.  Say, each hand suddenly acquired a charge of 20 million Coulombs.  The force between those two hands (spread apart, about 6 meters feet) would be $10^{24}$ Newtons, which translates to about $10^{23}$ pounds.  Needless to say, my body would not retain its structural integrity.

Of course, in addition to the forces pushing the extremities of my body apart, there would also be a force similar in magnitude pulling me toward the ground.  You may recall that when an electric charge is next to a grounded surface (like, say, the ground) it induces some opposite charge on that surface in a way that acts like an “image charge” of opposite sign.  In my case, the earth would accumulate a huge amount of negative charge around my feet so as to create a force like that of an “image me.”

Because of my 40 million Coulombs, the force between myself and my “image self” would be something like $10^{20}$ tons.  To give that some perspective, consider that $10^{20}$ tons is just a bit smaller than the weight of the entire planet earth.  So the force pulling me toward the earth would be something like the force of a collision between the earth and the planet Mars.

But my hypercharged self would not only crush the earth.  It would also break open the vacuum itself.  At the instant of losing those 1% of electrons, the electric potential at the edge of my body would be about 40 exavolts.  This is much larger than the voltage required to rip apart the vacuum and create electron-positron pairs.  So my erstwhile body would be the locus of a vacuum instability, in which electrons were sucked in while positrons were blasted out.

In short, if I lost 1% of my electrons, I would not be a person anymore.  I would be a bomb.  A Coulomb bomb, if you will, with an energy equivalent to that of ten trillion (modern) atomic bombs.  Which would surely destroy the planet.  All by removing just 1 out of every 100 of my electrons.

$\hspace{1mm}$

The moral of this story, of course, is that nothing of observable size will ever get 1% charged.  The Coulomb interaction cannot be thus toyed with.  All of chemistry and biology function by the interactions between just a few charges at a time, and their effects are plenty strong as they are.

$\hspace{1mm}$

$\hspace{1mm}$

## Footnote

As a PhD student, I worked on all sorts of problems that involved the Coulomb interaction, and occasionally my proposed solution would be very wrong.  The worst kind of wrong was the one that made my advisor remark “What you just created is a Coulomb bomb,” which meant that I had proposed something that wasn’t neutral on the large scale.

It’s one thing to feel like you just solved a problem incorrectly.  Its another to feel like your proposed solution would destroy the planet.

UPDATE: G&L reader Indumathi has pointed out a numerical error in a previous version of this post, and a bunch of the numbers (and the planet I used for comparison) have been updated.

9 Comments leave one →
1. May 31, 2013 12:19 pm

This is a well written post.

• Brian permalink*
May 31, 2013 4:37 pm

Thanks!

2. frint frinterson permalink
December 31, 2013 1:36 pm

Agree with first comment. You must have been one very entertaining and enlightening TA.

3. Ralph permalink
January 6, 2014 12:17 pm

I have always had the same question. Maybe you know the answer. Cosmical rays are mainly protons, so why isn’t the Earth a Coulomb bomb?

• Brian permalink*
January 6, 2014 1:08 pm

Unfortunately, I’m not educated enough to have a good answer for you. I do know that the earth tends to be negatively charged rather than positively charged, so likely the dominant source of ionization is not coming from cosmic rays but from solar radiation.

But, more generally, you don’t have to worry about the earth becoming a “Coulomb bomb.” Suppose, for example, that the earth starts out as neutral and is bombarded with only negatively-charged radiation. Then the earth will start to get charged, but the more charged it gets, the better it becomes at deflecting incoming negative particles. In particular, electric fields develop that push away like charges (and attract opposite charges). Eventually a stasis develops.

According to Wolfram Alpha, the stasis for our planet corresponds to about 500,000 Coulombs, which means the planet is something like $10^{-25}$% charged.

4. April 12, 2015 2:04 am

That was delightful! A Coulomb bomb!

• Brian permalink*
April 12, 2015 10:23 am

Thanks!

• April 12, 2015 1:41 pm

A couple of your posts have been real eye-openers about the electrical force. I had no idea!

I’m writing a series on Special Relativity (for beginners) right now, but down the road I may have to do a post about Coulomb bombs… it’s the sort of “Holy Cow! Wow!” science I like trying to bring to my readers (you know, all ten of them :) ).

### Trackbacks

1. How big is an electron? | Gravity and Levity