"Suppose you put two electrons (or two protons) a certain distance apart. They would be attracted by a gravitational force and repelled by an electric force. Is there a distance at which these two forces would balance?"

In the diagram at right, the (repulsive) electric force from the other electron opposes the (attractive) gravitational force from the same electron. The other electron is not shown in the diagram. |

We want the two forces to balance, that is:

F

_{elec}= F_{grav}

The electric force is given by Coulomb's Law, while the gravitational force is given by Newton's Law of Universal Gravitation. If:

- m = the mass of an electron = 9.1 x 10
^{-31}kg - q = the electric charge on an electron = 1.6 x 10
^{-19}C - r = the distance between the two electrons (which we want to figure out)
- G = Universal Gravitation Constant = 6.67 x 10
^{-11}N m^{2}/kg^{2} - k = 8.9 x 10
^{9}N m^{2}/C^{2}

this becomes:

Oops! The r^{2} factors cancel! We can't use this relationship to answer the original question (find the value of r) - or can we? Suppose we "shift gears" and come at this question from a different direction (and mix some metaphors while we're at it...). Let's examine the *ratio* of the two forces:

Notice that the r^{2} factors cancel in this ratio, also. However, we get an interesting result. This calculation says that the electric force between two electrons is always 4.1 x 10^{42} times as great as the gravitational force between them *at any distance!*

So, in fact, we have answered the original question - there is * no distance* at all at which the gravitational force between two electrons balances the electric force between them!

By the way, the mass of the proton is 1.7 x 10^{-27}kg. You should be able to show that the ratio of F_{elec}/F_{grav} is about 1.2 x 10^{36} for protons, regardless of the distance between them. Therefore, the gravitational force is certainly * not* the force that holds protons together in the nucleus of the atom!

"Ok, suppose we have two protons. We could (maybe) stick some neutrons to one of the protons. This would add mass to the system, which would increase the gravitational force between the single proton and the proton-plus-neutrons. How many neutrons would have to be added until the electric and gravitational forces would balance?"

Hmm.. It *sounds* possible. Suppose that we add n neutrons to one of the protons. The mass of a neutron is approximately the same as the mass of a proton, m, so the mass of the proton-plus-n-neutrons would be m + nm = m(1+n). Adding neutrons won't change the electric force between the objects, F_{elec}, since the charges are the same and the distances are the same. The gravitational force between the objects *will* change if we add neutrons, since the mass changes. Lets call this new gravitational force F_{+neutron grav}. If the electric and (new) gravitational forces balance, then:

F

_{elec}= F_{+neutrons grav}

Working on this equation gives:

Solving for n gives:

(The number 1.2 x 10^{36} comes from the calculation of the ratio F_{elec}/F_{grav} in the previous section.)

Wow! That's a lot of protons! Let's see. A mole of carbon has a mass of about 12 grams, and it contains Avogadro's number of carbon atoms - approximately 6 x 10^{23} carbon atoms. Each of these carbon atoms has about 6 neutrons, so there are about 1.2 x 10^{24} neutrons in 12 g of carbon. So, the number of grams of carbon required would be:

This would be a mass of carbon equal to:

So you would have to add the number of neutrons present in about 12 million tons of carbon to one of the protons in order to for the gravitational force between a single proton and that huge object to balance the electric force between the two protons. Interesting.

In connection with this, you might want to research the Millikan Oil Drop Experiment. In 1909, Robert Millikan and Harvey Fletcher were the first to measure the electric charge on an electron by balancing the electric force between charged oil drops in an electric field with the gravitational force on the oil drops due to the Earth. Millikan won the Nobel Prize in Physics in 1923 for this work.

last update April 3, 2008 by JL Stanbrough