always get the light and warmth we need.
What with all this, it generally comes as a rather sur prising shock to many people to learn that gravitation is not the strongest force in the universe. Suppose, for in stance, we compare it with the electromagnetic force that allows a magnet to attract iron or a proton to attract an electron. (The electromagnetic force also exhibits repul sion, which gravitational force does not, but that is a detail that need not distress us at this moment.)
How can we go about comparing the relative strengths of the electromagnetic force and the gravitational force? . Let's begin by considering two objects alone in the uni verse. The gravitational force between them, as was dis covered by Newton, can be expressed by the following equation (see also Chapter 7):
Gmr amp;
Fg = (Equation 1) d2 where F, is the gravitational force between the objects; m is the mass of one object; the mass of the other; d the distance between them; and G a universal "gravitational constant."
We must be careful about our units of measurement. If we measure mass in grams, distance in centimeters, and G in somewhat more complicated units, we will end up by determining the gravitational force in something called "dynes." (Before I'm through this chapter, the dynes will cancel out, so we need not, for present purposes, consider the dyne anything more than a one-syllable noise. It will be explained, however, in Chapter 13.)
Now let's get to work. The value of G is fixed (as far ,as we know) everywhere in the universe. Its value in the units I am using is 6.67 x 10-8. If you prefer long zero riddled decimals to exponential figures, you can express
G as 0.0000000667.
Let's suppose, next, that we are considering two objects of identical mass. This means that m = m'. so that mm' becomes mm, or M2. Furthermore, let's suppose the parti cles to be exactly I centimeter apart, center to center. In that case d = 1, and d2 = 1 also. Therefore, Equation 1 simplifies to the following:
F, = 0.0000000667 m2 (Equation 2)
We can now proceed to the electromagnetic force, which we can symbolize as F,.
Exactly one hundred years after Newton worked out the equation for gravitational forces, the French physicist Charles Augustin de Coulomb (1736-1806) was able'to show that a very similar equation could be used to deter mine the electromagnetic force between two electrically charged objects.
- Let us suppose, then, that the two objects for which we have been trying to calculate gravitational forces also carry electric charges, so that they also experience an electro magnetic force. In order to make sure that the electromag netic force is an attracting one and is therefore directly comparable to the gravitational force, let us suppose that one object carries a positive electric charge and the other a negative one. (The principle would remain even if we used like electric charges and measured the force of clec tromagnetic repulsion, but why introduce distractions?)
According to Coulomb, the electromagnetic force be 102 tween the two objects would be expressed by the foflo ' wmg equation:
F. (Equation 3) d2 where q is the charge on one object, q' on the other, and d is the distance between them.
If we let distance be measured in centimeters and elec tric charge in units called "electrostatic units" (usually abbreviated "esu7'), it is not necessary to insert a term analogous to the gravitational constant, provided the ob jects are separated by a vacuum. And, of course, since I started by assuming the objects were alone in the universe, there is necessarily a vacuunf between them.
Furthermore, if we use the units just mentioned, the value of the electromagnetic force will come out in dynes.
But lefs simplify matters by supposing that the positive electric charge on one object is exactly equal to the nega tive electric charge on the other, so that q = q,* which means that the objects . qq = qq = q2. Again, we can allow to be separated by just one centimeter, center to center, so that d2 = 1. Consequently, Equation 3 becomes:
Fe = q2 (Equation 4)
Let's summarize. We have two objects separated by one centimeter, center to center, each object possessing identi cal charge (positive in one case and negative in the other) and identical mass (no qualifications). There is both a gravitational and an electromagnetic attraction, between them.
The next problem is to determine how much stronger the electromagnetic force is than the gravitational