Comparison: Coulomb and Gravity

We have now been introduced to two forces that fall of like 1/r², the Coulomb force and the gravitational force given by Newton's law of gravitation. What are the relative strengths of these two forces?

The Coulomb force is:

and the gravitational force is:

Thus we immediately get:

Important:

This means that the ratio between the strength of the two forces is a constant, independent of distance.

Let's put in some numbers:

• Protons have a mass of 1.67·10^-27 kg, and a charge of 1.6·10^-19 C.
• Thus we get for the ratio of the strengths (in vacuum):

  Fc	=	9.0·109·(1.6·10-19)2	=  1.2·1036 for protons
  Fg		6.7·10-11·(1.67·10-27)2

• For two electrons, this ratio is even larger, roughly 10⁴².

Thus we see that:

• for the gravitational force to be appreciably large, one must have large masses, say something like 10¹⁸ to 10²⁰ kg or larger.
• the gravitational interaction is only relevant between nearly neutral bodies, when the bodies carry any appreciable net charge on them, the Coulomb force is much greater than the gravitational force.

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