Myths of Physics: 2. Gravity Is Much Weaker Than
Electromagnetism
Draft 9 of Saturday, February 6, 16. Do not copy, quote, or distribute.
This is one you will hear in physics classrooms and read in physics textbooks. It
even seems to be familiar experience. The magnetic repulsion between the like
poles of two small bar magnets easily overcomes their mutual gravitational
attraction.
But the sun and Earth have magnetic fields too, and their mutual gravitational
attraction easily overcomes their magnetic interaction. When Newton derived
Kepler’s laws of planetary motion he just needed his law of gravity and did not
have to take into account the magnetic and electric fields of the sun and planets.
So it’s not so obvious. Electromagnetism dominates at the atomic and subatomic
levels. But on the planetary level it’s the other way around.
The magnetic force results from electric currents that are moving electric charges.
It is part of the same phenomenon as static electricity, referred to as
electromagnetism. If you are in a reference frame in which a charge is at rest, you
see electricity. If you are in a reference frame in which a charge is moving, you
see magnetism.
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The static electric force between two charged bodies is given by Coulomb’s law,
which says that the force between two point charges is proportional to the
product of the charges and inversely proportional to the square of the distance
between them.
The gravitational force between two particles is given by Newton’s law of
gravity, which says that the force between two point masses is proportional to
the product of the masses and inversely proportional to the square of the
distance between them.
The electric and gravitational force laws are both inverse square laws, so if one
computes the ratio of the forces between two bodies, the distances cancel. For the
electron and proton, the gravitational force is 39 orders of magnitude weaker
than the electrical force. This is the source of the myth that gravity is a much
weaker force than electromagnetism.
But why base our estimate of the relative strength of gravity and
electromagnetism on these two particular particles? The proton is not even
elementary but composed of quarks.
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In fact, there is no universal way we can specify the absolute strength of the
gravitational force. Newton’s gravitational constant G is not dimensionless and
so is not a good measure of the strength of gravity since it depends on what units
you use.
An absolute strength of the electromagnetic force is specified by a dimensionless
parameter alpha called, for historical reasons, the fine structure constant. It is
actually not a constant but varies with energy. However that variation is very
gradual and for most practical purposes alpha can be taken to have a value of
1/137.
Conventionally a dimensionless parameter alpha-G is defined to represent the
gravitational force strength that is proportional to the square of the proton mass.
It has a value 23 orders of magnitude less than alpha so “officially” gravity is this
much weaker than electromagnetism.
However, as we have already noted, the proton is not even a fundamental
particle so it makes no sense to use it to define the strength of gravity. The only
“natural” mass that can be formed from the basic constants of physics is the
Planck mass, which is macroscopically large. It is about 22 micrograms, whereas a
speck of dust is only about 1 microgram. If you define the dimensionless strength
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of gravity using the Planck mass you get exactly 1. In the case, gravity is 137
times stronger than electromagnetism.
Gravity is so weak at the atomic and subatomic level because the masses of
atoms and subatomic particles are so small. It is strong on the everyday and
planetary scales because the masses of everyday objects and planets are so large.
However, a good question is: Why are the masses of elementary particles so
small compared to the Planck mass? This is a major puzzle called the hierarchy
problem that physicists have still not solved.
The lack of an absolute strength of gravity does not mean that its strength
relative to the other forces is not important. Changing the definition of the
strength parameter does not change the ratio of the forces between two bodies in
any specific situation. But, the point is, that ratio is not the same in all cases. In
fact, it can be almost anything, depending on the masses and charges of the
bodies being compared. In short, it makes no sense to even ask what is the
relative strength of gravity and electromagnetism.