Inverse Square Law

The concept of an Inverse Square Law sounds complicated but it's really just a natural phenomenon in 3D space.

There are many physical quantities that can spread out from a central point. The further you are away, the more spread out the quantity gets and so it diminishes with distance, a bit like waves on a pond.

In 2D space a quantity typically diminishes in inverse proportion to the distance, e.g. as the waves spread out along the circumference of an expanding circle.

In 3D space we find that a similar quantity diminshes in inverse proportion to the square of the distance, e.g. as the waves spread out along the surface of an expanding sphere.

In mathematical terms, we would expect a formula that involves a factor of 1 / r2 where r is the distance from the central point.

For our purposes, there are 2 laws of physics that are inverse square laws:

Newtons' law of gravitation:

F = -G mg1 mg2 / r2

Coulomb's law of electrostatics:

F = Ke q1 q2 / r2

The similarity between these formulae is one of the motivations for considering the possibility of a single unified force.

The main differences are in the sign and magnitude of the leading constant:

  • Weakly attractive for the gravitational constant G
  • Strongly repulsive for the electrostatic constant Ke

Clearly we need to provide explanations for these differences if our unified force theory is to be successful.

Furthermore, the experimental results from CERN show that the electromagnetic force is up to 10% stronger at the smallest scales (1000th of the width of an atomic nucleus). We will need an explanatation for that too.

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