# Inverse square law conundrum

We all know the equations for inverse square laws, such as force between two masses or between two charged particles. We were told the force is inversely propotional to the distance between the masses or charges, squared.

But what would the force equal if the distance between, say two point charges, was zero ie they are touching. Common sense says there would be no force, as neither will move, but this situation would mean that the force would equal the product of the two charges divided by zero squared. As far as my limited maths knowledge stretches, dividing by zero leads to an undefined number.

What does this then say about the force? Would it eqaul zero as you might expect? Or would it be undefined?

rcgldr
Homework Helper
Point charges are imaginary. Two charged objects with real mass cannot occupy the same space, although they could be "touching" each other, with a finite distance between their center of mass and charge.

Redbelly98
Staff Emeritus
Homework Helper
As the two charges are brought closer together, the electrostatic force between them becomes larger and larger. For hypothetical, classical point charges, it would be impossible to exert enough force to actually make them occupy the same point -- the required force would be infinite.

The inverse square law is a product of using Gauss' law in a 3-dimensional space. At/inside the surface of the object of mass or charge a new formula has to be derived. For spherically symmetric objects of ~uniform density (such as an ideal planet for example) the new field equation becomes linear.

Drakkith
Staff Emeritus
Does this keep the force from becoming infinite at small distances? On the scale of atomic particles i mean.

K^2
No, because elementary charge carriers are still point-particles as far as we can tell. In pure classical physics, electrostatic field diverges to infinity at point charge, and that's just the way it is. In quantum physics, fact that vacuum is not just empty space mostly takes care of that.

Drakkith
Staff Emeritus
No, because elementary charge carriers are still point-particles as far as we can tell. In pure classical physics, electrostatic field diverges to infinity at point charge, and that's just the way it is. In quantum physics, fact that vacuum is not just empty space mostly takes care of that.

What do you mean by the vacuum not being empy space?

Also, if you compare the strength of the attraction of an electron orbiting a proton, the hydrogen atom, is that attraction more or less than the two absolute electric charges of the proton and the electron? I guess I'm asking at what distance from a particle is their charge measured as it is?

Redbelly98
Staff Emeritus
Homework Helper
Does this keep the force from becoming infinite at small distances? On the scale of atomic particles i mean.
At a small but nonzero distance, the force is not infinite; just a lot larger than it is at larger distances.

As for the distance equaling zero, the uncertainty principle forbids two charges from having exactly the same location.

Also, if you compare the strength of the attraction of an electron orbiting a proton, the hydrogen atom, is that attraction more or less than the two absolute electric charges of the proton and the electron?
This question does not make a whole lot of sense. You can't ask if a force is greater than or less than a charge -- that's like asking if an inch is smaller than an ounce.

The charges on the proton and the electron are always the same value, if that helps.
I guess I'm asking at what distance from a particle is their charge measured as it is?
I don't understand.

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Drakkith
Staff Emeritus