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It has been more than a decade since I did all vector related math and physics so pardon me if my question does not make sense. I am reading some article that says it was inspired from Coulomb's law and gives the following expression to calculate the partial force that one particle is exerting on another at time $n$:

$f^{i,j}_n = (c - |p^{i}_n - p^{j}_n|)\frac{p^{j}_n - p^{i}_n}{|p^{i}_n - p^{j}_n|}$

where $c$ is the range of the particle over which it's force can spread and $p^{i}_n$ is the location of particle $i$ in the form of $(x,y)$.

I checked up the Wikipedia article on Coulomb's law but this formula does not make sense to me. What exactly is the formula trying to achieve? Can someone please help me understand this?

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up vote 7 down vote accepted

It's nothing you can rigorously derive from Coulomb's law, but the idea is probably the following: the rightmost factor $$\frac{p^j - p^i}{|p^j - p^i|}$$ is just the unit vector pointing from particle $i$ to particle $j$, just as in Coulomb's law or Newton's law. Normally, for Coulomb's law, you'd have a factor of $q_1 q_2/4\pi\varepsilon_0 |p^j - p^i|^2$ in front of that, but here they replaced the $1/r^2-$ behaviour by a linearly decreasing factor $$c - |p^j - p^i|,$$ which yields c when $p^i = p^j$, and vanishes as $|p^j - p^i| \rightarrow c.$ (Since you call c a range, they might also mean that whenever $|p^j - p^i| > c$, the force should vanish, but it's not clear from the notation alone.)

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It says that the force between two particles acts along the line joining them (with a magnitude that also depends on the distance between I and j) that last term is a unit vector pointing from i to j.

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