electromagnetism - Why does charge accumulate at points?

This is a very trivial question, but I can't seem to reason it out, again, as to why charges gather at points and edges.

Answer

I think, on would need a model to understand how it works in principle.

Let us consider the simplified system represented on the figure. You have two metallic spheres with radii $R_1$ and $R_2$ respectively.

They are linked via a metallic wire as well.

Because all these guys constitute a single metallic object, they must have the same potential that I noted $V$.

Let us assume for simplicity that the charges are uniformly distributed on the two spheres with two different surface charge densities $\sigma_1$ and $\sigma_2$.

Now, that the stage is set, there are two competing effects which have opposite influence on the increase of the charge density:

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  The charge on a metallic object, in the linear regime, is proportional to its potential which is denoted by $Q = CV$ where $C$ is the capacitance. The capacitance is such that, for simple convex shapes, it increase with the system size such that for a sphere $C \propto R$. From this, we can assert that $\frac{Q_1}{Q_2} = \frac{R_1}{R_2}$. Hence the total charge carried by sphere 1 is bigger than that of sphere 2.

  
  


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  There is a second effect (which is a simple one) that consists in noticing that the charge density is inversely proportional to the square of the size of the object essentially $\sigma \sim R^{-2} = \frac{Q}{4 \pi R^2}$ for a sphere. Hence $\frac{\sigma_1}{\sigma_2}=\frac{Q_1}{Q_2}\frac{R_2^2}{R_1^2}$

  
  
  

Overall, we finally get that $\frac{\sigma_1}{\sigma_2}=\frac{R_2}{R_1}$ and hence "points" have a larger charge density than bigger objects.