Suppose there are 13 equal charges at each corners of an $n=13$ regular polygon. The test charge $Q$ lies at the origin of the $n=13$ regular polygon.
In the case of an $n=12$ regular polygon, the net charge on $Q$ due to the 12 source charge is zero.
For an $n=12$ regular polygon:
But what happens if this is a case source charges positioned at each corners of an $n=13$ regular polygon with one source charge being removed? Does it depends on upon the source charge at which it is being positioned?
Evidently, in both cases, the source charges are equidistant from $Q$.
For an $n=13$ regular polygon:
I'm inclined to say it does not since in this example, the Coulomb's law is a function of the magnitude and the unit vector which both are dependent on the separation vector. The separation vector for any source charge in an $n=13$ regular polygon relative to $Q$ at the origin remains unchanged.
Some confirmation would be really helpful.
Best Answer
If you have 13 charges, each with charge $q$, the net force is zero. To see this, notice that your system has a discrete rotational symmetry. So the net force must be zero. If you remove one charge at a vertex, then it is just the same as adding a charge $-q$ at that vertex. The net force will be that due to a single charge $-q$ at that vertex.