Physics Force on Proton

Question

What would the force on a proton be at (0,0)m be if 6 electrons were arranged in a regular
hexagon centred on (0,0)m?

 

I was thinking along the lines of finding the electric field at the centre. Since all charges are equal would the total electric field be

 

E = 6 (KQ/r²)

 

The do F = EQ?

If this is correct  how do we know the distance (r) to work out the Electric field.

Answer 1

The net force is zero, because the proton is exactly in the middle of 3 pairs of electrons, with the electrons of each pair applying equal but opposite attractive forces on the proton.  You cannot just say that the field is 6 (KQ/r²), because the forces from the 6 electrons are not all along the same direction.

 

The above is an intuitive argument.  If you need a mathematical solution, then you would have to project the electric fields or forces due to the 6 electrons along on the x and y axes, add up algebraically the contributions from the 6, and see that the net x and y components are both zero.  But that is a tedious calculation.