Considering how closely the protons are held together in the nucleus, the electrical repulsion among them is incredibly strong. The attractive force that is holding them together is called, well, the nuclear force.

What are the characteristics of the nuclear force?

1. It does not depend on charge: the binding for neutron-neutron, proton-proton and neutron-proton are exactly the same.
2. It has an extremely short range of about one nucleon, i.e. 10^-15 m. Within this range, the nuclear force is much stronger than electrical forces. Beyond this range, the nuclear force abruptly drops to zero.

The stability (or instability) of a nucleus is thus determined by the contest between

1. the attractive nuclear force among the protons and neutrons and
2. the repulsive electrical force among the protons.

Note that electrical repulsion are long range forces compared to nuclear forces. While a proton repels every other proton in the nucleus, even if they are at opposite ends of the nucleus, a nucleon can only attract its immediate neighbours.

When the nucleus is small, adding another neutron or proton usually results in a more stable nucleus because of the attractive nuclear forces it brings. Beyond a certain size, adding another proton actually results in a less stable nucleus because the longer range electrical repulsion starts to dominate over the shorter range nuclear forces.

This explains why the most stable nuclides are not too big, not too small, with about 50 nucleons.

Why not we just keep adding neutrons? Since they contribute to the binding force without introducing any electrical repulsion? Unfortunately, neutrons themselves have a tendency to decay into a proton and an electron. In fact, it turns out the presence of protons is necessary to help keep the neutrons stable. This explains why there is a limit to how big a nucleus can be formed. In fact, lead-208 is the largest stable nuclide.

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